The Range Process in Random Walks: Theoretical Results and Applications

  • Pierre Vallois
  • Charles S. Tapiero
Part of the Advances in Computational Economics book series (AICE, volume 6)


This paper summarizes and apply some known results regarding the Range Run Length (RRL) in random walks. Explicitly, the inverse of the range process in random walks is evaluated for processes such as the symmetric and asymmetric discrete random walks, the discrete “birth death range process”, as well as Wiener processes with and without drift. Some of the results stated are proved elsewhere while a number of applications spanning the detection of outliers, R/S analysis and the estimation of the Hurst exponent for time series, as well as application to the control of ‘variability“ are suggested.


Random Walk Wiener Process GARCH Model Hurst Exponent Range Process 
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  1. Bollerslev, T., 1986, ‘Generalized autoregresive conditional heteroskedasticity’ Journal of Econometrics307–327.Google Scholar
  2. Chow Y.S., H. Robbins, and D. Siegmund, 1971, The Theory of Optimal Stopping, New York: Dover Publications.Google Scholar
  3. Daudin, J.J., 1995, `Etude de l’amplitude d’une marche aléatoire de Bernoulli’ Recherche Opérationelle/Operations Research (RAIRO)forthcoming. Google Scholar
  4. Dvoretzky, A. and P. Erdos, 1951, `Some problems on random walk in space’, Second Berkeley Symp. Mth. Stat. and Prob., pp. 353–368.Google Scholar
  5. Engle, R., 1987, ‘Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation’, Econometrica 55, 987–1008.Google Scholar
  6. Feller, W., 1951, `The asymptotic distribution of the range of sums of independent random variables’, Annals of Math. Stat. 22, 427–432CrossRefGoogle Scholar
  7. Hurst, H.E., 1951, `Long terms storage of reservoirs’, Transaction of the American Society of Civil Engineers 770–808.Google Scholar
  8. Imhoff, J.P., 1985, `On the range of brownian motion and its inverse process’, Ann. Prob. 13 (3), 1011–1017.CrossRefGoogle Scholar
  9. Imhoff, J.P., 1992, `A construction of the brownian motion path from BES (3) pieces’, Stochastic Processes and Applications 43, 345–353.CrossRefGoogle Scholar
  10. Jain, N.C. and S. Orey, 1968, `On the range of random walk’, Israel Journal of Mathematics 6, 373–380.CrossRefGoogle Scholar
  11. Jain, N.C. and W.E. Pruitt, 1972, `The range of random walk’, Sixth Berkeley Symp. Math. Stat. Prob. 3, 31–50.Google Scholar
  12. Peter Edgar E., 1995, Chaos and Order in Capital Markets, New York: Wiley.Google Scholar
  13. Tapiero, C.S., 1988, Applied Stochastic Models and Control in Management, New York: North-Holland.Google Scholar
  14. Tapiero, C.S., 1996, The Management of Quality and Its Control, London: Chapman and Hall.Google Scholar
  15. Tapiero, C.S. and P. Vallois, 1996, `Run length statistics and the Hurst exponent in random and birth-death random walks’, Chaos, Solitons and Fractals 7 (9), 1333–1341.CrossRefGoogle Scholar
  16. Troutman, B.M., 1983, `Weak convergence of the adjusted range of cumulative sums of exchangeable random variables’, J. Appl. Prob. 20, 297–304.CrossRefGoogle Scholar
  17. Vallois, P., 1993, `Diffusion arrêtée au premier instant où le processus de l’amplitude atteint un niveau donné’, Stochastics and Stochastic Reports 43, 93–115.Google Scholar
  18. Vallois, P., 1995, `On the range process of a Bernoulli random walk, in J. Janssen and C.H. Skiadas (Eds), Proceedings of the Sixth International Symposium on Applied Stochastic Models and Data Analysis,Vol II, Singapore: World Scientific, pp. 1020–1031.Google Scholar
  19. Vallois, P., 1996, `The range of a simple random walk on Z’, Adv. Appl. Prob. 28, 1014–1033.CrossRefGoogle Scholar
  20. Vallois, P. and C.S. Tapiero, 1995, `Moments of an amplitude process in a random walk’, Recherche Operationnelle/Operation Research (RAIRO), 29 (1), 1–17.Google Scholar
  21. Vallois, P. and C.S. Tapiero, 1996a, `The detection of of outliers and variability in random walks’, Working Paper, ESSEC.Google Scholar
  22. Vallois, P. and C.S. Tapiero, 1996b, `Range reliability in random walks’, Zeitschrift für Operations Research, forthcoming.Google Scholar

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© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Pierre Vallois
  • Charles S. Tapiero

There are no affiliations available

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