Electronic Communications in Probability

Conditional persistence of Gaussian random walks

Fuchang Gao, Zhenxia Liu, and Xiangfeng Yang

Full-text: Open access

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).

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465316772

Digital Object Identifier
doi:10.1214/ECP.v19-3587

Mathematical Reviews number (MathSciNet)
MR3269170

Zentralblatt MATH identifier
1307.60036

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

Keywords
Conditional persistence random walk integrated random walk

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

Citation

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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References

  • Aurzada, F., Dereich, S. and Lifshits, M.: Persistence probabilities for an integrated random walk bridge. Probab. Math. Statist. 34, (2014), 1–22.
  • Aurzada, F. and Simon, T.: Persistence probabilities & exponents, ARXIV1203.6554
  • Caravenna, Francesco; Deuschel, Jean-Dominique. Pinning and wetting transition for $(1+1)$-dimensional fields with Laplacian interaction. Ann. Probab. 36 (2008), no. 6, 2388–2433.
  • Dembo, Amir; Ding, Jian; Gao, Fuchang. Persistence of iterated partial sums. Ann. Inst. Henri PoincarÄ‚Å Probab. Stat. 49 (2013), no. 3, 873–884.
  • Denisov, D. and Wachtel, V.: Random walks in cones, ARXIV1110.1254
  • Denisov, D. and Wachtel, V.: Exit times for integrated random walks, ARXIV1207.2270
  • Gut, Allan. Probability: a graduate course. Second edition. Springer Texts in Statistics. Springer, New York, 2013. xxvi+600 pp. ISBN: 978-1-4614-4707-8; 978-1-4614-4708-5
  • Vysotsky, Vladislav. Positivity of integrated random walks. Ann. Inst. Henri PoincarÄ‚Å Probab. Stat. 50 (2014), no. 1, 195–213.