Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-08T17:34:02.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak convergence and first passage times

Published online by Cambridge University Press:  14 July 2016

Allan Gut
Allan Gut*
Affiliation:
University of Uppsala
Get access Rights & Permissions [Opens in a new window]

Abstract

Let Sn, n = 1, 2, ‥, denote the partial sums of i.i.d. random variables with the common distribution function F and positive, finite mean. Let N(c) = min [k; Sk > c‥kp], c ≥ 0, 0 ≤ p < 1. Under the assumption that F belongs to the domain of attraction of a stable law with index α, 1 < α ≤ 2, functional central limit theorems for the first passage time process N(nt), 0 ≤ t ≤ 1, when n → ∞, are derived in the function space D[0,1].

Keywords

WEAK CONVERGENCEFIRST PASSAGE TIMESKOROHOD J1-TOPOLOGYSKOROHOD M1-TOPOLOGYSUPREMUM FUNCTIONFIRST PASSAGE TIME FUNCTIONCONTINUOUS MAPPING THEOREMRANDOM CHANGE OF TIME
Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 2 , June 1975 , pp. 324 - 332
Copyright
Copyright © Applied Probability Trust 1975 

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

[1] Basu, A. K. (1972) Invariance theorems for first passage time random variables. Canad. Math. Bull. 15, 171–176.CrossRefGoogle Scholar
[2] Billingsley, P. (1962) Limit theorems for randomly selected partial sums. Ann. Math. Statist. 33, 85–92.Google Scholar
[3] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[4] Bingham, N. H. (1973) Maxima of sums of random variables and suprema of stable processes. Z. Wahrscheinlichkeitsth. 26, 273–296.Google Scholar
[5] Feller, W. (1971) An Introduction to Probability Theory and Its Applications II, 2nd. Ed. Wiley, New York.Google Scholar
[6] Gikhman, I. I. and Skorohod, A. V. (1969) Introduction to the Theory of Random Processes. Saunders, Philadelphia.Google Scholar
[7] Gut, A. (1974) On the moments and limit distributions of some first passage times. Ann. Probability 2, 277–308.Google Scholar
[8] Gut, A. (1973) A functional central limit theorem connected with extended renewal theory. Z. Wahrscheinlichkeitsth. 27, 123–129.Google Scholar
[9] Heyde, C. C. (1966) Some renewal theorems with application to a first passage problem. Ann. Math. Statist. 37, 699–710.Google Scholar
[10] Iglehart, D. L. and Whitt, W. (1971) The equivalence of functional central limit theorems for counting processes and associated partial sums. Ann. Math. Statist. 42, 1372–1378.CrossRefGoogle Scholar
[11] Skorohod, A. V. (1956) Limit theorems for stochastic processes. Theor. Probability Appl. I, 261–290.Google Scholar
[12] Teicher, H. (1973) A classical limit theorem without invariance or reflection. Ann. Probability 1, 702–704.Google Scholar
[13] Vervaat, W. (1972) Functional central limit theorems for processes with positive drift and their inverses. Z. Wahrscheinlichkeitsth. 23, 245–253.CrossRefGoogle Scholar
[14] Whitt, W. (1971) Weak convergence of first passage time processes. J. Appl. Prob. 8, 417–422.Google Scholar
[15] Whitt, W. (1972) (1973) Continuity of several functions on the function space D. Technical Reports, Yale University.Google Scholar