On the probability that integrated random walks stay positive

被引:15
|
作者
Vysotsky, Vladislav [1 ]
机构
[1] Univ Delaware, Newark, DE 19716 USA
关键词
Integrated random walk; Area of random walk; Unilateral small deviations; One-sided exit probability; Excursion; Area of excursion; CONVERGENCE;
D O I
10.1016/j.spa.2010.03.005
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
Let S-n be a centered random walk with a finite variance, and consider the sequence A(n) := Sigma(n)(i=1) S-i, which we call an integrated random walk. We are interested in the asymptotics of p(N) := P{min(1 <= k <= N) A(k) >= 0} as N -> infinity. Sinai (1992) [15] proved that p(N) asymptotic to N-1/4 if S-n is a simple random walk. We show that p(N) asymptotic to N-1/4 for some other kinds of random walks that include double-sided exponential and double-sided geometric walks, both not necessarily symmetric. We also prove that p(N) <= cN(-1/4) for integer-valued walks and upper exponential walks, which are the walks such that Law(S-1 vertical bar S-1 > 0) is an exponential distribution. (C) 2010 Elsevier B.V. All rights reserved.
引用
收藏
页码:1178 / 1193
页数:16
相关论文
共 50 条
  • [21] Positivity of integrated random walks
    Vysotsky, Vladislav
    ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, 2014, 50 (01): : 195 - 213
  • [22] Sharp probability estimates for random walks with barriers
    Ford, Kevin
    PROBABILITY THEORY AND RELATED FIELDS, 2009, 145 (1-2) : 269 - 283
  • [23] Sharp probability estimates for random walks with barriers
    Kevin Ford
    Probability Theory and Related Fields, 2009, 145 : 269 - 283
  • [24] Transition probability estimates for subordinate random walks
    Cygan, Wojciech
    Sebek, Stjepan
    MATHEMATISCHE NACHRICHTEN, 2021, 294 (03) : 518 - 558
  • [25] Return Probability of Quantum and Correlated Random Walks
    Kiumi, Chusei
    Konno, Norio
    Tamura, Shunya
    ENTROPY, 2022, 24 (05)
  • [26] PROBABILITY DENSITIES OF RANDOM-WALKS IN RANDOM-SYSTEMS
    HAVLIN, S
    BUNDE, A
    PHYSICA D-NONLINEAR PHENOMENA, 1989, 38 (1-3) : 184 - 191
  • [27] Some finitely additive probability: Random walks
    Gangopadhyay, S
    Rao, BV
    JOURNAL OF THEORETICAL PROBABILITY, 1997, 10 (03) : 643 - 657
  • [28] Some Finitely Additive Probability: Random Walks
    Sreela Gangopadhyay
    B. V. Rao
    Journal of Theoretical Probability, 1997, 10 : 643 - 657
  • [29] Positive hulls of random walks and bridges
    Godland, Thomas
    Kabluchko, Zakhar
    STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 2022, 147 : 327 - 362
  • [30] Exit times for integrated random walks
    Denisov, Denis
    Wachtel, Vitali
    ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, 2015, 51 (01): : 167 - 193