## Abstract

Let S_{n} be a centered random walk with a finite variance, and consider the sequence A_{n}:=∑_{i=1}^{n} S _{i}, which we call an integrated random walk. We are interested in the asymptotics of PN:=P{min_{1≤k≤N} A_{k}≥0 as N→∞. Sinai (1992) [15] proved that pN equivalent to N^{-1/4} if S_{n} is a simple random walk. We show that pN equivalent 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 pN ≤ cN^{-1/4} for integer-valued walks and upper exponential walks, which are the walks such that Law(S_{1}|S _{1}>0) is an exponential distribution.

Original language | English (US) |
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Pages (from-to) | 1178-1193 |

Number of pages | 16 |

Journal | Stochastic Processes and their Applications |

Volume | 120 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2010 |

Externally published | Yes |

## Keywords

- Area of excursion
- Area of random walk
- Excursion
- Integrated random walk
- One-sided exit probability
- Unilateral small deviations

## ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics