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3 papers
Transient phenomena for random walks with nonidential jumps having nonidetically infinite variances
A. A. Borovkov Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
Abstract:
Let
$\zeta_1,\zeta_2,\dots$ be independent random variables,
$$
Z_n=\sum_{i=1}^n\zeta_i,\qquad \overline{Z}_n=\max_{k\leq n}Z_k,\qquad Z=\overline{Z}_\infty.
$$
It is well known that if
$\zeta_i$ are identically distributed, then
$Z$ is a proper random variable when
${\mathbf{E}\zeta_i=-a<0}$, and
$Z=\infty$ a.s. when
$a=0$. The limiting distribution for
$\overline{Z}_n$
as
$n\to\infty$,
$a\to 0$ (in the triangular array scheme) when
$\mathbf{E}\zeta_i^2<\infty$
is well studied (see, e.g., [J. F. C. Kingman,
J. Roy. Statist. Soc. Ser. B, 24 (1962), pp. 383–392],
[Yu. V. Prokhorov,
Litov. Math. Sb., 3 (1963), pp. 199–204 (in Russian)], and [A. A. Borovkov,
Stochastic Process in Queueing Theory, Springer-Verlag, New York, 1976]).
In the present paper, we study the limiting distribution for
$\overline{Z}_n$ as
$\mathbf{E}\zeta_i\to 0$,
$n\to\infty$, in the case when
$\mathbf{E}\zeta_i^2=\infty$ and the summands
$\zeta_i$ are nonidentically distributed.
Keywords:
random walks, maxima of partial sums, transient phenomena, convergence to stable processes, nonidentically distributed summands, infinite variance. Received: 16.09.2004
DOI:
10.4213/tvp105