Statistics of the occupation time for a random walk in the presence of a moving boundary

Considering a simple symmetric random walk in dimension d ≧ 3, we study the almost sure joint asymptotic behavior of two objects: first the local times of a pair of neighboring points, then the local time of a point and the occupation time of the surface of the unit ball around it.

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Oscillating random walk with a moving boundary

Israel Journal of Mathematics ◽

10.1007/bf02937518 ◽

1994 ◽

Vol 88 (1-3) ◽

pp. 333-365

Author(s):

Neal Madras ◽

David Tanny

Keyword(s):

Random Walk ◽

Moving Boundary ◽

Oscillating Random Walk

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Small drift limit theorems for random walks

Journal of Applied Probability ◽

10.1017/jpr.2016.95 ◽

2017 ◽

Vol 54 (1) ◽

pp. 199-212

Author(s):

Ernst Schulte-Geers ◽

Wolfgang Stadje

Keyword(s):

Brownian Motion ◽

Random Walk ◽

Random Walks ◽

Limit Theorems ◽

Occupation Time ◽

Functional Limit ◽

Brownian Motion With Drift ◽

Alternative Approach ◽

New Form ◽

Drift Limit

AbstractWe show analogs of the classical arcsine theorem for the occupation time of a random walk in (−∞,0) in the case of a small positive drift. To study the asymptotic behavior of the total time spent in (−∞,0) we consider parametrized classes of random walks, where the convergence of the parameter to 0 implies the convergence of the drift to 0. We begin with shift families, generated by a centered random walk by adding to each step a shift constant a>0 and then letting a tend to 0. Then we study families of associated distributions. In all cases we arrive at the same limiting distribution, which is the distribution of the time spent below 0 of a standard Brownian motion with drift 1. For shift families this is explained by a functional limit theorem. Using fluctuation-theoretic formulae we derive the generating function of the occupation time in closed form, which provides an alternative approach. We also present a new form of the first arcsine law for the Brownian motion with drift.

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

Journal of Applied Probability ◽

10.1017/jpr.2018.39 ◽

2018 ◽

Vol 55 (2) ◽

pp. 627-651 ◽

Cited By ~ 1

Author(s):

Fiona Sloothaak ◽

Vitali Wachtel ◽

Bert Zwart

Keyword(s):

Random Walk ◽

Moving Boundary ◽

First Passage Time ◽

Passage Time ◽

Finite Variance ◽

Tail Behavior ◽

First Passage ◽

The Impact ◽

Return Point ◽

Asymptotic Tail

Abstract We study the asymptotic tail behavior 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 -½, 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, leading to a possible phase transition depending on the order of the distance between zero and the moving boundary.

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A random walk on Z with drift driven by its occupation time at zero

Stochastic Processes and their Applications ◽

10.1016/j.spa.2009.03.002 ◽

2009 ◽

Vol 119 (8) ◽

pp. 2682-2710

Author(s):

Iddo Ben-Ari ◽

Mathieu Merle ◽

Alexander Roitershtein

Keyword(s):

Random Walk ◽

Occupation Time

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Wald's identity and random walk models for neuron firing

Advances in Applied Probability ◽

10.1017/s0001867800042130 ◽

1976 ◽

Vol 8 (02) ◽

pp. 257-277 ◽

Cited By ~ 3

Author(s):

V. I. Kryukov

Keyword(s):

Random Walk ◽

Sequential Analysis ◽

Moving Boundary ◽

First Passage Time ◽

Model Parameters ◽

First Passage ◽

Fundamental Identity ◽

Main Technique ◽

Special Cases ◽

Wide Range

A class of probability models for the firing of neurons is introduced and treated analytically. The cell membrane potential is assumed to be a one-dimensional random walk on the continuum; the first passage of the moving boundary triggers a nerve spike, an all-or-none event. The interspike interval distribution of first-passage time is shown to satisfy a Volterra integral equation suitable for numerical evaluation. Explicit solutions as well as their Laplace (Mellin) transforms are obtained for some special cases. The main technique used in this paper is the extension of Wald's fundamental identity of sequential analysis to a wide range of additive stochastic processes with an (eventually) linear boundary function. This identity is also useful in evaluating model parameters in terms of observed firing times, as well as providing a unified exposition of many earlier results.

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Wald's identity and random walk models for neuron firing

Advances in Applied Probability ◽

10.2307/1425904 ◽

1976 ◽

Vol 8 (2) ◽

pp. 257-277 ◽

Cited By ~ 11

Author(s):

V. I. Kryukov

Keyword(s):

Random Walk ◽

Sequential Analysis ◽

Moving Boundary ◽

First Passage Time ◽

Model Parameters ◽

First Passage ◽

Fundamental Identity ◽

Main Technique ◽

Special Cases ◽

Wide Range

A class of probability models for the firing of neurons is introduced and treated analytically. The cell membrane potential is assumed to be a one-dimensional random walk on the continuum; the first passage of the moving boundary triggers a nerve spike, an all-or-none event. The interspike interval distribution of first-passage time is shown to satisfy a Volterra integral equation suitable for numerical evaluation. Explicit solutions as well as their Laplace (Mellin) transforms are obtained for some special cases. The main technique used in this paper is the extension of Wald's fundamental identity of sequential analysis to a wide range of additive stochastic processes with an (eventually) linear boundary function. This identity is also useful in evaluating model parameters in terms of observed firing times, as well as providing a unified exposition of many earlier results.

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