Trimmed Lévy processes and their extremal components

Yuguang Ipsen, Ross Maller and Sidney Resnick

Stochastic Processes and their Applications, 2020, vol. 130, issue 4, 2228-2249

Abstract: We analyze a stochastic process of the form (r)Xt=Xt−∑i=1rΔt(i), where (Xt)t≥0 is a driftless, infinite activity, subordinator on R+ with its jumps on [0,t] ordered as Δt(1)≥Δt(2)⋯. The r largest of these are “trimmed” from Xt to give (r)Xt. When r→∞, both (r)Xt↓0 and Δt(r)↓0 a.s. for each t>0, and it is interesting to study the weak limiting behavior of ((r)Xt,Δt(r)) in this case. We term this “large-trimming” behavior, and study the joint convergence of ((r)Xt,Δt(r)) as r→∞ under linear normalization, assuming extreme value-related conditions on the Lévy measure of Xt which guarantee that Δt(r) has a limit distribution with linear normalization. Allowing (r)Xt to have random centering and norming in a natural way, we first show that ((r)Xt,Δt(r)) has a bivariate normal limiting distribution as r→∞; then replacing the random normalizations with deterministic normings produces normal, and in some cases, non-normal, limits whose parameters we can specify.

Keywords: Trimmed Lévy process; Trimmed subordinator; Subordinator large jumps; Extreme value-related conditions; Large-trimming limits (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:130:y:2020:i:4:p:2228-2249

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DOI: 10.1016/j.spa.2019.06.018

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