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Functional limit theorems for random walks perturbed by positive alpha-stable jumps
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Iksanov, Alexander, Pilipenko, Andrey and Povar, Ben (2023) Functional limit theorems for random walks perturbed by positive alpha-stable jumps. Bernoulli, 29 (2). doi:10.3150/22-bej1515 ISSN 1350-7265.
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Official URL: https://doi.org/10.3150/22-bej1515
Abstract
Let ξ1, ξ2,… be i.i.d. random variables of zero mean and finite variance and η1, η2,… positive i.i.d. random variables whose distribution belongs to the domain of attraction of an α-stable distribution, α∈(0,1). The two collections are assumed independent. We consider a Markov chain with jumps of two types. If the present position of the Markov chain is positive, then the jump ξk occurs; if the present position of the Markov chain is nonpositive, then the jump ηk occurs. We prove functional limit theorems for this and two closely related Markov chains under Donsker’s scaling. The weak limit is a nonnegative process (X(t))t≥0 satisfying a stochastic equation dX(t)=dW(t)+dUα(L(0)X(t)), where W is a Brownian motion, Uα is an α-stable subordinator which is independent of W, and
L(0)X is a local time of X at 0. Also, we explain that X is a Feller Brownian motion with a ‘jump-type’ exit from 0.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||
SWORD Depositor: | Library Publications Router | ||||
Journal or Publication Title: | Bernoulli | ||||
Publisher: | Int Statistical Institute | ||||
ISSN: | 1350-7265 | ||||
Official Date: | 1 May 2023 | ||||
Dates: |
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Volume: | 29 | ||||
Number: | 2 | ||||
DOI: | 10.3150/22-bej1515 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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