Probability and Mathematical Statistics: Vol.

CONDITIONED RANDOM WALKS WITH RANDOM INDICES

A. Szubarga
D. Szynal

Abstract: Let $(X ,k > 1) k$ be a sequence of i.i.d. random variables with $EX = 0, 1$ $EX2 = s2 < oo , 1$ and let $(N ,m > 0), m$ $N = 0 0$ a.s., be a sequence of positive integer-valued random variables. Let $(S ,n > 0) n$ and $(S ,m > 0) Nm$ be defined by $S = 0 0$ a.s., $S = X + ...+ X , n 1 n$ $n > 1,$ $S = 0 N0$ a.s., $S = X + X + ...+ X , Nm 1 2 Nm$ $m > 1.$ Put

N = inf(n : S < 0), M = max(S : n < N ). n n

In this note, under additional conditions on sequences $(Xk, k > 1)$ and $(Nm, m > 0),$ we investigate the limit behaviour of $V~ -- P [M/s Nm < n| N > Nm],$ $V~ -- P [max0 <k<Nm Sk/s Nm < n|N > Nm],$ and $V~ --- P[N > Nm |M > ns Nm],$ where $n > 0.$

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

Key words and phrases: -