# First-passage time asymptotics over moving boundaries for random walk bridges

F. Sloothaak, B. Zwart, V. Wachtel

### Abstract

We study the asymptotic tail probability of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -1/2, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge. In the latter case, a phase transition appears in the asymptotics, of which the precise nature depends on the order of distance between zero and the moving boundary.
Original language English 1708.02408 22 arXiv 1708.02408 Published - 9 Aug 2017

### Fingerprint

First Passage Time
Moving Boundary
Random walk
Regularly Varying Function
Slowly Varying Function
Tail Behavior
Tail Probability
Zero
Increment
Tail
Phase Transition
Asymptotic Behavior
Exponent

### Cite this

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title = "First-passage time asymptotics over moving boundaries for random walk bridges",
abstract = "We study the asymptotic tail probability of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -1/2, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge. In the latter case, a phase transition appears in the asymptotics, of which the precise nature depends on the order of distance between zero and the moving boundary.",
author = "F. Sloothaak and B. Zwart and V. Wachtel",
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language = "English",
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number = "1708.02408",

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In: arXiv, No. 1708.02408, 1708.02408, 09.08.2017.

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AU - Zwart, B.

AU - Wachtel, V.

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N2 - We study the asymptotic tail probability of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -1/2, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge. In the latter case, a phase transition appears in the asymptotics, of which the precise nature depends on the order of distance between zero and the moving boundary.

AB - We study the asymptotic tail probability of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -1/2, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge. In the latter case, a phase transition appears in the asymptotics, of which the precise nature depends on the order of distance between zero and the moving boundary.

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