On the probability that integrated random walks stay positive

Let S-n be a centered random walk with a finite variance, and consider the sequence A(n) := Sigma(n)(i=1) S-i, which we call an integrated random walk. We are interested in the asymptotics of p(N) := P{min(1 <= k <= N) A(k) >= 0} as N -> infinity. Sinai (1992) [15] proved that p(N) asymptotic to N-1/4 if S-n is a simple random walk. We show that p(N) asymptotic to N-1/4 for some other kinds of random walks that include double-sided exponential and double-sided geometric walks, both not necessarily symmetric. We also prove that p(N) <= cN(-1/4) for integer-valued walks and upper exponential walks, which are the walks such that Law(S-1 vertical bar S-1 > 0) is an exponential distribution. (C) 2010 Elsevier B.V. All rights reserved.