Hostname: page-component-cb9f654ff-p5m67 Total loading time: 0 Render date: 2025-08-22T16:14:36.848Z Has data issue: false hasContentIssue false
  • English
  • Français

On the moments of some first-passage times

Published online by Cambridge University Press:  14 July 2016

Valeri T. Stefanov
Valeri T. Stefanov*
Affiliation:
Institute of Mathematics, Sofia
*
∗ Postal address: Institute of Mathematics, Bulgarian Academy of Sciences, P.O. Box 373 1090-Sofia, Bulgaria.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let {Xt }t≧0 (t may be discrete or continuous) be a random process whose finite-dimensional distributions are of exponential type. The first-passage time inf{t:Xt ≧f(t)}, where f(t) is a positive, continuous function, such that f(t)= o(t) as t↑∞, is considered. The problem of finiteness of its moments is solved for both the case that {Xt }t≧0 has stationary independent increments as well as the case in which no assumptions are made about stationarity and independence for the increments of the process. Applications to sequential estimation are also given.

Keywords

SEQUENTIAL ESTIMATION

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 22 , Issue 2 , June 1985 , pp. 461 - 466
Copyright
Copyright © Applied Probability Trust 1985 

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.)

Article purchase

Temporarily unavailable

References

Borovkov, A. A. (1965) On the first passage time for one class of processes with independent increments. Theory Prob. Appl. 10, 331–334.CrossRefGoogle Scholar
Chow, Y. S., Robbins, H. and Teicher, H. (1965) Moments of randomly stopped sums. Ann. Math. Statist. 36, 789–799.Google Scholar
Chow, Y. S. (1966) On the moments of some one-sided stopping rules. Ann. Math. Statist. 37, 382–387.CrossRefGoogle Scholar
Chow, Y. S. and Teicher, H. (1966) On second moments of stopping rules. Ann. Math. Statist. 37, 388–392.CrossRefGoogle Scholar
Gut, A. (1974a) On the moments and limit distributions of some first passage times. Ann. Prob. 2, 277–308.Google Scholar
Gut, A. (1974b) On the moments of some first passage times for sums of dependent random variables. Stoch. Proc. Appl. 2, 115–126.CrossRefGoogle Scholar
Gut, A. (1975) On a.s. and r-mean convergence of random processes with an application to first passage times. Z. Wahrscheinlichkeitsth. 31, 333–341.Google Scholar
Heyde, C. C. (1964) Two probability theorems and their application to some first passage problems. J. Austral. Math. Soc. 4, 214–222.Google Scholar
Heyde, C. C. (1966) Some renewal theorems with application to a first passage problem. Ann. Math. Statist. 37, 699–710.Google Scholar
Janson, S. (1983) Renewal theory for m-dependent variables. Ann. Prob. 11, 558–568.CrossRefGoogle Scholar
Musiela, M. (1981) On sequential estimation of parameters of continuous Gaussian Markov processes. Prob. Math. Statist. 2, 37–53.Google Scholar
Siegmund, D. (1969) The variance of one-sided stopping rules. Ann. Math. Statist. 40, 1074–1077.CrossRefGoogle Scholar
Stefanov, V. T. (1985) On efficient stopping times. Stoch. Proc. Appl. To appear.CrossRefGoogle Scholar
Sudakov, V. N. (1969) On measures defined by Markov stopping times (in Russian). Zap. Nauc. Sem. LOMI 12, 157–164.Google Scholar