Trimmed Levy processes and their extremal components

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:130

Article

Ipsen, Yuguang¹  Maller, Ross¹  Resnick, Sidney² 

[1] Australian Natl Univ, Res Sch Finance Actuarial Studies & Stat, Canberra, ACT, Australia

[2] Cornell Univ, Sch Operat Res & Informat Engn, Ithaca, NY 14853 USA

关键词: Trimmed Levy process;    Trimmed subordinator;    Subordinator large jumps;    Extreme value-related conditions;    Large-trimming limits;   

DOI  :  10.1016/j.spa.2019.06.018

来源: Elsevier

【 摘 要 】

We analyze a stochastic process of the form X-(r)(t) = X-t - Sigma(r)(i) =1 Delta((i))(t), where (X-t)(t >= 0) is a driftless, infinite activity, subordinator on R+ with its jumps on [0, t] ordered as Delta((l))(t) >= Delta(()(2))(t).... The r largest of these are trimmed from X-t to give (r) X-t. When r -> infinity, both (r) X-t down arrow 0 and Delta((r))(t) down arrow 0 a.s. for each t > 0, and it is interesting to study the weak limiting behavior of (((r)) X-t, Delta((r))(t)) in this case. We term this large-trimming behavior, and study the joint convergence of (((r)) X-t, Delta((r))(t)) as r -> infinity under linear normalization, assuming extreme value-related conditions on the Levy measure of X-t which guarantee Delta((r))(t) that A, has a limit distribution with linear normalization. Allowing X-(r)(t) to have random centering and forming in a natural way, we first show that ((r)(X)(t), Delta((r))(l)) has a bivariate normal limiting distribution as r -> infinity; then replacing the random normalizations with deterministic normings produces normal, and in some cases, non-normal, limits whose parameters we can specify. (C) 2019 Elsevier B.V. All rights reserved.

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