2010
DOI: 10.1016/j.spa.2010.03.005
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On the probability that integrated random walks stay positive

Abstract: 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… Show more

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Cited by 21 publications
(44 citation statements)
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“…where B k and B k (t) are Bernoulli numbers and Bernoulli polynomials, respectively. Noting that ψ n (n) = ψ(1) = 0 = ψ(0) = ψ n (0), we conclude that the first correction term in (15) disappears. Furthermore, by the definition of…”
Section: Proof Of the Chebyshev-type Estimate (7)mentioning
confidence: 73%
“…where B k and B k (t) are Bernoulli numbers and Bernoulli polynomials, respectively. Noting that ψ n (n) = ψ(1) = 0 = ψ(0) = ψ n (0), we conclude that the first correction term in (15) disappears. Furthermore, by the definition of…”
Section: Proof Of the Chebyshev-type Estimate (7)mentioning
confidence: 73%
“…(It was some moment inequalities, which were already known in the literature.) For integrated random walks we do not have such information and, therefore, should find an alternative way of justification of (14). This is done in Section 2.…”
Section: Main Resultmentioning
confidence: 99%
“…Item 1 uses arguments similar to that previously developed by Vysotsky (c.f. Lemma 2 in [Vys10].) It is based on the observation that the law of a negative excursion of a right-continuous random walk is invariant by time reversal.…”
Section: First Termmentioning
confidence: 99%