RECURRENCE RATES AND HITTING-TIME DISTRIBUTIONS FOR RANDOM WALKS ON THE LINE

We consider random walks on the line given by a sequence of independent identically distributed jumps belonging to the strict domain of attraction of a stable distribution, and first determine the almost sure exponential divergence rate, as ε → 0, of the return time to (-ε, ε). We then refine this r...

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Bibliographic Details
Published in:The Annals of probability 2013-03, Vol.41 (2), p.619-635
Main Authors: Pène, Françoise, Saussol, Benoît, Zweimüller, Roland
Format: Article
Language:English
Subjects:
60E07
60F05
60G50
Conditional convergence
hitting time
Integers
Laplace transformation
Mathematical theorems
Mathematics
Perceptron convergence procedure
Probability
Probability distribution
quantitative recurrence
Random variables
Random walk
Random walk theory
recurrence
stable distribution
Studies
Theorems
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Summary:We consider random walks on the line given by a sequence of independent identically distributed jumps belonging to the strict domain of attraction of a stable distribution, and first determine the almost sure exponential divergence rate, as ε → 0, of the return time to (-ε, ε). We then refine this result by establishing a limit theorem for the hitting-time distributions of (x - ε, x + ε) with arbitrary x ∈ ℝ.
ISSN:0091-1798
2168-894X
DOI:10.1214/11-AOP698