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Stability of overshoots of zero mean random walks
journal contribution
posted on 2023-06-07, 07:13 authored by Aleksandar Mijatovic, Vladislav VysotskiyVladislav VysotskiyWe prove that for a random walk on the real line whose increments have zero mean and are either integer-valued or spread out (i.e. the distributions of steps of the walk are eventually non-singular), the Markov chain of overshoots above a fixed level converges in total variation to its stationary distribution. We find the explicit form of this distribution heuristically and then prove its invariance using a time-reversal argument. If, in addition, the increments of the walk are in the domain of attraction of a non-one-sided a-stable law with index a?(1,2) (resp. have finite variance), we establish geometric (resp. uniform) ergodicity for the Markov chain of overshoots. All the convergence results above are also valid for the Markov chain obtained by sampling the walk at the entrance times into an interval.
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Publication status
- Published
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- Published version
Journal
Electronic Journal of ProbabilityISSN
1083-6489Publisher
Institute of Mathematical StatisticsExternal DOI
Issue
a63Volume
25Page range
1-22Pages
22.0Department affiliated with
- Mathematics Publications
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- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2020-06-10First Open Access (FOA) Date
2020-06-10First Compliant Deposit (FCD) Date
2020-06-09Usage metrics
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