Probability and Mathematical Statistics: Vol. 6, Fasc. 2

RANDOM WALKS WITH RANDOM INDICES AND NEGATIVE DRIFT CONDITIONED TO STAY POSITIVE

A. Szubarga
D. Szynal

Abstract: Let $(X ,k > 1) k$ be a sequence of independent, identically distributed random variables with $E|X |= m < 0, 1$ and let $(N ,n > 0), n$ $N = 0 0$ a.s., be a sequence of positive integer-valued random variables. Form the random walk $(S ,n > 0) Nn$ by setting $S = 0, 0$ $S = X + ...+X , Nn 1 Nn$ $n > 1.$

The main result in this paper shows (under appropriate conditions on $(N ,n > 0) n$ and $(X ,k > 1) k$ ) that $S Nn$ conditioned on $[S > 0,...,S > 0] 1 Nn$ converges weakly to a random variable $S*$ considered by Iglehart [4].

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -