Positivity of integrated random walks

Vladislav Vysotsky

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 1, page 195-213
  • ISSN: 0246-0203

Abstract

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Take a centered random walk S n and consider the sequence of its partial sums A n : = ∑ i = 1 n S i . Suppose S 1 is in the domain of normal attraction of an α -stable law with 1 l t ; α ≤ 2 . Assuming that S 1 is either right-exponential (i.e. ℙ ( S 1 g t ; x | S 1 g t ; 0 ) = e - a x for some a g t ; 0 and all x g t ; 0 ) or right-continuous (skip free), we prove that ℙ { A 1 g t ; 0 , ⋯ , A N g t ; 0 } ∼ C α N 1 / ( 2 α ) - 1 / 2 as N → ∞ , where C α g t ; 0 depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

How to cite

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Vysotsky, Vladislav. "Positivity of integrated random walks." Annales de l'I.H.P. Probabilités et statistiques 50.1 (2014): 195-213. <http://eudml.org/doc/271949>.

@article{Vysotsky2014,
abstract = {Take a centered random walk $S_\{n\}$ and consider the sequence of its partial sums $A_\{n\}:=\sum _\{i=1\}^\{n\}S_\{i\}$. Suppose $S_\{1\}$ is in the domain of normal attraction of an $\alpha $-stable law with $1&lt;\alpha \le 2$. Assuming that $S_\{1\}$ is either right-exponential (i.e. $\mathbb \{P\}(S_\{1\}&gt;x|S_\{1\}&gt;0)=\mathrm \{e\}^\{-ax\}$ for some $a&gt;0$ and all $x&gt;0$) or right-continuous (skip free), we prove that \[\mathbb \{P\}\lbrace A\_\{1\}&gt;0,\dots ,A\_\{N\}&gt;0\rbrace \sim C\_\{\alpha \}N^\{\{1\}/\{(2\alpha )\}-1/2\}\] as $N\rightarrow \infty $, where $C_\{\alpha \}&gt;0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.},
author = {Vysotsky, Vladislav},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {integrated random walk; persistence; one-sided exit probability; unilateral small deviations; area of random walk; Sparre-Andersen theorem; stable excursion; area of excursion},
language = {eng},
number = {1},
pages = {195-213},
publisher = {Gauthier-Villars},
title = {Positivity of integrated random walks},
url = {http://eudml.org/doc/271949},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Vysotsky, Vladislav
TI - Positivity of integrated random walks
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 1
SP - 195
EP - 213
AB - Take a centered random walk $S_{n}$ and consider the sequence of its partial sums $A_{n}:=\sum _{i=1}^{n}S_{i}$. Suppose $S_{1}$ is in the domain of normal attraction of an $\alpha $-stable law with $1&lt;\alpha \le 2$. Assuming that $S_{1}$ is either right-exponential (i.e. $\mathbb {P}(S_{1}&gt;x|S_{1}&gt;0)=\mathrm {e}^{-ax}$ for some $a&gt;0$ and all $x&gt;0$) or right-continuous (skip free), we prove that \[\mathbb {P}\lbrace A_{1}&gt;0,\dots ,A_{N}&gt;0\rbrace \sim C_{\alpha }N^{{1}/{(2\alpha )}-1/2}\] as $N\rightarrow \infty $, where $C_{\alpha }&gt;0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.
LA - eng
KW - integrated random walk; persistence; one-sided exit probability; unilateral small deviations; area of random walk; Sparre-Andersen theorem; stable excursion; area of excursion
UR - http://eudml.org/doc/271949
ER -

References

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