Passage times of random walks and Lévy processes across power law boundaries

We establish an integral test involving only the distribution of the increments of a random walk S which determines whether [?] is almost surely zero, finite or infinite when 1/2 < x < 1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Maller [9] co...

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Bibliographic Details
Published in:Probability theory and related fields 2005-09, Vol.133 (1), p.57-70
Main Authors: DONEY, R. A, MAILER, R. A
Format: Article
Language:English
Subjects:
Boundaries
Exact sciences and technology
Laplace transforms
Limit theorems
Markov processes
Mathematics
Power law
Probability
Probability and statistics
Probability theory and stochastic processes
Random variables
Random walk
Random walk theory
Sciences and techniques of general use
Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications)
Stochastic processes
Studies
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Summary:We establish an integral test involving only the distribution of the increments of a random walk S which determines whether [?] is almost surely zero, finite or infinite when 1/2 < x < 1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Maller [9] concerning finiteness of one-sided passage times over power law boundaries, so that we now have quite explicit criteria for all values of x >= 0. The results, and those of [9], are also extended to Levy processes.
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-004-0414-3