A generalization result regarding the small and large scale behavior of infinitely divisible processes
type: Item
instance of: scholarly article
title: A generalization result regarding the small and large scale behavior of infinitely divisible processes
zbMATH Open document ID: 1269.60065
author: Jennifer L. Sinclair
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-12-04T00:00:00Z
review text: Rather general conditions for the convergence of normed time-scaled stochastic integrals of independently scattered L\(\acute{e}\)vy random measures to a limit are considered. The aim of the paper is to generalize previous results to the greater class of stochastically continuous infinitely divisible processes with no Gaussian part and expand the results to the multi-dimensional case. Let \(X_t\) be a stochastic process \[ X_t: = \int_S f_t(s) \left( N(ds) - \frac{\nu (ds)}{1 \vee \parallel f_t(s) \parallel} \right), \] where \({N}\) is a Poisson random measure on a measurable space (S, s) with intensity \(\nu\) and \(\{f_t(s)\}\) are measurable deterministic functions in \(\mathbb{R}^d\). In the paper, the convergence of properly normed and centered time-scaled process \(X_{\beta t}\) to a stable or Gaussian limit as \(\beta \rightarrow \beta_0\) is considered. In other words, the aim of the paper is to determine rather wide conditions on norming sequences \(n_\beta\) and centers \(c_{\beta, t}\) in order for the processes \(n_\beta X_{\beta t} - c_{\beta, t}\) to converge in the sense of finite-dimensional distributions (\(\overset{\mathrm{fdd}}{}\)) as the scaling factor \(\beta\) tends to some value \(\beta_0\). Both small (\(\beta_0 = 0\)) and large (\(\beta_0 = \infty\)) scaling factors are of interest. The main result is Theorem 3.1. Under the conditions that there exist one-to-one invertible measurable maps \(\phi_\beta\) from \(S\) into \(S\) with \(\phi^{-1}_\beta\) measurable and normalized sequences \(n_\beta\) and \(m_\beta\) satisfying \(\text{a.e.-}\nu\), for any \(t \in \mathbb{R}\) and for some functions \(k_t\), \(h_t\), \(g\) and \(l\) on the space \(S\), \[ m_\beta f_{\beta t} (\phi_\beta(s)) \rightarrow h_t(s); \quad \parallel m_\beta f_{\beta t} (\phi_\beta(s))\parallel \leq k_t(s); \] \[ \frac{d(\nu \circ \phi_\beta)}{d \nu} (s) \left( \frac{n_\beta}{m_\beta} \right)^2 \rightarrow g(s); \quad \frac{d(\nu \circ \phi_\beta)}{d \nu} (s) \left( \f…
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60J50
zbMATH DE Number: 6112310
zbMATH Keywords: infinitely divisible processes
zbMATH Keywords: Lévy processes
zbMATH Keywords: tempered stable processes
reviewed by: Nijolė Kalinauskaitė
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.08.039
OpenAlex ID: W2057992466
cites work: Small and large scale behavior of the Poissonized telecom process
cites work: On layered stable processes
cites work: Generalized tempered stable processes
cites work: Q3633248
cites work: Q3583833
cites work: Distributions of subadditive functionals of sample paths of infinitely divisible processes
cites work: Spectral representations of infinitely divisible processes
cites work: Tempering stable processes
cites work: Q2774021
DOI: 10.1016/J.JMAA.2012.08.039
description: scientific article; zbMATH DE number 6112310