On the probability that integrated random walks stay positive

Details On the probability that integrated random walks stay positive On the probability that integrated random walks stay positive

2009-11-29

arxiv.org/abs/0911.5456v2

http://dx.doi.org/10.1016/j.spa.2010.03.005

Mathematics

Probability

Theorems 2 and 3 were restated and merged into one theorem; a new lemma (Lemma 1) added; Lemma 3 and Remark 1 were restated an

Scientific paper

10.1016/j.spa.2010.03.005

Let $S_n$ be a centered random walk with a finite variance, and define the new sequence $A_n:=\sum_{i=1}^n S_i$, which we call an integrated random walk. We are interested in the asymptotics of $$p_N:=P(\min_{1 \le k \le N} A_k \ge 0)$$ as $N \to \infty$. Sinai (1992) proved that $p_N \asymp N^{-1/4}$ if $S_n$ is a simple random walk. We show that $p_N \asymp N^{-1/4}$ for some other types of random walks that include double-sided exponential and double-sided geometric walks, both not necessarily symmetric. We also prove that $p_N \le c N^{-1/4}$ for lattice walks and for upper exponential walks, that are the walks such that $Law (S_1 | S_1>0)$ is an exponential distribution.

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