Abstract
This paper introduces a definition of reliability based on a process range. Thus, process failure is defined when the range of a process first reaches a given and unacceptable level. The Mean Time To Failure (MTTF) which is denned as the mean of the first time for a range to attain a given amplitude is then calculated for an asymmetric random walk process. The probability distribution of the range is then given and the process reliability over long periods of system operations are then calculated. Applications such as the control of wings movements, stock price and exchange rates volatility (defined in terms of reliability) are also used to motivate the usefulness of range processes in reliability studies. Finally, we point out that there is necessarily a relationship between the range reliability and the propensity of a series to become chaotic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agin MA, Godbole AP (1992) A new exact runs test for randomness. In: Page C, Lepage R (eds) Computing Science and Statistics, Springer-Verlag, New York, pp. 281–285
Choisid L, Isaac R (1978) On the range of recurrent Markov chains. Ann. Probability 6:680–687
Chow YS, Robbins H, Siegmund D (1971) The theory of optimal stopping. Dover Publications, New York
Dvoretzki A, Erdos P (1951) Some problems on random walk in space. Second Berkeley Symp. Math. Stat. and Prob.: 353–368
Feller W (1951) The asymptotic distribution of the range of sums of independent random variables. Annals of Math. Stat. 22:427–432
Fou JC, Koutras MV (1994) Distribution theory of runs: A Markov chain approach. JASA 89:1050–1058
Imhof JP (1985) On the range of Brownian motion and its inverse process. Ann Prob. 13, 3:1011–1017
Hurst HE (1951) Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers: 770–808
Jain NC, Orey S (1968) On the range of random walk. Israel Journal of Mathematics 6:373–380
Jain NC, Pruitt WE (1972) Tne range of random walk. Sixth Berkeley Symp. Math. Stat. Prob. 3:31–50
Mood AM (1940) The distribution theory of runs. Annals of Mathematical Statistics 11:367–392
Mosteller F (1941) Note on an application of runs to quality control charts. Annals of Mathematical Statistics 12:228–232
Nelson DB, Foster DP (1994) Asymptotic filtering theory for univariate ARCH model. Econometrica 62:1–41
Otway TH (1995) Records of the Florentine proveditori degli cambiatori: An example of an antipersistent time series in economics. Chaos, Solitons and Fractals 5:103–107
Peters EE (1991) Chaos and order in the capital markets. Wiley, New York
Prabhu NU (1980) Stochastic storage processes. Springer Verlag, New York
Schwager SJ (1983) Run probabilities in sequences of Markov dependent trials. JASA 78:168–175
Spitzer F (1964) Principles of random walk. Van Nostrand, New York
Tapiero CS (1988) Applied stochastic models and control in management. North Holland, New York
Tapiero CS (1996) The management of quality and its control. Chapman and Hall, London
Tapiero CS, Vallois P (1996) Run length statistics and the Hurst exponent. Chaos, Solitons and Fractals 7, 9:1333–1341
Troutman BM (1983) Weak convergence of the adjusted range of cumulative sums of exchangeable random variables. J. Appl. Prob. 20:297–304
Vallois P (1993) Diffusion arreté au premier instant où le processus de l'amplitude atteint un niveau donné. Stochastics and Stochastic Reports 43:93–115
Vallois P (1995) On the range process of a Bernoulli random walk. In: Jansen J, Skiadas CH (eds.) Proceedings of the Sixth International Symposium on Applied Stochastic Models and Data Analysis, vol II., Word Scientific: 1020–1031
Vallois P (1996) The range of a simple random walk on Z. Journal of Applied Probability. Adv. Appl. Prob. 28:1014–1033
Vallois P, Tapiero CS (1995a) Moments of an amplitude process in a random walk and approximations: Computations and applications. Recherche Operationnelle/Operation Research (RAIRO) 29, 1:1–17
Vallois P, Tapiero CS (1995b) The average run length of the range in birth and death random walks. Proceedings of the Conference on Applied Stochastic Models and Data Analysis, Dublin
Vallois P, Tapiero CS (1997) The range process in random walks: Theoretical results and applications. In: Ammans H, Rustern B, Whinston A (eds.) Advances in Computational Economics, Kluwer Publications, pp. 291–307
Wald A, Wolfowitz J (1940) On a test whether two populations are from the same population. Annals of Mathematical Statistics 11:147–162
Wolfowitz J (1943) On the theory of runs with some applications to quality control. Annals of Mathematical Statistics 14:280–288
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vallois, P., Tapiero, C.S. Range reliability in random walks. Mathematical Methods of Operations Research 45, 325–345 (1997). https://doi.org/10.1007/BF01194783
Issue Date:
DOI: https://doi.org/10.1007/BF01194783