Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T23:52:27.793Z Has data issue: false hasContentIssue false

A limit theorem for two-dimensional conditioned random walk

Published online by Cambridge University Press:  22 January 2016

Michio Shimura*
Affiliation:
Institute of Mathematics, University of Tsukuba, Sakura-mura Niihari-gun Ibaraki, 305, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let {(Sn, Tn), n = 0, 1, 2, …} be a two-dimensional random walk with stationary independent increments starting at the origin 0. Throughout the paper we always assume the following condition:

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1984

References

[ 1 ] Billingsley, P., Convergence of probability measures, Wiley, New York, 1968.Google Scholar
[ 2 ] Bolthausen, E., On a functional central limit theorem for random walks conditioned to stay positive, Ann. Probab., 4 (1976), 480485.CrossRefGoogle Scholar
[ 3 ] Hardy, G.H., Divergent series, Clarendon Press, Oxford, England, 1949.Google Scholar
[ 4 ] Iglehart, D.L., Functional central limit theorems for random walks conditioned to stay positive, Ann. Probab., 2 (1974), 608619.Google Scholar
[ 5 ] Ito, K. and McKean, H. P. Jr., Diffusion processes and their sample paths, second printing, Springer, Berlin, 1974.Google Scholar
[ 6 ] Kozlov, M.V., On the asymptotic behavior of the probability of non-extinction for critical branching processes in a random environment, Theory Probab. Appl., 21 (1976), 791804.Google Scholar
[ 7 ] Lindvall, T., Weak convergence of probability measures and random functions in the function space D[0, ∞), J. Appl. Probab., 10 (1973), 109121.CrossRefGoogle Scholar
[ 8 ] MoguFskii, A.A. and Pécher skii, E. A., On the first exit time from a semigroup in Rm for a random walk, Theory Probab. Appl., 22 (1977), 818825.Google Scholar
[ 9 ] Ritter, G.A., Growth of random walks conditioned to stay positive, Ann. Probab., 9 (1981), 699704.Google Scholar
[10] Rosen, B., On the asymptotic distribution of sums of independent identically distributed random variables, Ark. Mat., 4 (1962), 323332.Google Scholar
[11] Shimura, M., A class of conditional limit theorems related to ruin problem, Ann. Probab., 11 (1983), 4045.Google Scholar
[12] Spitzer, F., A Tauberian theorem and its probability interpretation, Trans. Amer. Math. Soc, 94 (1960), 150169.CrossRefGoogle Scholar
[13] Williams, D., Decomposing the Brownian path, Bull. Amer. Math. Soc, 76 (1970), 871873.Google Scholar