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2014 Conditional persistence of Gaussian random walks
Fuchang Gao, Zhenxia Liu, Xiangfeng Yang
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Electron. Commun. Probab. 19: 1-9 (2014). DOI: 10.1214/ECP.v19-3587

Abstract

Let $\{X_n\}_{n\geq1}$ be a sequence of i.i.d. standard Gaussian random variables, let $S_n=\sum_{i=1}^nX_i$ be the Gaussian random walk, and let $T_n=\sum_{i=1}^nS_i$ be the integrated (or iterated) Gaussian random walk. In this paper we derive the following upper and lower bounds for the conditional persistence:$$\mathbb{P}\left\{\max_{1\leq k \leq n}T_{k} \leq 0\,\,\Big|\,\,T_n=0,S_n=0\right\}\lesssim n^{-1/2},\ \mathbb{P}\left\{\max_{1\leq k \leq 2n}T_{k} \leq 0\,\,\Big|\,\,T_{2n}=0,S_{2n}=0\right\}\gtrsim\frac{n^{-1/2}}{\log n},$$for $n\rightarrow\infty,$ which partially proves a conjecture by Caravenna and Deuschel (2008).

Citation

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Fuchang Gao. Zhenxia Liu. Xiangfeng Yang. "Conditional persistence of Gaussian random walks." Electron. Commun. Probab. 19 1 - 9, 2014. https://doi.org/10.1214/ECP.v19-3587

Information

Accepted: 10 October 2014; Published: 2014
First available in Project Euclid: 7 June 2016

zbMATH: 1307.60036
MathSciNet: MR3269170
Digital Object Identifier: 10.1214/ECP.v19-3587

Subjects:
Primary: 60G50
Secondary: 60F99

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