On the probability that integrated random walks stay positive

Let SnSn be a centered random walk with a finite variance, and consider the sequence An:=∑i=1nSi, which we call an integrated random walk. We are interested in the asymptotics of pN≔P{min1≤k≤NAk≥0} as N→∞N→∞. Sinai (1992) [15] proved that pN≍N−1/4pN≍N−1/4 if SnSn is a simple random walk. We show that pN≍N−1/4pN≍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 pN≤cN−1/4pN≤cN−1/4 for integer-valued walks and upper exponential walks, which are the walks such that Law(S1|S1>0)Law(S1|S1>0) is an exponential distribution.