Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T23:58:00.411Z Has data issue: false hasContentIssue false
  • English
  • Français

A Central Limit Theorem for Cumulative Processes

Part of: Limit theorems Special processes

Published online by Cambridge University Press:  01 July 2016

Allen L. Roginsky
Allen L. Roginsky*
Affiliation:
IBM Corporation
*
* Postal address; E95/B673, IBM Corporation, P.O. Box 12195, Research Triangle Park, NC 27709, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.

Keywords

RATE OF CONVERSIONRENEWAL PROCESS

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 1 , March 1994 , pp. 104 - 121
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahmad, I. A. (1981) The exact order of normal approximation in bivariate renewal theory. Adv. Appl. Prob. 13, 113–128.Google Scholar
Doob, J. L. (1948) Renewal theorem from the point of view of the theory of probability. Trans. Amer. Math. Soc. 63, 422–438.Google Scholar
Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
Englund, G. (1980) Remainder term estimate for the asymptotic normality of the number of renewals. J. Appl. Prob. 17, 1108–1113.CrossRefGoogle Scholar
Feller, W. (1949) Fluctuation theory of renewal events. Trans. Amer. Math. Soc. 67, 98–119.Google Scholar
Glynn, P. W. and Whitt, W. (1988) Ordinary CLT and WLLN versions of L = ?W. Math. Operat. Res. 13, 673–692.CrossRefGoogle Scholar
Gut, A. (1988) Stopped Random Walks. Springer-Verlag, New York.CrossRefGoogle Scholar
Gut, A. and Janson, S. (1983) The limiting behavior of certain stopped sums and some applications. Scand. J. Statist. 10, 281–292.Google Scholar
Heyde, C. C. and Leslie, J. R. (1972) On the influence of moments on approximations by portions of a Chebyshev series in central limit theorem. Z. Wahrscheinlichkeitsth. 21, 255–268.Google Scholar
Hunter, J. J. (1974) Renewal theory in two dimensions: asymptotic results. Adv. Appl. Prob. 6, 546–562.Google Scholar
Ibragimov, I. A. (1967) On the Chebyshev-Craer asymptotic expansions. Theory Prob. Appl. 12, 455–470.Google Scholar
Murphree, E. and Smith, W. L. (1986) On transient regenerative processes. J. Appl. Prob. 23, 52–70.Google Scholar
Niculescu, S. P. (1984) On the asymptotic distribution of multivariate renewal processes. J. Appl. Prob. 21, 639–645.Google Scholar
Niculescu, S. P. and Omey, E. (1985) On the exact order of normal approximation in multivariate renewal theory. J. Appl. Prob. 22, 280–287.Google Scholar
Petrov, V. V. (1975) Sums of Independent Random Variables. Springer-Verlag, New York.Google Scholar
Rényi, A. (1957) On the asymptotic distribution of the sum of a random number of independent random variables. Acta Math. Acad. Sci. Hungar. 8, 193–199.Google Scholar
Roginsky, A. L. (1989) On the Central Limit Theorems for the Renewal and Cumulative Processes. Ph.D. dissertation, University of North Carolina.Google Scholar
Roginsky, A. L. (1992) A central limit theorem with remainder term for renewal processes. Adv. Appl. Prob. 24, 267–287.Google Scholar
Siegmund, O. (1975) The time until ruin in collective risk theory. Mitt. Verein Schweiz. Versicherungsmath. 75, 157–165.Google Scholar
Smith, W. L. (1955) Regenerative stochastic processes. Proc. Royal Soc. Edinburgh. A 232, 6–31.Google Scholar
Wald, A. (1944). On cumulative sums of random variables. Ann. Math. Statist. 15, 283–296.Google Scholar
Woodroofe, M. and Keener, R. (1987) Asymptotic expansions in boundary crossing problems. Ann. Prob. 15, 102–114.CrossRefGoogle Scholar