[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content

Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes

  • Conference paper
  • First Online:
Engineering Mathematics II

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 179))

Abstract

New algorithms for computing asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes are presented. The algorithms are based on special techniques of sequential phase space reduction, which can be applied to processes with asymptotically coupled and uncoupled finite phase spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
GBP 19.95
Price includes VAT (United Kingdom)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
GBP 103.50
Price includes VAT (United Kingdom)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
GBP 129.99
Price includes VAT (United Kingdom)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
GBP 129.99
Price includes VAT (United Kingdom)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Abadov, Z.A.: Asymptotical expansions with explicit estimation of constants for exponential moments of sums of random variables defined on a markov chain and their applications to limit theorems for first hitting times. Candidate of Science dissertation, Kiev State University (1984)

    Google Scholar 

  2. Abbad, M., Filar, J.A.: Algorithms for singularly perturbed Markov control problems: a survey. In: Leondes, C.T. (ed.) Techniques in Discrete-Time Stochastic Control Systems. Control and Dynamic Systems, vol. 73, pp. 257–289. Academic Press, New York (1995)

    Google Scholar 

  3. Albeverio, S., Koroliuk, V.S., Samoilenko, I.V.: Asymptotic expansion of semi-Markov random evolutions. Stochastics 81(5), 477–502 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Albrecht, A.R., Howlett, P.G., Pearce, C.E.M.: Necessary and sufficient conditions for the inversion of linearly-perturbed bounded linear operators on Banach space using Laurent series. J. Math. Anal. Appl. 383(1), 95–110 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Albrecht, A.R., Howlett, P.G., Pearce, C.E.M.: The fundamental equations for inversion of operator pencils on Banach space. J. Math. Anal. Appl. 413(1), 411–421 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alimov, D., Shurenkov, V.M.: Markov renewal theorems in triangular array model. Ukr. Mat. Zh. 42, 1443–1448 (1990) (English translation in Ukr. Math. J. 42, 1283–1288)

    Google Scholar 

  7. Alimov, D., Shurenkov, V.M.: Asymptotic behavior of terminating Markov processes that are close to ergodic. Ukr. Mat. Zh. 42, 1701–1703 (1990) (English translation in Ukr. Math. J. 42 1535–1538)

    Google Scholar 

  8. Allen, B., Anderssen, R.S., Seneta, E.: Computation of stationary measures for infinite Markov chains. In: Neuts, M.F. (ed.) Algorithmic Methods in Probability. Studies in the Management Sciences, vol. 7, pp. 13–23. North-Holland, Amsterdam (1977)

    Google Scholar 

  9. Altman, E., Avrachenkov, K.E., Núñez-Queija, R.: Perturbation analysis for denumerable Markov chains with application to queueing models. Adv. Appl. Probab. 36(3), 839–853 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Andersson, F., Silvestrov, S.: The mathematics of Internet search engines. Acta Appl. Math. 104, 211–242 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Anisimov, V.V. Limit theorems for semi-Markov processes. I, II. Teor. Veroyatn. Mat. Stat. 2, I: 3–12; II: 13–21 (1970)

    Google Scholar 

  12. Anisimov, V.V.: Limit theorems for sums of random variables on a Markov chain, connected with the exit from a set that forms a single class in the limit. Teor. Veroyatn. Mat. Stat. 4, 3–17 (1971) (English translation in Theory Probab. Math. Statist. 4, 1–13)

    Google Scholar 

  13. Anisimov, V.V.: Limit theorems for sums of random variables that are given on a subset of states of a Markov chain up to the moment of exit, in a series scheme. Teor. Veroyatn. Mat. Stat. 4, 18–26 (1971) (English translation in Theory Probab. Math. Statist. 4, 15–22)

    Google Scholar 

  14. Anisimov, V.V.: Limit theorems for sums of random variables that are given on a countable subset of the states of a Markov chain up to the first exit time. Teor. Veroyatn. Mat. Stat. 8, 3–13 (1973)

    MathSciNet  Google Scholar 

  15. Anisimov, V.V.: Asymptotical consolidation of states for random processes. Kibernetika (3), 109–117 (1973)

    Google Scholar 

  16. Anisimov, V.V.: Limit theorems for processes admitting the asymptotical consolidation of states. Theor. Veroyatn. Mat. Stat. 22, 3–15 (1980) (English translation in Theory Probab. Math. Statist. 22, 1–13)

    Google Scholar 

  17. Anisimov, V.V.: Approximation of Markov processes that can be asymptotically lumped. Teor. Veroyatn. Mat. Stat. 34, 1–12 (1986) (English translation in Theory Probab. Math. Statist. 34, 1–11)

    Google Scholar 

  18. Anisimov, V.V.: Random Processes with Discrete Components, 183 pp. Vysshaya Shkola and Izdatel’stvo Kievskogo Universiteta, Kiev (1988)

    Google Scholar 

  19. Anisimov, V.V.: Switching Processes in Queueing Models. Applied Stochastic Methods. ISTE, London and Wiley, Hoboken, NJ (2008). 345 pp

    Book  MATH  Google Scholar 

  20. Anisimov, V.V., Chernyak, A.V.: Limit theorems for certain rare functionals on Markov chains and semi-Markov processes. Teor. Veroyatn. Mat. Stat. 26, 3–8 (1982) (English translation in Theory Probab. Math. Statist. 26, 1–6)

    Google Scholar 

  21. Anisimov, V.V., Voĭna, A.A., Lebedev, E.A.: Asymptotic estimation of integral functionals and consolidation of stochastic systems. Vestnik Kiev. Univ. Model. Optim. Slozhn. Sist. (2), 41–50 (1983)

    Google Scholar 

  22. Anisimov, V.V., Zakusilo, O.K., Donchenko, V.S.: Elements of Queueing and Asymptotical Analysis of Systems, 248 pp. Lybid’, Kiev (1987)

    Google Scholar 

  23. Asmussen, S.: Busy period analysis, rare events and transient behavior in fluid flow models. J. Appl. Math. Stoch. Anal. 7(3), 269–299 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Asmussen, S.: Applied Probability and Queues. Second edition, Applications of Mathematics, Stochastic Modelling and Applied Probability, vol. 51, xii+438 pp. Springer, New York (2003)

    Google Scholar 

  25. Asmussen, S., Albrecher, H.: Ruin Probabilities. Second edition, Advanced Series on Statistical Science & Applied Probability, vol. 14, xviii+602 pp. World Scientific, Hackensack, NJ (2010)

    Google Scholar 

  26. Avrachenkov, K.E.: Analytic perturbation theory and its applications. Ph.D. thesis, University of South Australia (1999)

    Google Scholar 

  27. Avrachenkov, K.E.: Singularly perturbed finite Markov chains with general ergodic structure. In: Boel, R., Stremersch, G. (eds.) Discrete Event Systems. Analysis and Control. Kluwer International Series in Engineering and Computer Science, vol. 569, pp. 429–432. Kluwer, Boston (2000)

    Google Scholar 

  28. Avrachenkov, K., Borkar, V., Nemirovsky, D.: Quasi-stationary distributions as centrality measures for the giant strongly connected component of a reducible graph. J. Comput. Appl. Math. 234(11), 3075–3090 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Avrachenkov, K.E., Filar, J., Haviv, M.: Singular perturbations of Markov chains and decision processes. In: Feinberg, E.A., Shwartz, A. (eds.) Handbook of Markov Decision Processes. Methods and Applications. International Series in Operations Research & Management Science, vol. 40, pp. 113–150. Kluwer, Boston (2002)

    Google Scholar 

  30. Avrachenkov, K.E., Filar, J.A., Howlett, P.G.: Analytic Perturbation Theory and Its Applications, xii+372 pp. SIAM, Philadelphia, PA (2013)

    Google Scholar 

  31. Avrachenkov, K.E., Haviv, M.: Perturbation of null spaces with application to the eigenvalue problem and generalized inverses. Linear Algebr. Appl. 369, 1–25 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Avrachenkov, K.E., Haviv, M.: The first Laurent series coefficients for singularly perturbed stochastic matrices. Linear Algebr. Appl. 386, 243–259 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Avrachenkov, K.E., Haviv, M., Howlett, P.G.: Inversion of analytic matrix functions that are singular at the origin. SIAM J. Matrix Anal. Appl. 22(4), 1175–1189 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Avrachenkov, K.E., Lasserre, J.B.: The fundamental matrix of singularly perturbed Markov chains. Adv. Appl. Probab. 31(3), 679–697 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  35. Avrachenkov, K.E., Lasserre, J.B.: Analytic perturbation of generalized inverses. Linear Algebr. Appl. 438(4), 1793–1813 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Avrachenkov, K., Litvak, N., Son Pham, K.: Distribution of PageRank mass among principle components of the web. In: Chung, F.R.K., Bonato, A. (eds.) Algorithms and Models for the Web-Graph. Lecture Notes in Computer Science, vol. 4863, pp. 16–28. Springer, Berlin (2007)

    Google Scholar 

  37. Avrachenkov, K., Litvak, N., Son Pham, K.: A singular perturbation approach for choosing the PageRank damping factor. Internet Math. 5(1–2), 47–69 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Barbour, A.D., Pollett, P.K.: Total variation approximation for quasi-stationary distributions. J. Appl. Probab. 47(4), 934–946 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Barbour, A.D., Pollett, P.K.: Total variation approximation for quasi-equilibrium distributions II. Stoch. Process. Appl. 122(11), 3740–3756 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Barlow, J.: Perturbation results for nearly uncoupled Markov chains with applications to iterative methods. Numer. Math. 65(1), 51–62 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  41. Baumgärtel, H.: Analytic Perturbation Theory for Matrices and Operators. Operator Theory: Advances and Applications, vol. 15, 427 pp. Birkhäuser, Basel (1985)

    Google Scholar 

  42. Benois, O., Landim, C., Mourragui, M.: Hitting times of rare events in Markov chains. J. Stat. Phys. 153(6), 967–990 (2013)

    Google Scholar 

  43. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences.Classics in Applied Mathematics, vol. 9, xx+340 pp. SIAM, Philadelphia (1994). (A revised reprint ofNonnegative Matrices in the Mathematical Sciences. Computer Science and Applied Mathematics, xviii+316 pp. Academic Press, New York, 1979)

    Google Scholar 

  44. Bielecki, T., Stettner, Ł.: On ergodic control problems for singularly perturbed Markov processes. Appl. Math. Optim. 20(2), 131–161 (1989)

    Google Scholar 

  45. Blanchet, J., Zwart, B.: Asymptotic expansions of defective renewal equations with applications to perturbed risk models and processor sharing queues. Math. Methods Oper. Res. 72, 311–326 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  46. Bini, D.A., Latouche, G., Meini, B.: Numerical Methods for Structured Markov Chains. Numerical Mathematics and Scientific Computation, p. xii+327. Oxford Science Publications, Oxford University Press, New York (2005)

    Google Scholar 

  47. Bobrova, A.F.: Estimates of accuracy of an asymptotic consolidation of countable Markov chains. Stability Problems for Stochastic Models, pp. 16–24. Trudy Seminara, VNIISI, Moscow (1983)

    Google Scholar 

  48. Borovkov, A.A.: Ergodicity and Stability of Stochastic Processes. Wiley Series in Probability and Statistics, vol. 314, p. xxiv+585. Wiley, Chichester (1998) (Translation from the 1994 Russian original)

    Google Scholar 

  49. Burnley, C.: Perturbation of Markov chains. Math. Mag. 60(1), 21–30 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  50. Cao, X.R.: The Maclaurin series for performance functions of Markov chains. Adv. Appl. Probab. 30, 676–692 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  51. Cao, W.L., Stewart, W.J.: Iterative aggregation/disaggregation techniques for nearly uncoupled Markov chains. J. Assoc. Comput. Mach. 32, 702–719 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  52. Chatelin, F., Miranker, W.L.: Aggregation/disaggregation for eigenvalue problems. SIAM J. Numer. Anal. 21(3), 567–582 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  53. Coderch, M., Willsky, A.S., Sastry, S.S., Castañon, D.A.: Hierarchical aggregation of singularly perturbed finite state Markov processes. Stochastics 8, 259–289 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  54. Cole, J.D.: Perturbation Methods in Applied Mathematics, p. vi+260. Blaisdell, Waltham, Mass (1968)

    Google Scholar 

  55. Collet, P., Martínez, S., San Martín, J.: Quasi-Stationary Distributions. Markov Chains, Diffusions and Dynamical Systems. Probability and its Applications, p. xvi+280. Springer, Heidelberg (2013)

    Google Scholar 

  56. Courtois, P.J.: Error analysis in nearly-completely decomposable stochastic systems. Econometrica 43(4), 691–709 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  57. Courtois, P.J.: Decomposability: Queueing and Computer System Applications. ACM Monograph Series, p. xiii+201. Academic Press, New York (1977)

    Google Scholar 

  58. Courtois, P.J.: Error minimization in decomposable stochastic models. In: Ralph, L.D., Teunis, J.O. (eds.) Applied Probability - Computer Science: the Interface. Progress in Computer Science, 2, vol. I, pp. 189–210. Birkhäuser, Boston (1982)

    Google Scholar 

  59. Courtois, P.J., Louchard, G.: Approximation of eigen characteristics in nearly-completely decomposable stochastic systems. Stoch. Process. Appl. 4, 283–296 (1976)

    Article  MATH  Google Scholar 

  60. Courtois, P.J., Semal, P.: Error bounds for the analysis by decomposition of non-negative matrices. In: Iazeolla, G., Courtois, P.J., Hordijk, A. (eds.) Mathematical Computer Performance and Reliability, pp. 209–224. North-Holland, Amsterdam (1984)

    Google Scholar 

  61. Courtois, P.J., Semal, P.: Block decomposition and iteration in stochastic matrices. Philips J. Res. 39(4–5), 178–194 (1984)

    MathSciNet  MATH  Google Scholar 

  62. Courtois, P.J., Semal, P.: Bounds for the positive eigenvectors of nonnegative matrices and for their approximations by decomposition. J. Assoc. Comput. Mach. 31(4), 804–825 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  63. Courtois, P.J., Semal, P. Bounds for transient characteristics of large or infinite Markov chains. In: Stewart, W.J. (ed.) Numerical Solution of Markov Chains. Probability: Pure and Applied, vol. 8, pp. 413–434. Marcel Dekker, New York (1991)

    Google Scholar 

  64. Craven, B.D.: Perturbed Markov processes. Stoch. Models 19(2), 269–285 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  65. Darroch, J., Seneta, E.: On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Probab. 2, 88–100 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  66. Darroch, J., Seneta, E.: On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probab. 4, 192–196 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  67. Delebecque, F.: A reduction process for perturbed Markov chains. SIAM J. Appl. Math. 43, 325–350 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  68. Delebecque, F.: On the resolvent approach to the spectral decomposition of a regular matrix pencil. Linear Algebr. Appl. 129, 63–75 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  69. Delebecque, F., Quadrat, J.P.: Optimal control of Markov chains admitting strong and weak interactions. Automatica 17(2), 281–296 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  70. Drozdenko, M.: Weak convergence of first-rare-event times for semi-Markov processes I. Theory Stoch. Process. 13(4), 29–63 (2007)

    MathSciNet  MATH  Google Scholar 

  71. Drozdenko, M.: Weak convergence of first-rare-event times for semi-Markov processes, vol. 49. Doctoral dissertation, Mälardalen University, Västerås (2007)

    Google Scholar 

  72. Drozdenko, M.: Weak convergence of first-rare-event times for semi-Markov processes II. Theory Stoch. Process. 15(2), 99–118 (2009)

    MathSciNet  MATH  Google Scholar 

  73. Englund, E.: Perturbed renewal equations with application to M/M queueing systems. 1. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 60, 31–37 (1999) (Also in Theory Probab. Math. Stat. 60, 35–42)

    Google Scholar 

  74. Englund, E.: Perturbed renewal equations with application to M/M queueing systems. 2. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 61, 21–32 (1999) (Also in Theory Probab. Math. Stat. 61, 21–32)

    Google Scholar 

  75. Englund, E.: Nonlinearly perturbed renewal equations with applications to a random walk. In: Silvestrov, D., Yadrenko, M., Olenko A., Zinchenko, N. (eds.) Proceedings of the Third International School on Applied Statistics, Financial and Actuarial Mathematics, Feodosiya (2000). (Theory Stoch. Process. 6(22)(3–4), 33–60)

    Google Scholar 

  76. Englund, E.: Nonlinearly perturbed renewal equations with applications. Doctoral dissertation, Umeå University (2001)

    Google Scholar 

  77. Englund, E., Silvestrov, D.S.: Mixed large deviation and ergodic theorems for regenerative processes with discrete time. In: Jagers, P., Kulldorff, G., Portenko, N., Silvestrov, D. (eds.) Proceedings of the Second Scandinavian–Ukrainian Conference in Mathematical Statistics, Vol. I, Umeå, 1997. (Theory Stoch. Process. 3(19)(1–2), 164–176)

    Google Scholar 

  78. Engström, C., Silvestrov, S.: Generalisation of the damping factor in PageRank for weighted networks. In: Silvestrov, D., Martin-Löf, A. (eds.) Modern Problems in Insurance Mathematics. Chap. 19. EAA series, pp. 313–334. Springer, Cham (2014)

    Google Scholar 

  79. Engström, C., Silvestrov, S.: PageRank, a look at small changes in a line of nodes and the complete graph. In: Silvestrov S., Rančić M. (eds.) Engineering Mathematics II. Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization. Springer, Heidelberg (2016)

    Google Scholar 

  80. Engström, C., Silvestrov, S.: PageRank, connecting a line of nodes with a complete graph. In: Silvestrov S., Rančić M. (eds.) Engineering Mathematics II. Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization. Springer, Heidelberg (2016)

    Google Scholar 

  81. Erdélyi, A.: Asymptotic Expansions, vi+108 pp. Dover, New York (1956)

    Google Scholar 

  82. Feinberg, B.N., Chiu, S.S.: A method to calculate steady-state distributions of large Markov chains by aggregating states. Oper. Res. 35(2), 282–290 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  83. Feller, W.: An Introduction to Probability Theory and Its Applications, vol. I, xviii+509 pp. Wiley, New York (1968). (3rd edition of An Introduction to Probability Theory and Its Applications, vol. I, xii+419 pp. Wiley, New York, 1950)

    Google Scholar 

  84. Fuh, C.D.: Uniform Markov renewal theory and ruin probabilities in Markov random walks. Ann. Appl. Probab. 14(3), 1202–1241 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  85. Fuh, C.D.: Asymptotic expansions on moments of the first ladder height in Markov random walks with small drift. Adv. Appl. Probab. 39, 826–852 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  86. Fuh, C.D., Lai, T.L.: Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks. Adv. Appl. Probab. 33, 652–673 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  87. Funderlic, R.E., Meyer, C.D., Jr.: Sensitivity of the stationary distribution vector for an ergodic Markov chain. Linear Algebr. Appl. 76, 1–17 (1985)

    Google Scholar 

  88. Gaĭtsgori, V.G., Pervozvanskiĭ, A.A.: Aggregation of states in a Markov chain with weak interaction. Kibernetika (3), 91–98 (1975). (English translation in Cybernetics 11(3), 441–450)

    Google Scholar 

  89. Gaĭtsgori, V.G., Pervozvanskiy, A.A.: Decomposition and aggregation in problems with a small parameter. Izv. Akad. Nauk SSSR, Tekhn. Kibernet. (1), 33–46 (1983) (English translation in Eng. Cybern. 2(1), 26–38)

    Google Scholar 

  90. Gibson, D., Seneta, E.: Augmented truncations of infinite stochastic matrices. J. Appl. Probab. 24(3), 600–608 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  91. Glynn, P.: On exponential limit laws for hitting times of rare sets for Harris chains and processes. In: Glynn, P., Mikosch, T., Rolski, T. (eds). New Frontiers in Applied Probability: a Festschrift for Søren Asmussen. vol. 48A, 319–326 (2011). (J. Appl. Probab. Spec.)

    Google Scholar 

  92. Golub, G.H., Seneta, E.: Computation of the stationary distribution of an infinite Markov matrix. Bull. Aust. Math. Soc. 8, 333–341 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  93. Golub, G.H., Van Loan, C.F: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, xiv+756 pp. Johns Hopkins University Press, Baltimore, MD (2013). (4th edition of Matrix Computations. Johns Hopkins Series in the Mathematical Sciences, vol. 3, xvi+476 pp. Johns Hopkins University Press, Baltimore, MD, 1983)

    Google Scholar 

  94. Grassman, W.K., Taksar, M.I., Heyman, D.P.: Regenerative analysis and steady state distributions for Markov chains. Oper. Res. 33, 1107–1116 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  95. Guo, D.Z.: On the sensitivity of the solution of nearly uncoupled Markov chains. SIAM J. Matrix Anal. Appl. 14(4), 1112–1123 (2006)

    MathSciNet  Google Scholar 

  96. Gusak, D.V., Korolyuk, V.S.. Asymptotic behaviour of semi-Markov processes with a decomposable set of states. Teor. Veroyatn. Mat. Stat. 5, 43–50 (1971) (English translation in Theory Probab. Math. Stat. 5, 43–51)

    Google Scholar 

  97. Gut, A., Holst, L.: On the waiting time in a generalized roulette game. Stat. Probab. Lett. 2(4), 229–239 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  98. Gyllenberg, M., Silvestrov, D.S.: Quasi-stationary distributions of a stochastic metapopulation model. J. Math. Biol. 33, 35–70 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  99. Gyllenberg, M., Silvestrov, D.S.: Quasi-stationary phenomena in semi-Markov models. In: Janssen, J., Limnios, N. (eds.) Proceedings of the Second International Symposium on Semi-Markov Models: Theory and Applications, Compiègne, pp. 87–93 (1998)

    Google Scholar 

  100. Gyllenberg, M., Silvestrov, D.S.: Quasi-stationary phenomena for semi-Markov processes. In: Janssen, J., Limnios, N. (eds.) Semi-Markov Models and Applications, pp. 33–60. Kluwer, Dordrecht (1999)

    Chapter  Google Scholar 

  101. Gyllenberg, M., Silvestrov, D.S.: Cramér-Lundberg and diffusion approximations for nonlinearly perturbed risk processes including numerical computation of ruin probabilities. In: Silvestrov, D., Yadrenko, M., Borisenko, O., Zinchenko, N. (eds.) Proceedings of the Second International School on Actuarial and Financial Mathematics, Kiev (1999). (Theory Stoch. Process., 5(21)(1–2), 6–21)

    Google Scholar 

  102. Gyllenberg, M., Silvestrov, D.S.: Nonlinearly perturbed regenerative processes and pseudo-stationary phenomena for stochastic systems. Stoch. Process. Appl. 86, 1–27 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  103. Gyllenberg, M., Silvestrov, D.S.: Cramér-Lundberg approximation for nonlinearly perturbed risk processes. Insur. Math. Econ. 26, 75–90 (2000)

    Article  MATH  Google Scholar 

  104. Gyllenberg, M., Silvestrov, D.S.: Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems. De Gruyter Expositions in Mathematics, vol. 44, ix+579 pp. Walter de Gruyter, Berlin (2008)

    Google Scholar 

  105. Häggström, O.: Finite Markov Chains and Algorithmic Applications. London Mathematical Society Student Texts, vol. 52, 126 pp. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  106. Hanen, A.: Probème central limite dans le cas Markovien fini. La matrice limite n’a qu’une seule classe ergodique et pas d’état transitoire. C. R. Acad. Sci. Paris 256, 68–70 (1963)

    MathSciNet  MATH  Google Scholar 

  107. Hanen, A.: Problème central limite dans le cas Markovien fini. II. La matrice limite a plusieurs classes ergodiques et pas d’états transitoires. C. R. Acad. Sci. Paris 256, 362–364 (1963)

    MathSciNet  MATH  Google Scholar 

  108. Hanen, A.: Problème central limite dans le cas Markovien fini. Cas général. C. R. Acad. Sci. Paris 256, 575–577 (1963)

    MathSciNet  MATH  Google Scholar 

  109. Hanen, A.: Théorèmes limites pour une suite de chaînes de Markov. Ann. Inst. H. Poincaré 18, 197–301 (1963)

    MathSciNet  MATH  Google Scholar 

  110. Harrod, W.J., Plemmons, R.J.: Comparison of some direct methods for computing stationary distributions of Markov chains. SIAM J. Sci. Stat. Comput. 5, 453–469 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  111. Hassin, R., Haviv, M.: Mean passage times and nearly uncoupled Markov chains. SIAM J. Discret. Math. 5, 386–397 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  112. Haviv, M.: An approximation to the stationary distribution of a nearly completely decomposable Markov chain and its error analysis. SIAM J. Algebr. Discret. Methods 7(4), 589–593 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  113. Haviv, M.: Aggregation/disaggregation methods for computing the stationary distribution of a Markov chain. SIAM J. Numer. Anal. 24(4), 952–966 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  114. Haviv, M.: Error bounds on an approximation to the dominant eigenvector of a nonnegative matrix. Linear Multilinear Algebr. 23(2), 159–163 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  115. Haviv, M.: An aggregation/disaggregation algorithm for computing the stationary distribution of a large Markov chain. Commun. Stat. Stoch. Models 8(3), 565–575 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  116. Haviv, M.: On censored Markov chains, best augmentations and aggregation/disaggregation procedures. Aggregation and disaggregation in operations research. Comput. Oper. Res. 26(10–11), 1125–1132 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  117. Haviv, M.: More on Rayleigh-Ritz refinement technique for nearly uncoupled stochastic matrices. SIAM J. Matrix Anal. Appl. 10(3), 287–293 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  118. Haviv, M., Ritov, Y.: An approximation to the stationary distribution of a nearly completely decomposable Markov chain and its error bound. SIAM J. Algebr. Discret. Methods 7(4), 583–588 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  119. Haviv, M., Ritov, Y.: On series expansions and stochastic matrices. SIAM J. Matrix Anal. Appl. 14(3), 670–676 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  120. Haviv, M., Ritov, Y.: Bounds on the error of an approximate invariant subspace for non-self-adjoint matrices. Numer. Math. 67(4), 491–500 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  121. Haviv, M., Ritov, Y., Rothblum, U.G.: Iterative methods for approximating the subdominant modulus of an eigenvalue of a nonnegative matrix. Linear Algebr. Appl. 87, 61–75 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  122. Haviv, M., Ritov, Y., Rothblum, U.G.: Taylor expansions of eigenvalues of perturbed matrices with applications to spectral radii of nonnegative matrices. Linear Algebr. Appl. 168, 159–188 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  123. Haviv, M., Rothblum, U.G.: Bounds on distances between eigenvalues. Linear Algebr. Appl. 63, 101–118 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  124. Haviv, M., Van der Heyden, L.: Perturbation bounds for the stationary probabilities of a finite Markov chain. Adv. Appl. Probab. 16, 804–818 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  125. Hernández-Lerma, O. Lasserre, J.B.: Markov Chains and Invariant Probabilities. Progress in Mathematics, vol. 211, xvi+205 pp. Birkhäuser, Basel (2003)

    Google Scholar 

  126. Ho, Y.C., Cao, X.R.: Perturbation Analysis of Discrete Event Dynamic Systems. The Springer International Series in Engineering and Computer Science, 433 pp. Springer, New York (1991)

    Google Scholar 

  127. Hoppensteadt, F., Salehi, H., Skorokhod, A.: On the asymptotic behavior of Markov chains with small random perturbations of transition probabilities. In: Gupta, A.K. (ed.) Multidimensional Statistical Analysis and Theory of Random Matrices: Proceedings of the Sixth Eugene Lukacs Symposium. Bowling Green, OH, 1996, pp. 93–100. VSP, Utrecht (1996)

    Google Scholar 

  128. Hoppensteadt, F., Salehi, H., Skorokhod, A.: Markov chain with small random perturbations with applications to bacterial genetics. Random Oper. Stoch. Equ. 4(3), 205–227 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  129. Hoppensteadt, F., Salehi, H., Skorokhod, A.: Discrete time semigroup transformations with random perturbations. J. Dyn. Differ. Equ. 9(3), 463–505 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  130. Hössjer, O.: Coalescence theory for a general class of structured populations with fast migration. Adv. Appl. Probab. 43(4), 1027–1047 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  131. Hössjer, O.: Spatial autocorrelation for subdivided populations with invariant migration schemes. Methodol. Comput. Appl. Probab. 16(4), 777–810 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  132. Hössjer, O., Ryman, N.: Quasi equilibrium, variance effective size and fixation index for populations with substructure. J. Math. Biol. 69(5), 1057–1128 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  133. Howlett, P., Albrecht, A., Pearce, C.: Laurent series for inversion of linearly perturbed bounded linear operators on Banach space. J. Math. Anal. Appl. 366(1), 112–123 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  134. Howlett, P., Avrachenkov, K.: Laurent series for the inversion of perturbed linear operators on Hilbert space. In: Glover, B.M., Rubinov, A.M. (eds.) Optimization and Related Topics. Applied Optimisation, vol. 47, pp. 325–342. Kluwer, Dordrecht (2001)

    Google Scholar 

  135. Howlett, P., Avrachenkov, K., Pearce, C., Ejov, V.: Inversion of analytically perturbed linear operators that are singular at the origin. J. Math. Anal. Appl. 353(1), 68–84 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  136. Howlett, P., Pearce, C., Torokhti, A.: On nonlinear operator approximation with preassigned accuracy. J. Comput. Anal. Appl. 5(3), 273–297 (2003)

    MathSciNet  MATH  Google Scholar 

  137. Hunter, J.J.: Stationary distributions of perturbed Markov chains. Linear Algebr. Appl. 82, 201–214 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  138. Hunter, J.J.: The computation of stationary distributions of Markov chains through perturbations. J. Appl. Math. Stoch. Anal. 4(1), 29–46 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  139. Hunter, J.J.: A survey of generalized inverses and their use in applied probability. Math. Chron. 20, 13–26 (1991)

    MathSciNet  MATH  Google Scholar 

  140. Hunter, J.J.: Stationary distributions and mean first passage times of perturbed Markov chains. Linear Algebr. Appl. 410, 217–243 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  141. Hunter, J.J.: Generalized inverses of Markovian kernels in terms of properties of the Markov chain. Linear Algebr. Appl. 447, 38–55 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  142. Kalashnikov, V.V.: Qualitative Analysis of the Behaviour of Complex Systems by the Method of Test Functions. Theory and Methods of Systems Analysis, 247 pp. Nauka, Moscow (1978)

    Google Scholar 

  143. Kalashnikov, V.V.: Solution of the problem of approximating a denumerable Markov chain. Eng. Cybern. (3), 92–95 (1997)

    Google Scholar 

  144. Kalashnikov, V.V.: Geometric Sums: Bounds for Rare Events with Applications. Mathematics and its Applications, vol. 413, ix+265 pp. Kluwer, Dordrecht (1997)

    Google Scholar 

  145. Kalashnikov, V.V., Anichkin, S.A.: Continuity of random sequences and approximation of Markov chains. Adv. Appl. Probab. 13(2), 402–414 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  146. Kaplan, E.I.: Limit theorems for exit times of random sequences with mixing. Teor. Veroyatn. Mat. Stat. 21, 53–59 (1979). (English translation in Theory Probab. Math. Stat. 21, 59–65)

    Google Scholar 

  147. Kaplan, E.I.: Limit Theorems for Sum of Switching Random Variables with an Arbitrary Phase Space of Switching Component. Candidate of Science dissertation, Kiev State University (1980)

    Google Scholar 

  148. Kartashov, N.V.: Inequalities in stability and ergodicity theorems for Markov chains with a general phase space. I. Teor. Veroyatn. Primen. vol. 30, 230–240 (1985). (English translation in Theory Probab. Appl. 30, 247–259)

    Google Scholar 

  149. Kartashov, N.V.: Inequalities in stability and ergodicity theorems for Markov chains with a general phase space. II. Teor. Veroyatn. Primen. 30, 478–485 (1985). (English translation in Theory Probab. Appl. 30, 507–515)

    Google Scholar 

  150. Kartashov, N.V.: Asymptotic representations in an ergodic theorem for general Markov chains and their applications. Teor. Veroyatn. Mat. Stat. 32, 113–121 (1985). (English translation in Theory Probab. Math. Stat. 32, 131–139)

    Google Scholar 

  151. Kartashov, N.V.: Asymptotic expansions and inequalities in stability theorems for general Markov chains under relatively bounded perturbations. In: Stability Problems for Stochastic Models, Varna, 1985, pp. 75–85. VNIISI, Moscow (1985). (English translation in J. Soviet Math. 40(4), 509–518)

    Google Scholar 

  152. Kartashov, N.V.: Inequalities in theorems of consolidation of Markov chains. Theor. Veroyatn. Mat. Stat. 34, 62–73 (1986). (English translation in Theory Probab. Math. Stat. 34, 67–80)

    Google Scholar 

  153. Kartashov, N.V.: Estimates for the geometric asymptotics of Markov times on homogeneous chains. Teor. Veroyatn. Mat. Stat. 37, 66–77 (1987). (English translation in Theory Probab. Math. Stat. 37, 75–88)

    Google Scholar 

  154. Kartashov, M.V.: Computation and estimation of the exponential ergodicity exponent for general Markov processes and chains with recurrent kernels. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 54, 47–57 (1996). (English translation in Theory Probab. Math. Stat. 54, 49–60)

    Google Scholar 

  155. Kartashov, M.V.: Strong Stable Markov Chains, 138 pp. VSP, Utrecht and TBiMC, Kiev (1996)

    Google Scholar 

  156. Kartashov, M.V.: Calculation of the spectral ergodicity exponent for the birth and death process. Ukr. Mat. Zh. 52, 889–897 (2000) (English translation in Ukr. Math. J. 52, 1018–1028)

    Google Scholar 

  157. Kartashov, M.V.: Ergodicity and stability of quasihomogeneous Markov semigroups of operators. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 72, 54–62 (2005). (English translation in Theory Probab. Math. Stat. 72, 59–68)

    Google Scholar 

  158. Kartashov, M.V.: Quantitative and qualitative limits for exponential asymptotics of hitting times for birth-and-death chains in a scheme of series. Teor. Imovirn. Mat. Stat. 89, 40–50 (2013). (English translation in Theory Probab. Math. Stat. 89, 45–56 (2014))

    Google Scholar 

  159. Kato, T.: Perturbation Theory for Linear Operators. Classics of Mathematics, 623 pp. Springer, Berlin (2013). (2nd edition of Perturbation Theory for Linear Operators, xix+592 pp. Springer, New York)

    Google Scholar 

  160. Keilson, J.: A limit theorem for passage times in ergodic regenerative processes. Ann. Math. Stat. 37, 866–870 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  161. Keilson, J.: Markov Chain Models – Rarity and Exponentiality. Applied Mathematical Sciences, vol. 28, xiii+184 pp. Springer, New York (1979)

    Google Scholar 

  162. Kemeny, J.G., Snell, J.L.: Finite Markov Chains. The University Series in Undergraduate Mathematics, viii+210 pp. D. Van Nostrand, Princeton, NJ (1960)

    Google Scholar 

  163. Kevorkian, J., Cole, J.D.: Perturbation Methods in Applied Mathematics. Applied Mathematical Sciences, vol. 34, x+558 pp. Springer, New York (1981)

    Google Scholar 

  164. Kevorkian, J., Cole, J.D.: Multiple Scale and Singular Perturbation Methods. Applied Mathematical Sciences, vol. 114, viii+632 pp. Springer, New York (1996)

    Google Scholar 

  165. Khasminskii, R.Z., Yin, G., Zhang, Q.: Singularly perturbed Markov chains: quasi-stationary distribution and asymptotic expansion. In: Proceedings of Dynamic Systems and Applications, vol. 2, pp. 301–308. Atlanta, GA, 1995. Dynamic, Atlanta, GA (1996)

    Google Scholar 

  166. Khasminskii, R.Z., Yin, G., Zhang, Q.: Asymptotic expansions of singularly perturbed systems involving rapidly fluctuating Markov chains. SIAM J. Appl. Math. 56(1), 277–293 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  167. Kijima, M.: Markov Processes for Stochastic Modelling. Stochastic Modeling Series, x+341 pp. Chapman & Hall, London (1997)

    Google Scholar 

  168. Kim, D.S., Smith, R.L.: An exact aggregation/disaggregation algorithm for large scale Markov chains. Nav. Res. Logist. 42(7), 1115–1128 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  169. Kingman, J.F.: The exponential decay of Markovian transition probabilities. Proc. Lond. Math. Soc. 13, 337–358 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  170. Kokotović, P.V., Phillips, R.G., Javid, S.H.: Singular perturbation modeling of Markov processes. In: Bensoussan, A., Lions, J.L. (eds.) Analysis and optimization of systems: Proceedings of the Fourth International Conference on Analysis and Optimization. Lecture Notes in Control and Information Science, vol. 28, pp. 3–15. Springer, Berlin (1980)

    Google Scholar 

  171. Konstantinov, M. Gu, D.W., Mehrmann, V., Petkov, P.: Perturbation Theory for Matrix Equations. Studies in Computational Mathematics, vol. 9, xii+429 pp. North-Holland, Amsterdam (2003)

    Google Scholar 

  172. Konstantinov, M.M., Petkov, P.H.: Perturbation methods in linear algebra and control. Appl. Comput. Math. 7(2), 141–161 (2008)

    MathSciNet  MATH  Google Scholar 

  173. Kontoyiannis, I., Meyn, S.P.: Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13(1), 304–362 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  174. Korolyuk, D.V.: Limit theorems for hitting time type functionals defined on processes with semi-Markov switchings. Candidate of Science Dissertation, Kiev State University (1983)

    Google Scholar 

  175. Korolyuk, D.V., Silvestrov D.S.: Entry times into asymptotically receding domains for ergodic Markov chains. Teor. Veroyatn. Primen. 28, 410–420 (1983). (English translation in Theory Probab. Appl. 28, 432–442)

    Google Scholar 

  176. Korolyuk, D.V., Silvestrov D.S.: Entry times into asymptotically receding regions for processes with semi-Markov switchings. Teor. Veroyatn. Primen. 29, 539–544 (1984). (English translation in Theory Probab. Appl. 29, 558–563)

    Google Scholar 

  177. Korolyuk, V.S.: On asymptotical estimate for time of a semi-Markov process being in the set of states. Ukr. Mat. Zh. 21, 842–845 (1969). (English translation in Ukr. Math. J. 21, 705–710)

    Google Scholar 

  178. Korolyuk, V.S.: Stochastic Models of Systems, 208 pp. Naukova Dumka, Kiev (1989)

    Google Scholar 

  179. Korolyuk, V.S., Brodi, S.M., Turbin, A.F.: Semi-Markov processes and their application. Probability Theory. Mathematical Statistics. Theoretical Cybernetics, vol. 11, pp. 47–97. VINTI, Moscow (1974)

    Google Scholar 

  180. Korolyuk, V.S., Korolyuk, V.V.: Stochastic Models of Systems. Mathematics and its Applications, vol. 469, xii+185 pp. Kluwer, Dordrecht (1999)

    Google Scholar 

  181. Korolyuk, V.S., Limnios, N.: Diffusion approximation of integral functionals in merging and averaging scheme. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 59, 99–105 (1998). (English translation in Theory Probab. Math. Stat. 59, 101–107)

    Google Scholar 

  182. Korolyuk, V.S., Limnios, N.: Diffusion approximation for integral functionals in the double merging and averaging scheme. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 60, 77–84 (1999). (English translation in Theory Probab. Math. Stat. 60, 87–94)

    Google Scholar 

  183. Korolyuk, V.S., Limnios, N.: Evolutionary systems in an asymptotic split phase space. In: Limnios, N., Nikulin, M. (eds.) Recent Advances in Reliability Theory: Methodology, Practice and Inference, pp. 145–161. Birkhäuser, Boston (2000)

    Chapter  Google Scholar 

  184. Korolyuk, V.S., Limnios, N.: Markov additive processes in a phase merging scheme. In: Korolyuk, V., Prokhorov, Yu., Khokhlov, V., Klesov, O. (eds.) Proceedings of the Conference Dedicated to the 90th Anniversary of Boris Vladimirovich Gnedenko, Kiev (2002). (Theory Stoch. Process., 8(3–4), 213–225)

    Google Scholar 

  185. Korolyuk, V.S., Limnios, N.: Average and diffusion approximation of stochastic evolutionary systems in an asymptotic split state space. Ann. Appl. Probab. 14, 489–516 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  186. Korolyuk, V.S., Limnios, N.: Diffusion approximation of evolutionary systems with equilibrium in asymptotic split phase space. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 70, 63–73 (2004). (English translation in Theory Probab. Math. Stat. 70, 71–82)

    Google Scholar 

  187. Koroliuk, V.S., Limnios, N.: Stochastic Systems in Merging Phase Space, xv+331 pp. World Scientific, Singapore (2005)

    Google Scholar 

  188. Koroliuk, V.S., Limnios, N.: Reliability of semi-Markov systems with asymptotic merging phase space. In: Rykov, V.V., Balakrishnan, N., Nikulin, M.S. (eds.) Mathematical and Statistical Models and Methods in Reliability, pp. 3–18. Birkhäuser, Boston (2010)

    Chapter  Google Scholar 

  189. Korolyuk, V.S., Penev, I.P., Turbin, A.F.: The asymptotic behavior of the distribution of the absorption time of a Markov chain. Kibernetika (2), 20–22 (1972)

    Google Scholar 

  190. Korolyuk, V.S., Penev, I.P., Turbin, A.F.: Asymptotic expansion for the distribution of the absorption time of a weakly inhomogeneous Markov chain. In: Korolyuk, V.S. (ed.) Analytic Methods of Investigation in Probability Theory, pp. 97–105. Akad. Nauk Ukr. SSR, Inst. Mat., Kiev (1981)

    Google Scholar 

  191. Korolyuk, V., Swishchuk, A.: Semi-Markov Random Evolutions, 254 pp. Naukova Dumka, Kiev (1992). (English revised edition of Semi-Markov Random Evolutions. Mathematics and its Applications, vol. 308, x+310 pp. Kluwer, Dordrecht, 1995)

    Google Scholar 

  192. Korolyuk, V.V., Tadzhiev, A.: Asymptotic behavior of Markov evolutions prior to the time of absorption. Ukr. Mat. Zh. 38, 248–251 (1986). (English translation in Ukr. Math. J. 38, 219–222)

    Google Scholar 

  193. Korolyuk, V.S., Turbin, A.F.: On the asymptotic behaviour of the occupation time of a semi-Markov process in a reducible subset of states. Teor. Veroyatn. Mat. Stat. 2, 133–143 (1970). (English translation in Theory Probab. Math. Stat. 2, 133–143)

    Google Scholar 

  194. Korolyuk, V.S., Turbin, A.F.: A certain method of proving limit theorems for certain functionals of semi-Markov processes. Ukr. Mat. Zh. 24, 234–240 (1972)

    MathSciNet  MATH  Google Scholar 

  195. Korolyuk, V.S., Turbin, A.F.: Semi-Markov Processes and its Applications, 184 pp. Naukova Dumka, Kiev (1976)

    Google Scholar 

  196. Korolyuk, V.S., Turbin, A.F.: Mathematical Foundations of the State Lumping of Large Systems, 218 pp.. Naukova Dumka, Kiev (1978). (English edition: Mathematical Foundations of the State Lumping of Large Systems. Mathematics and its Applications, vol. 264, x+278 pp. Kluwer, Dordrecht, 1993)

    Google Scholar 

  197. Korolyuk, V.S., Turbin, A.F., Tomusjak, A.A.: Sojourn time of a semi-Markov process in a decomposing set of states. Analytical Methods of Probability Theory, vol. 152, pp. 69–79. Naukova Dumka, Kiev (1979)

    Google Scholar 

  198. Koury, J.R., McAllister, D.F., Stewart, W.J.: Iterative methods for computing stationary distributions of nearly completely decomposable Markov chains. SIAM J. Algebr. Discret. Methods 5, 164–186 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  199. Kovalenko, I.N.: An algorithm of asymptotic analysis of a sojourn time of Markov chain in a set of states. Dokl. Acad. Nauk Ukr. SSR, Ser. A 6, 422–426 (1973)

    Google Scholar 

  200. Kovalenko, I.N.: Studies in the Reliability Analysis of Complex Systems, 210 pp. Naukova Dumka, Kiev (1975)

    Google Scholar 

  201. Kovalenko, I.N.: Rare events in queuing theory - a survey. Queuing Syst. Theory Appl. 16(1–2), 1–49 (1994)

    Article  MATH  Google Scholar 

  202. Kovalenko, I.N., Kuznetsov, M.Ju.: Renewal process and rare events limit theorems for essentially multidimensional queueing processes. Math. Oper. Stat. Ser. Stat. 12(2), 211–224 (1981)

    Google Scholar 

  203. Kovalenko, I.N., Kuznetsov, N.Y., Pegg, P.A.: Mathematical Theory of Reliability of Time Dependent Systems with Practical Applications. Wiley Series in Probability and Statistics, 316 pp. Wiley, New York (1997)

    Google Scholar 

  204. Kupsa, M., Lacroix, Y.: Asymptotics for hitting times. Ann. Probab. 33(2), 610–619 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  205. Langville, A.N., Meyer, C.D.: Updating Markov chains with an eye on Google’s PageRank. SIAM J. Matrix Anal. Appl. 27(4), 968–987 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  206. Lasserre, J.B.: A formula for singular perturbations of Markov chains. J. Appl. Probab. 31, 829–833 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  207. Latouche, G.: Perturbation analysis of a phase-type queue with weakly correlated arrivals. Adv. Appl. Probab. 20, 896–912 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  208. Latouche, G.: First passage times in nearly decomposable Markov chains. In: Stewart, W.J. (ed.) Numerical Solution of Markov Chains. Probability: Pure and Applied, vol. 8, pp. 401–411. Marcel Dekker, New York (1991)

    Google Scholar 

  209. Latouche, G., Louchard, G.: Return times in nearly decomposable stochastic processes. J. Appl. Probab. 15, 251–267 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  210. Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM Series on Statistics and Applied Probability, xiv+334 pp. SIAM, Philadelphia, PA and ASA, Alexandria, VA (1999)

    Google Scholar 

  211. Leadbetter, M.R.: On series expansion for the renewal moments. Biometrika 50(1–2), 75–80 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  212. Li, R.C., Stewart, G.W.: A new relative perturbation theorem for singular subspaces. Linear Algebr. Appl. 313(1–3), 41–51 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  213. Li, X., Yin, G., Yin, K., Zhang, Q.: A numerical study of singularly perturbed Markov chains: quasi-equilibrium distributions and scaled occupation measures. Dyn. Contin. Discret. Impuls. Syst. 5(1–4), 295–304 (1999)

    MathSciNet  MATH  Google Scholar 

  214. Louchard, G., Latouche, G.: Random times in nearly-completely decomposable, transient Markov chains. Cahiers Centre Études Rech. Opér. 24(2–4), 321–352 (1982)

    MathSciNet  MATH  Google Scholar 

  215. Louchard, G., Latouche, G.: Geometric bounds on iterative approximations for nearly completely decomposable Markov chains. J. Appl. Probab. 27(3), 521–529 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  216. Marek, I., Mayer, P.: Convergence analysis of an iterative aggregation/disaggregation method for computing stationary probability vectors of stochastic matrices. Numer. Linear Algebr. Appl. 5(4), 253–274 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  217. Marek, I., Mayer, P., Pultarová, I.: Convergence issues in the theory and practice of iterative aggregation/disaggregation methods. Electron. Trans. Numer. Anal. 35, 185–200 (2009)

    MathSciNet  MATH  Google Scholar 

  218. Marek, I., Pultarová, I.: A note on local and global convergence analysis of iterative aggregation-disaggregation methods. Linear Algebr. Appl. 413(2–3), 327–341 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  219. Mattingly, R.B., Meyer, C.D.: Computing the stationary distribution vector of an irreducible Markov chain on a shared-memory multiprocessor. In: Stewart, W.J. (ed.) Numerical Solution of Markov Chains. Probability: Pure and Applied, vol. 8, pp. 491–510. Marcel Dekker, New York (1991)

    Google Scholar 

  220. McAllister, D.F., Stewart, G.W., Stewart, W.J.: On a Rayleigh-Ritz refinement technique for nearly uncoupled stochastic matrices. Linear Algebr. Appl. 60, 1–25 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  221. Meshalkin, L.D.: Limit theorems for Markov chains with a finite number of states. Teor. Veroyatn. Primen. 3, 361–385 (1958). (English translation in Theory Probab. Appl. 3, 335–357)

    Google Scholar 

  222. Masol, V.I., Silvestrov, D.S.: Record values of the occupation time of a semi-Markov process. Visnik Kiev. Univ. Ser. Mat. Meh. 14, 81–89 (1972)

    MathSciNet  Google Scholar 

  223. Motsa, A.I., Silvestrov, D.S.: Asymptotics of extremal statistics and functionals of additive type for Markov chains. In: Klesov, O., Korolyuk, V., Kulldorff, G., Silvestrov, D. (eds.) Proceedings of the First Ukrainian–Scandinavian Conference on Stochastic Dynamical Systems, Uzhgorod, 1995 (1996). (Theory Stoch. Proces., 2(18)(1–2), 217–224)

    Google Scholar 

  224. Meyer, C.D.: Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems. SIAM Rev. 31(2), 240–272 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  225. Meyer, C.D.: Sensitivity of the stationary distribution of a Markov chain. SIAM J. Matrix Anal. Appl. 15(3), 715–728 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  226. Meyer, C.D. (2000). Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia, PA, xii+718 pp

    Google Scholar 

  227. Meyer, C.D.: Continuity of the Perron root. Linear Multilinear Algebr. 63(7), 1332–1336 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  228. Meyer Jr., C.D.: The condition of a finite Markov chain and perturbation bounds for the limiting probabilities. SIAM J. Algebr. Discret. Methods 1(3), 273–283 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  229. Meyer, C.D., Stewart, G.W.: Derivatives and perturbations of eigenvectors. SIAM J. Numer. Anal. 25(3), 679–691 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  230. Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability, xxviii+594 pp. Cambridge University Press, Cambridge (2009). (2nd edition of Markov Chains and Stochastic Stability. Communications and Control Engineering Series, xvi+ 548 pp. Springer, London, 1993)

    Google Scholar 

  231. Mitrophanov, A.Y.: Stability and exponential convergence of continuous-time Markov chains. J. Appl. Probab. 40, 970–979 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  232. Mitrophanov, A.Y.: Sensitivity and convergence of uniformly ergodic Markov chains. J. Appl. Probab. 42, 1003–1014 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  233. Mitrophanov, A.Y.: Ergodicity coefficient and perturbation bounds for continuous-time Markov chains. Math. Inequal. Appl. 8(1), 159–168 (2005)

    MathSciNet  MATH  Google Scholar 

  234. Mitrophanov, A.Y.: Stability estimates for finite homogeneous continuous-time Markov chains. Theory Probab. Appl. 50(2), 319–326 (2006)

    Article  MathSciNet  Google Scholar 

  235. Mitrophanov, A.Y., Lomsadze, A., Borodovsky, M.: Sensitivity of hidden Markov models. J. Appl. Probab. 42, 632–642 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  236. Nagaev, S.V.: Some limit theorems for stationary Markov chains. Teor. Veroyatn. Primen. 2, 389–416 (1957). (English translation in Theory Probab. Appl. 2, 378–406)

    Google Scholar 

  237. Nagaev, S.V.: A refinement of limit theorems for homogeneous Markov chains. Teor. Veroyatn. Primen. 6, 67–86 (1961). (English translation in Theory Probab. Appl. 6, 62–81)

    Google Scholar 

  238. Ni, Y.: Perturbed renewal equations with multivariate nonpolynomial perturbations. In: Frenkel, I., Gertsbakh, I., Khvatskin, L., Laslo, Z., Lisnianski, A. (eds.) Proceedings of the International Symposium on Stochastic Models in Reliability Engineering, Life Science and Operations Management, pp. 754–763. Beer Sheva, Israel (2010)

    Google Scholar 

  239. Ni, Y.: Analytical and numerical studies of perturbed renewal equations with multivariate non-polynomial perturbations. J. Appl. Quant. Methods 5(3), 498–515 (2010)

    Google Scholar 

  240. Ni, Y.: Nonlinearly perturbed renewal equations: asymptotic results and applications. Doctoral Dissertation, 106, Mälardalen University, Västerås (2011)

    Google Scholar 

  241. Ni, Y.: Nonlinearly perturbed renewal equations: the non-polynomial case. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 84, 111–122 (2012). (Also in Theory Probab. Math. Stat. 84, 117–129)

    Google Scholar 

  242. Ni, Y.: Exponential asymptotical expansions for ruin probability in a classical risk process with non-polynomial perturbations. In: Silvestrov, D., Martin-Löf, A. (eds.) Modern Problems in Insurance Mathematics. Chap. 6, EAA series, pp. 67–91. Springer, Cham (2014)

    Google Scholar 

  243. Ni, Y., Silvestrov, D., Malyarenko, A.: Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations. J. Numer. Appl. Math. 1(96), 173–197 (2008)

    MATH  Google Scholar 

  244. Nåsell, I.: Extinction and Quasi-Stationarity in the Stochastic Logistic SIS Model. Lecture Notes in Mathematics, Mathematical Biosciences Subseries, vol. 2022, xii+199 pp. Springer, Heidelberg (2011)

    Google Scholar 

  245. Paige, C.C., Styan, G.P.H., Wachter, P.G.: Computation of the stationary distribution of a Markov chain. J. Stat. Comput. Simul. 4, 173–186 (1975)

    Article  MATH  Google Scholar 

  246. Pervozvanskiĭ, A.A., Gaitsgori, V.G.: Theory of Suboptimal Decisions Decomposition and Aggregation. Mathematics and Its Applications (Soviet Series), vol. 12, xviii+384 pp. Kluwer, Dordrecht (1988)

    Google Scholar 

  247. Pervozvanskiĭ, A.A., Smirnov, I.N.: An estimate of the steady state of a complex system with slowly varying constraints. Kibernetika (4), 45–51 (1974). (English translation in Cybernetics 10(4), 603–611)

    Google Scholar 

  248. Petersson, M.: Quasi-stationary distributions for perturbed discrete time regenerative processes. Teor. \(\check{\rm I}{\rm movirn}\). Mat. Stat. 89, 140–155 (2013). (Also in Theor. Probab. Math. Stat. 89, 153–168)

    Google Scholar 

  249. Petersson, M.: Asymptotics of ruin probabilities for perturbed discrete time risk processes. In: Silvestrov, D., Martin-Löf, A. (eds.) Modern Problems in Insurance Mathematics. Chapter 7, EAA series, pp. 93–110. Springer, Cham (2014)

    Google Scholar 

  250. Petersson, M.: Asymptotic expansions for moment functionals of perturbed discrete time semi-Markov processes. In: Silvestrov, S., Rančić, M. (eds.) Engineering Mathematics II. Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization. Springer, Heidelberg (2016)

    Google Scholar 

  251. Petersson, M.: Asymptotic for quasi-stationary distributions of perturbed discrete time semi-Markov processes. In: Silvestrov, S., Rančić, M. (eds.) Engineering Mathematics II. Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization. Springer, Heidelberg (2016)

    Google Scholar 

  252. Petersson, M.: Perturbed discrete time stochastic models. Doctoral Dissertation, Stockholm University (2016)

    Google Scholar 

  253. Phillips, R.G., Kokotović, P.V.: A singular perturbation approach to modeling and control of Markov chains. IEEE Trans. Autom. Control 26, 1087–1094 (1981)

    Article  MATH  Google Scholar 

  254. Plotkin, J.D., Turbin, A.F.: Inversion of linear operators that are perturbed on the spectrum. Ukr. Mat. Zh. 23, 168–176 (1971)

    MathSciNet  Google Scholar 

  255. Plotkin, J.D., Turbin, A.F.: Inversion of normally solvable linear operators that are perturbed on the spectrum. Ukr. Mat. Zh. 27(4), 477–486 (1975)

    MathSciNet  MATH  Google Scholar 

  256. Poliščuk, L.I., Turbin, A.F.: Asymptotic expansions for certain characteristics of semi-Markov processes. Teor. Veroyatn. Mat. Stat. 8, 122–127 (1973). (English translation in Theory Probab. Math. Stat. 8, 121–126)

    Google Scholar 

  257. Pollett, P.K., Stewart, D.E.: An efficient procedure for computing quasi-stationary distributions of Markov chains with sparse transition structure. Adv. Appl. Probab. 26, 68–79 (1994)

    MathSciNet  MATH  Google Scholar 

  258. Quadrat, J.P.: Optimal control of perturbed Markov chains: the multitime scale case. In: Ardema, M.D. (ed.) Singular Perturbations in Systems and Control. CISM Courses and Lectures, vol. 280, pp. 215–239. Springer, Vienna (1983)

    Google Scholar 

  259. Rohlichek, J.R.: Aggregation and time scale analysis of perturbed markov systems. Ph.D. thesis, Massachusetts Inst. Tech., Cambridge, MA (1987)

    Google Scholar 

  260. Rohlicek, J.R., Willsky, A.S.: Multiple time scale decomposition of discrete time Markov chains. Syst. Control Lett. 11(4), 309–314 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  261. Rohlicek, J.R., Willsky, A.S.: The reduction of perturbed Markov generators: an algorithm exposing the role of transient states. J. Assoc. Comput. Mach. 35(3), 675–696 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  262. Romanovskiĭ, V.I.: Discrete Markov Chains, 436 pp. Gostehizdat, Moscow-Leningrad (1949)

    Google Scholar 

  263. Samoĭlenko, \(\bar{\rm I}\).V.: Asymptotic expansion of Markov random evolution. Ukr. Mat. Visn. 3(3), 394–407 (2006). (English translation in Ukr. Math. Bull. 3(3), 381–394)

    Google Scholar 

  264. Samoĭlenko, \(\bar{\rm I}\).V.: Asymptotic expansion of semi-Markov random evolution. Ukr. Mat. Zh. 58, 1234–1248 (2006). (English translation in Ukr. Math. J. 58, 1396–1414)

    Google Scholar 

  265. Schweitzer, P.J.: Perturbation theory and finite Markov chains. J. Appl. Probab. 5, 401–413 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  266. Schweitzer, P.J.: Aggregation methods for large Markov chains. In: Iazeolla, G., Courtois, P.J., Hordijk, A. (eds.) Mathematical Computer Performance and Reliability, pp. 275–286. North-Holland, Amsterdam (1984)

    Google Scholar 

  267. Schweitzer, P.J.: Posterior bounds on the equilibrium distribution of a finite Markov chain. Commun. Stat. Stoch. Models 2(3), 323–338 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  268. Schweitzer, P.J.: Dual bounds on the equilibrium distribution of a finite Markov chain. J. Math. Anal. Appl. 126(2), 478–482 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  269. Schweitzer, P.J.: A survey of aggregation-disaggregation in large Markov chains. In: Stewart, W.J. (ed.) Numerical solution of Markov chains. Probability: Pure and Applied, vol. 8, pp. 63–88. Marcel Dekker, New York (1991)

    Google Scholar 

  270. Schweitzer, P.J., Kindle, K.W.: Iterative aggregation for solving undiscounted semi-Markovian reward processes. Commun. Stat. Stoch. Models 2(1), 1–41 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  271. Schweitzer, P.J., Puterman, M.L., Kindle, K.W.: Iterative aggregation-disaggregation procedures for discounted semi-Markov reward processes. Oper. Res. 33(3), 589–605 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  272. Schweitzer, P., Stewart, G.W.: The Laurent expansion of pencils that are singular at the origin. Linear Algebr. Appl. 183, 237–254 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  273. Seneta, E.: Finite approximations to infinite non-negative matrices. Proc. Camb. Philos. Soc. 63, 983–992 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  274. Seneta, E.: Finite approximations to infinite non-negative matrices. II. Refinements and applications. Proc. Camb. Philos. Soc. 64, 465–470 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  275. Seneta, E.: The principle of truncations in applied probability. Comment. Math. Univ. Carolinae 9, 237–242 (1968)

    MathSciNet  MATH  Google Scholar 

  276. Seneta, E.: Nonnegative Matrices. An Introduction to Theory and Applications, x+214 pp. Wiley, New York (1973)

    Google Scholar 

  277. Seneta, E.: Iterative aggregation: convergence rate. Econ. Lett. 14(4), 357–361 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  278. Seneta, E.: Sensitivity to perturbation of the stationary distribution: some refinements. Linear Algebr. Appl. 108, 121–126 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  279. Seneta, E.: Perturbation of the stationary distribution measured by ergodicity coefficients. Adv. Appl. Probab. 20, 228–230 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  280. Seneta, E.: Sensitivity analysis, ergodicity coefficients, and rank-one updates for finite Markov chains. In: Stewart, W.J. (ed.) Numerical Solution of Markov Chains. Probability: Pure and Applied, vol. 8, pp. 121–129. Marcel Dekker, New York (1991)

    Google Scholar 

  281. Seneta, E.: Sensitivity of finite Markov chains under perturbation. Stat. Probab. Lett. 17(2), 163–168 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  282. Seneta, E.: Nonnegative Matrices and Markov chains. Springer Series in Statistics, xvi+287 pp. Springer, New York (2006). (A revised reprint of 2nd edition of Nonnegative Matrices and Markov Chains. Springer Series in Statistics, xiii+279 pp. Springer, New York, 1981)

    Google Scholar 

  283. Serlet, L.: Hitting times for the perturbed reflecting random walk. Stoch. Process. Appl. 123(1), 110–130 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  284. Sheskin, T.J.: A Markov chain partitioning algorithm for computing steady state probabilities. Oper. Res. 33, 228–235 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  285. Shurenkov, V.M.: Transition phenomena of the renewal theory in asymptotical problems of theory of random processes 1. Mat. Sbornik, 112, 115–132 (1980). (English translation in Math. USSR: Sbornik, 40(1), 107–123)

    Google Scholar 

  286. Shurenkov, V.M.: Transition phenomena of the renewal theory in asymptotical problems of theory of random processes 2. Mat. Sbornik 112, 226–241 (1980). (English translation in Math. USSR: Sbornik, 40(2), 211–225)

    Google Scholar 

  287. Silvestrov, D.S.: Limit theorems for a non-recurrent walk connected with a Markov chain. Ukr. Mat. Zh. 21, 790–804 (1969). (English translation in Ukr. Math. J. 21, 657–669)

    Google Scholar 

  288. Silvestrov, D.S.: Limit theorems for semi-Markov processes and their applications. 1, 2. Teor. Veroyatn. Mat. Stat. 3, 155–172, 173–194 (1970). (English translation in Theory Probab. Math. Stat. 3, 159–176, 177–198)

    Google Scholar 

  289. Silvestrov, D.S.: Limit theorems for semi-Markov summation schemes. 1. Teor. Veroyatn. Mat. Stat. 4, 153–170 (1971). (English translation in Theory Probab. Math. Stat. 4, 141–157)

    Google Scholar 

  290. Silvestrov, D.S.: Uniform estimates of the rate of convergencxe for sums of random variables defined on a finite homogeneous Markov chain with absorption. Theor. Veroyatn. Mat. Stat. 5, 116–127 (1971). (English translation in Theory Probab. Math. Stat. 5, 123–135)

    Google Scholar 

  291. Silvestrov, D.S.: Limit distributions for sums of random variables that are defined on a countable Markov chain with absorption. Dokl. Acad. Nauk Ukr. SSR, Ser. A (4), 337–340 (1972)

    Google Scholar 

  292. Silvestrov, D.S.: Estimation of the rate of convergence for sums of random variables that are defined on a countable Markov chain with absorption. Dokl. Acad. Nauk Ukr. SSR, Ser. A 5, 436–438 (1972)

    MathSciNet  Google Scholar 

  293. Silvestrov, D.S.: Limit Theorems for Composite Random Functions, 318 pp. Vysshaya Shkola and Izdatel’stvo Kievskogo Universiteta, Kiev (1974)

    Google Scholar 

  294. Silvestrov, D.S.: A generalization of the renewal theorem. Dokl. Akad. Nauk Ukr. SSR, Ser. A 11, 978–982 (1976)

    MathSciNet  Google Scholar 

  295. Silvestrov, D.S.: The renewal theorem in a series scheme. 1. Teor. Veroyatn. Mat. Stat. 18, 144–161 (1978). (English translation in Theory Probab. Math. Stat. 18, 155–172)

    Google Scholar 

  296. Silvestrov, D.S.: The renewal theorem in a series scheme 2. Teor. Veroyatn. Mat. Stat. 20, 97–116 (1979). (English translation in Theory Probab. Math. Stat. 20, 113–130)

    Google Scholar 

  297. Silvestrov, D.S.: A remark on limit distributions for times of attainment for asymptotically recurrent Markov chains. Theory Stoch. Process. 7, 106–109 (1979)

    MathSciNet  MATH  Google Scholar 

  298. Silvestrov, D.S.: Semi-Markov Processes with a Discrete State Space, 272 pp. Library for the Engineer in Reliability, Sovetskoe Radio, Moscow (1980)

    Google Scholar 

  299. Silvestrov, D.S.: Mean hitting times for semi-Markov processes, and queueing networks. Elektron. Infor. Kybern. 16, 399–415 (1980)

    MathSciNet  Google Scholar 

  300. Silvestrov, D.S.: Theorems of large deviations type for entry times of a sequence with mixing. Teor. Veroyatn. Mat. Stat. 24, 129–135 (1981). (English translation in Theory Probab. Math. Stat. 24, 145–151)

    Google Scholar 

  301. Silvestrov, D.S.: Exponential asymptotic for perturbed renewal equations. Teor. Imovirn. Mat. Stat. 52, 143–153 (1995). (English translation in Theory Probab. Math. Stat. 52, 153–162)

    Google Scholar 

  302. Silvestrov, D.S.: Recurrence relations for generalised hitting times for semi-Markov processes. Ann. Appl. Probab. 6, 617–649 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  303. Silvestrov, D.S.: Nonlinearly perturbed Markov chains and large deviations for lifetime functionals. In: Limnios, N., Nikulin, M. (eds.) Recent Advances in Reliability Theory: Methodology, Practice and Inference, pp. 135–144. Birkhäuser, Boston (2000)

    Chapter  Google Scholar 

  304. Silvestrov, D.S.: Asymptotic expansions for quasi-stationary distributions of nonlinearly perturbed semi-Markov processes. Theory Stoch. Process. 13(1–2), 267–271 (2007)

    MathSciNet  MATH  Google Scholar 

  305. Silvestrov D.S.: Nonlinearly perturbed stochastic processes and systems. In: Rykov, V., Balakrishnan, N., Nikulin, M. (eds.) Mathematical and Statistical Models and Methods in Reliability, Chapter 2, pp. 19–38. Birkhäuser (2010)

    Google Scholar 

  306. Silvestrov, D.S.: Improved asymptotics for ruin probabilities. In: Silvestrov, D., Martin-Löf, A. (eds.) Modern Problems in Insurance Mathematics, Chap. 5. EAA series, pp. 93–110. Springer, Cham (2014)

    Google Scholar 

  307. Silvestrov D.S.: American-Type Options. Stochastic Approximation Methods, Volume 1. De Gruyter Studies in Mathematics, vol. 56, x+509 pp. Walter de Gruyter, Berlin (2014)

    Google Scholar 

  308. Silvestrov D.S.: American-Type Options. Stochastic Approximation Methods, Volume 2. De Gruyter Studies in Mathematics, vol. 57, xi+558 pp. Walter de Gruyter, Berlin (2015)

    Google Scholar 

  309. Silvestrov D.: Necessary and sufficient conditions for convergence of first-rare-event times for perturbed semi-Markov processes. Research Report 2016:4, Department of Mathematics, Stockholm University, 39 pp. (2016). arXiv:1603.04344

  310. Silvestrov, D.S., Abadov, Z.A.: Asymptotic behaviour for exponential moments of sums of random variables defined on exponentially ergodic Markov chains. Dokl. Acad. Nauk Ukr. SSR, Ser. A (4), 23–25 (1984)

    Google Scholar 

  311. Silvestrov, D.S., Abadov, Z.A.: Uniform asymptotic expansions for exponential moments of sums of random variables defined on a Markov chain and distributions of passage times. 1. Teor. Veroyatn. Mat. Stat. 45, 108–127 (1991). (English translation in Theory Probab. Math. Stat. 45, 105–120)

    Google Scholar 

  312. Silvestrov, D.S., Abadov, Z.A.: Uniform representations of exponential moments of sums of random variables defined on a Markov chain and of distributions of passage times. 2. Teor. Veroyatn. Mat. Stat. 48, 175–183 (1993). (English translation in Theory Probab. Math. Stat. 48, 125–130)

    Google Scholar 

  313. Silvestrov, D.S., Drozdenko, M.O.: Necessary and sufficient conditions for the weak convergence of the first-rare-event times for semi-Markov processes. Dopov. Nac. Akad. Nauk Ukr., Mat. Prirodozn. Tekh Nauki (11), 25–28 (2005)

    Google Scholar 

  314. Silvestrov, D.S., Drozdenko, M.O.: Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes. Theory Stoch. Process., 12(28), no. 3–4, Part I: 151–186. Part II, 187–202 (2006)

    MATH  Google Scholar 

  315. Silvestrov, D., Manca, R.: Reward algorithms for semi-Markov processes. Research Report 2015:16, Department of Mathematics, Stockholm University, 23 pp. (2015). arXiv:1603.05693

  316. Silvestrov, D., Manca, R.: Reward algorithms for exponential moments of hitting times for semi-Markov processes. Research Report 2016:1, Department of Mathematics, Stockholm University, 23 pp. (2016)

    Google Scholar 

  317. Silvestrov, D., Manca, R., Silvestrova, E.: Computational algorithms for moments of accumulated Markov and semi-Markov rewards. Commun. Stat. Theory, Methods 43(7), 1453–1469 (2014)

    Google Scholar 

  318. Silvestrov, D.S., Petersson, M.: Exponential expansions for perturbed discrete time renewal equations. In: Karagrigoriou, A., Lisnianski, A., Kleyner, A., Frenkel, I. (eds.) Applied Reliability Engineering and Risk Analysis. Probabilistic Models and Statistical Inference. Chapter 23, pp. 349–362. Wiley, New York (2013)

    Google Scholar 

  319. Silvestrov, D.S., Petersson, M., Hössjer, O.: Nonlinearly perturbed birth-death-type models. Research Report 2016:6, Department of Mathematics, Stockholm University, 63 pp. (2016). arXiv:1604.02295

  320. Silvestrov, D., Silvestrov, S.: Asymptotic expansions for stationary distributions of perturbed semi-Markov processes. Research Report 2015:9, Department of Mathematics, Stockholm University, 75 pp. (2015). arXiv:1603.03891

  321. Silvestrov, D., Silvestrov, S.: Asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes. I, II. (2016). Part I: arXiv:1603.04734, 30 pp., Part II: arXiv:1603.04743, 33 pp

  322. Silvestrov, D.S., Velikii, Y.A.: Necessary and sufficient conditions for convergence of attainment times. In: Zolotarev, V.M., Kalashnikov, V.V. (eds.) Stability Problems for Stochastic Models. Trudy Seminara, VNIISI, Moscow, 129–137 (1988). (English translation in J. Soviet. Math. 57, 3317–3324)

    Google Scholar 

  323. Simon, H.A., Ando, A.: Aggregation of variables in dynamic systems. Econometrica 29, 111–138 (1961)

    Article  MATH  Google Scholar 

  324. Sirl, D., Zhang, H., Pollett, P.: Computable bounds for the decay parameter of a birth-death process. J. Appl. Probab. 44(2), 476–491 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  325. Stewart, G.W.: On the continuity of the generalized inverse. SIAM J. Appl. Math. 17, 33–45 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  326. Stewart, G.W.: Error and perturbation bounds for subspaces associated with certain eigenvalue problems. SIAM Rev. 15, 727–764 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  327. Stewart, G.W.: Perturbation bounds for the definite generalized eigenvalue problem. Linear Algebr. Appl. 23, 69–85 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  328. Stewart, G.W.: Computable error bounds for aggregated Markov chains. J. Assoc. Comput. Mach. 30(2), 271–285 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  329. Stewart, G.W.: On the structure of nearly uncoupled Markov chains. In: Iazeolla, G., Courtois, P.J., Hordijk, A. (eds.) Mathematical Computer Performance and Reliability, pp. 287–302. North-Holland, Amsterdam (1984)

    Google Scholar 

  330. Stewart, G.W.: A second order perturbation expansion for small singular values. Linear Algebr. Appl. 56, 231–235 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  331. Stewart, G.W.: Stochastic perturbation theory. SIAM Rev. 32(4), 579–610 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  332. Stewart, G.W.: On the sensitivity of nearly uncoupled Markov chains. In: Stewart, W.J. (ed.) Numerical Solution of Markov Chains. Probability: Pure and Applied, vol. 8, pp. 105–119. Marcel Dekker, New York (1991)

    Google Scholar 

  333. Stewart, G.W.: Gaussian elimination, perturbation theory, and Markov chains. In: Meyer, C.D., Plemmons, R.J. (eds). Linear Algebra, Markov Chains, and Queueing Models. IMA Volumes in Mathematics and its Applications, vol. 48, pp. 59–69. Springer, New York (1993)

    Google Scholar 

  334. Stewart, G.W.: On the perturbation of Markov chains with nearly transient states. Numer. Math. 65(1), 135–141 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  335. Stewart, G.W.: Matrix Algorithms. Vol. I. Basic Decompositions, xx+458 pp. SIAM, Philadelphia, PA (1998)

    Google Scholar 

  336. Stewart, G.W.: Matrix Algorithms. Vol. II. Eigensystems, xx+469 pp. SIAM, Philadelphia, PA (2001)

    Google Scholar 

  337. Stewart, G.W.: On the powers of a matrix with perturbations. Numer. Math. 96(2), 363–376 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  338. Stewart, G.W., Stewart, W.J., McAllister, D.F.: A two-stage iteration for solving nearly completely decomposable Markov chains. In: Golub, G., Greenbaum, A., Luskin, M. (eds.) Recent Advances in Iterative Methods. IMA Volumes in Mathematics and its Applications, vol. 60, pp. 201–216. Springer, New York (1994)

    Google Scholar 

  339. Stewart, G.W., Sun, J.G.: Matrix Perturbation Theory. Computer Science and Scientific Computing, xvi+365 pp. Academic Press, Boston (1990)

    Google Scholar 

  340. Stewart, G.W., Zhang, G.: On a direct method for the solution of nearly uncoupled Markov chains. Numer. Math. 59(1), 1–11 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  341. Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains, xx+539 pp. Princeton University Press, Princeton, NJ (1994)

    Google Scholar 

  342. Sumita, U., Reiders, M.: A new algorithm for computing the ergodic probability vector for large Markov chains: Replacement process approach. Probab. Eng. Inf. Sci. 4, 89–116 (1988)

    Article  Google Scholar 

  343. Torokhti, A., Howlett, P., Pearce, C.: Method of best successive approximations for nonlinear operators. J. Comput. Anal. Appl. 5(3), 299–312 (2003)

    MathSciNet  MATH  Google Scholar 

  344. Turbin, A.F.: On asymptotic behavior of time of a semi-Markov process being in a reducible set of states. Linear case. Teor. Veroyatn. Mat. Stat. 4, 179–194 (1971). (English translation in Theory Probab. Math. Stat. 4, 167–182)

    Google Scholar 

  345. Turbin, A.F.: An application of the theory of perturbations of linear operators to the solution of certain problems that are connected with Markov chains and semi-Markov processes. Teor. Veroyatn. Mat. Stat. 6, 118–128 (1972). (English translation in Theory Probab. Math. Stat. 6, 119–130)

    Google Scholar 

  346. Van Doorn, E.A., Pollett, P.K.: Quasi-stationary distributions for discrete-state models. Eur. J. Oper. Res. 230, 1–14 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  347. Vantilborgh, H.: Aggregation with an error of O(\(\varepsilon ^2\)). J. Assoc. Comput. Mach. 32(1), 162–190 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  348. Verhulst, F.: Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics. Texts in Applied Mathematics, vol. 50, xvi+324 pp. Springer, New York (2005)

    Google Scholar 

  349. Vishik, M.I., Lyusternik, L.A.: The solution of some perturbation problems in the case of matrices and self-adjoint and non-self-adjoint differential equations. Uspehi Mat. Nauk 15, 3–80 (1960)

    MathSciNet  Google Scholar 

  350. Wentzell, A.D.: Asymptotic expansions in limit theorems for stochastic processes I. Probab. Theory Relat. Fields 106(3), 331–350 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  351. Wentzell, A.D.: Asymptotic expansions in limit theorems for stochastic processes II. Probab. Theory Relat. Fields 113(2), 255–271 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  352. Wentzell, A.D., Freidlin, M.I. (1979). Fluctuations in Dynamical Systems Subject to Small Random Perturbations. Probability Theory and Mathematical Statistics, Nauka, Moscow, 424 pp. (English edition: Random Perturbations of Dynamical Systems. Fundamental Principles of Mathematical Sciences, 260, Springer, New York (1998, 2012), xxviii+458 pp)

    Google Scholar 

  353. Wilkinson, J.H.: Error analysis of direct method of matrix inversion. J. Assoc. Comput. Mach. 8, 281–330 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  354. Yin, G.G., Zhang, Q.: Continuous-time Markov Chains and Applications. A Singular Perturbation Approach. Applications of Mathematics, vol. 37, ivx+349 pp. Springer, New York (1998)

    Google Scholar 

  355. Yin, G., Zhang, Q.: Discrete-time singularly perturbed Markov chains. In: Yao, D.D., Zhang, H., Zhou, X.Y. (eds.) Stochastic Modeling and Optimization, pp. 1–42. Springer, New York (2003)

    Chapter  Google Scholar 

  356. Yin, G.G., Zhang, Q.: Discrete-time Markov chains. Two-time-scale methods and applications. Stochastic Modelling and Applied Probability, xix+348 pp. Springer, New York (2005)

    Google Scholar 

  357. Yin, G.G., Zhang, Q.: Continuous-Time Markov Chains and Applications. A Two-Time-Scale Approach. Stochastic Modelling and Applied Probability, vol. 37, xxii+427 pp. Springer, New York (2013). (2nd revised edition of Continuous-Time Markov Chains and Applications. A Singular Perturbation Approach. Applications of Mathematics, vol. 37, xvi+349 pp. Springer, New York, 1998)

    Google Scholar 

  358. Yin, G., Zhang, Q., Badowski, G.: Discrete-time singularly perturbed Markov chains: aggregation, occupation measures, and switching diffusion limit. Adv. Appl. Probab. 35(2), 449–476 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  359. Yin, G., Zhang, Q., Yang, H., Yin, K.: Discrete-time dynamic systems arising from singularly perturbed Markov chains. In: Proceedings of the Third World Congress of Nonlinear Analysts, Part 7, Catania, 2000 (2001). (Nonlinear Anal., 47(7), 4763–4774)

    Google Scholar 

  360. Zakusilo, O.K.: Thinning semi-Markov processes. Teor. Veroyatn. Mat. Stat. 6, 54–59 (1972). (English translation in Theory Probab. Math. Stat. 6, 53–58)

    Google Scholar 

  361. Zakusilo, O.K.: Necessary conditions for convergence of semi-Markov processes that thin. Teor. Veroyatn. Mat. Stat. 7, 65–69 (1972). (English translation in Theory Probab. Math. Stat. 7, 63–66)

    Google Scholar 

  362. Zhang, Q., Yin, G.: Exponential bounds for discrete-time singularly perturbed Markov chains. J. Math. Anal. Appl. 293(2), 645–662 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dmitrii Silvestrov .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Silvestrov, D., Silvestrov, S. (2016). Asymptotic Expansions for Stationary Distributions of Perturbed Semi-Markov Processes. In: Silvestrov, S., Rančić, M. (eds) Engineering Mathematics II. Springer Proceedings in Mathematics & Statistics, vol 179. Springer, Cham. https://doi.org/10.1007/978-3-319-42105-6_10

Download citation

Publish with us

Policies and ethics