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

Maller, Ross; Schindler, Tanja ✉

Angol nyelvű Tudományos Szakcikk (Folyóiratcikk)
Megjelent: STOCHASTIC PROCESSES AND THEIR APPLICATIONS 0304-4149 129 (10) pp. 4144-4162 2019
  • Gazdaságtudományi Doktori Minősítő Bizottság: B
  • SJR Scopus - Modeling and Simulation: D1
    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.
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    2020-09-19 10:36