Open Access
2018 On the ladder heights of random walks attracted to stable laws of exponent 1
Kôhei Uchiyama
Electron. Commun. Probab. 23: 1-12 (2018). DOI: 10.1214/18-ECP122

Abstract

Let $Z$ be the first ladder height of a one dimensional random walk $S_n=X_1+\cdots + X_n$ with i.i.d. increments $X_j$ which are in the domain of attraction of a stable law of exponent $\alpha $, $0<\alpha \leq 1$. We show that $P[Z>x]$ is slowly varying at infinity if and only if $\lim _{n\to \infty } n^{-1}\sum _1^n P[S_k>0]=0$. By a known result this provides a criterion for $S_{T(R)} /R \stackrel{{\rm P}} \longrightarrow \infty $ as $R\to \infty $, where $T(R)$ is the time when $S_n$ crosses over the level $R$ for the first time. The proof mostly concerns the case $\alpha =1$.

Citation

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Kôhei Uchiyama. "On the ladder heights of random walks attracted to stable laws of exponent 1." Electron. Commun. Probab. 23 1 - 12, 2018. https://doi.org/10.1214/18-ECP122

Information

Received: 22 February 2018; Accepted: 28 February 2018; Published: 2018
First available in Project Euclid: 30 March 2018

zbMATH: 1390.60170
MathSciNet: MR3785397
Digital Object Identifier: 10.1214/18-ECP122

Subjects:
Primary: 60G50
Secondary: 60J45

Keywords: domain of attraction , ladder height , large deviation , Random walk , slowly varying , stable law of exponent 1

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