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Trimmed Lévy processes and their extremal components

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  • Ipsen, Yuguang
  • Maller, Ross
  • Resnick, Sidney

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.

Suggested Citation

  • Ipsen, Yuguang & Maller, Ross & Resnick, Sidney, 2020. "Trimmed Lévy processes and their extremal components," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2228-2249.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:4:p:2228-2249
    DOI: 10.1016/j.spa.2019.06.018
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    References listed on IDEAS

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    1. De Haan, Laurens, 1974. "Equivalence classes of regularly varying functions," Stochastic Processes and their Applications, Elsevier, vol. 2(3), pages 243-259, July.
    2. Ipsen, Yuguang & Maller, Ross & Resnick, Sidney, 2019. "Ratios of ordered points of point processes with regularly varying intensity measures," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 205-222.
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    Cited by:

    1. David M. Mason, 2021. "Self-Standardized Central Limit Theorems for Trimmed Lévy Processes," Journal of Theoretical Probability, Springer, vol. 34(4), pages 2117-2144, December.

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