Probability and Mathematical Statistics: Vol.

CONDITIONED LIMIT THEOREMS FOR FUNCTIONS OF THE AVERAGE OF I.I.D. RANDOM VARIABLES

A. Szubarga
D. Szynal

Abstract: Let $(q ,k > 1) k$ be a sequence of i.i.d. random variables with $Eq = 0, 1$ $0 < Eq2 = s2 < oo . 1$ Form the random walk $(S ,n > 0) n$ by setting $S = 0, 0$ $S = q + ...+ q , n 1 n$ $n > 1.$ Let $T$ denote the hitting time of the set $(- oo ,0]$ by the random walk. Put $X (t) = S /s V~ n, n [nt]$ $0 < t < 1.$ Let $h$ be a real-valued, right-continuous function on $R,$ having left limits, with $h(0) = 1,$ and continuous at $0.$ For $b > 0$ we define the map $H : D[0,1]-- > D[0,1] n$ by $H (f) = fh(n-bf), n$ $f (- D[0,1],$ $n > 1.$ Put $Y = H (X ). n n n$ This note deals with the asymptotic behaviour of $Y n$ conditioned on $[T > n].$ Moreover, we investigate the asymptotic behaviour in the question when n is replaced by $N , n$ where $(N ,n > 1) n$ is a sequence of positive integer-valued random variables.

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

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