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 : = ∑ i = 1 n 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 → ∞ . Sinai (1992) [15] proved that p N ≍ N − 1 / 4 if S n is a simple random walk. We show that p N ≍ 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 ≤ c N − 1 / 4 for integer-valued walks and upper exponential walks, which are the walks such that Law ( S 1 | S 1 > 0 ) is an exponential distribution.