Probability Bounds for First Exits Through Moving Boundaries

Abstract

Let $S_1, S_2,\cdots$ be partial sums of independent and identically distributed random variables and let $f(n)$ and $g(n)$ be increasing positive sequences. Nearly sharp bounds are presented for the probabilities $P\{S_i \geqq g(i), i = 1,\cdots, n\}$ and $P\{- f(i) \leqq S_i \leqq f(i), i = 1,\cdots, n\}$ under conditions on $f$ and $g$. The most difficult results are the lower bounds in the normal case. Results are obtained by an embedding method which approximates Brownian motion by sums of independent random variables taking on only two or three values.