Electronic Communications in Probability

Conditional persistence of Gaussian random walks

Fuchang Gao, Zhenxia Liu, and Xiangfeng Yang

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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).

Article information

Electron. Commun. Probab., Volume 19 (2014), paper no. 70, 9 pp.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F99: None of the above, but in this section

Conditional persistence random walk integrated random walk

This work is licensed under a Creative Commons Attribution 3.0 License.


Gao, Fuchang; Liu, Zhenxia; Yang, Xiangfeng. Conditional persistence of Gaussian random walks. Electron. Commun. Probab. 19 (2014), paper no. 70, 9 pp. doi:10.1214/ECP.v19-3587. https://projecteuclid.org/euclid.ecp/1465316772

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