Small time convergence of subordinators with regularly or slowly varying canonical measure

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:129

Article

Maller, Ross¹  Schindler, Tanja¹ 

[1] Australian Natl Univ, Res Sch Finance Actuarial Studies & Stat, GPO Box 4, Canberra, ACT 2601, Australia

关键词: Trimmed subordinator;    Levy process;    Maximal jump process;    Functional convergence;    Regular variation;    Extremal process;    Cauchy process;   

DOI  :  10.1016/j.spa.2018.11.016

来源: Elsevier

【 摘 要 】

We consider subordinators X-alpha = (X-alpha(t))(t >= 0) in the domain of attraction at 0 of a stable subordinator (S-alpha(t))(t >= 0) (where alpha is an element of (0, 1)); thus, with the property that (Pi) over bar (alpha), the tail function of the canonical measure of X-alpha, is regularly varying of index -alpha is an element of (-1, 0) as x down arrow 0. We also analyse the boundary case, alpha = 0, when (Pi) over bar (alpha) is slowly varying at 0. When alpha is an element of (0, 1), we show that (t (Pi) over bar (alpha)(X-alpha(t)))(-1) converges in distribution, as t down arrow 0, to the random variable (S-alpha(1))(alpha). This latter random variable, as a function of alpha, converges in distribution as alpha down arrow 0 to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in D[0, 1]), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from the process. The alpha = 0 case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe. (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

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