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Stability of overshoots of zero mean random walks

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posted on 2023-06-07, 07:13 authored by Aleksandar Mijatovic, Vladislav VysotskiyVladislav Vysotskiy
We 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.

History

Publication status

  • Published

File Version

  • Published version

Journal

Electronic Journal of Probability

ISSN

1083-6489

Publisher

Institute of Mathematical Statistics

Issue

a63

Volume

25

Page range

1-22

Pages

22.0

Department affiliated with

  • Mathematics Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2020-06-10

First Open Access (FOA) Date

2020-06-10

First Compliant Deposit (FCD) Date

2020-06-09

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