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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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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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