On the probability that integrated random walks stay positive

Vladislav Vysotsky

doi:10.1016/j.spa.2010.03.005

On the probability that integrated random walks stay positive

Vladislav Vysotsky

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

Let S_n be a centered random walk with a finite variance, and consider the sequence A_n:=∑_i=1ⁿ S _i, which we call an integrated random walk. We are interested in the asymptotics of PN:=P{min_1≤k≤N A_k≥0 as N→∞. Sinai (1992) [15] proved that pN equivalent to N^-1/4 if S_n is a simple random walk. We show that pN equivalent 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 pN ≤ cN^-1/4 for integer-valued walks and upper exponential walks, which are the walks such that Law(S₁|S ₁>0) is an exponential distribution.

Original language	English (US)
Pages (from-to)	1178-1193
Number of pages	16
Journal	Stochastic Processes and their Applications
Volume	120
Issue number	7
DOIs	https://doi.org/10.1016/j.spa.2010.03.005
State	Published - Jul 2010
Externally published	Yes

Keywords

Area of excursion
Area of random walk
Excursion
Integrated random walk
One-sided exit probability
Unilateral small deviations

ASJC Scopus subject areas

Statistics and Probability
Modeling and Simulation
Applied Mathematics

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10.1016/j.spa.2010.03.005

Cite this

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title = "On the probability that integrated random walks stay positive",

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

keywords = "Area of excursion, Area of random walk, Excursion, Integrated random walk, One-sided exit probability, Unilateral small deviations",

author = "Vladislav Vysotsky",

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AU - Vysotsky, Vladislav

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N2 - Let Sn be a centered random walk with a finite variance, and consider the sequence An:=∑i=1n S i, which we call an integrated random walk. We are interested in the asymptotics of PN:=P{min1≤k≤N Ak≥0 as N→∞. Sinai (1992) [15] proved that pN equivalent to N-1/4 if Sn is a simple random walk. We show that pN equivalent 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 pN ≤ cN-1/4 for integer-valued walks and upper exponential walks, which are the walks such that Law(S1|S 1>0) is an exponential distribution.

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

KW - Area of excursion

KW - Area of random walk

KW - Excursion

KW - Integrated random walk

KW - One-sided exit probability

KW - Unilateral small deviations

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On the probability that integrated random walks stay positive

Abstract

Keywords

ASJC Scopus subject areas

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