On tail probability of local times of Gaussian processes
type: Item
instance of: scholarly article
title: On tail probability of local times of Gaussian processes
zbMATH Open document ID: 0997.60037
DOI: 10.1016/S0304-4149(99)00011-3
author: Toshiko Ogawa
author: Yuji Kasahara
author: Norio Kôno
published in: Stochastic Processes and their Applications
publication date: 2002-08-29T00:00:00Z
review text: Let \(X = \{X_t, t\geq 0\}\) be a Gaussian process with stationary increments, zero mean and such that \(X_0 = 0.\) Let \( \sigma ^2 (h) = E(X_{t+h} - X_t)^2 \) be the covariance function of the increments and \(l(t,x)\) be the local time of \(X\) defined by \[ l(t,x) = \lim _{\varepsilon \downarrow 0} \frac {1}{2\varepsilon } \int _0^t I_{[x-\varepsilon , x+\varepsilon ]} X_s ds . \] Asymptotic properties of \(l(1,0)\) are studied in connection with the behaviour of \(\sigma ^2.\) Especially, it is proved that under some conditions on \(\sigma ^2,\) the exponential rate of the tail probability \(P[l(0,1)> x] \) is asymptotically \(1/ {\sigma ^{-1}(1/x)}\) as \(x\to \infty ,\) where \(\sigma ^{-1}\) is the inverse function of \(\sigma .\) Results for Brownian and fractional Brownian processes follow then as corollaries.
reviewed by: Zuzana Prášková
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 1792509
zbMATH Keywords: Gaussian processes
zbMATH Keywords: tail probability
zbMATH Keywords: local time
zbMATH Keywords: Tauberian theorems
MaRDI profile type: Publication
cites work: Local Times and Sample Function Properties of Stationary Gaussian Processes
cites work: Q4726487
cites work: On moduli of continuity for local times of Gaussian processes
cites work: Q5624436
cites work: Tauberian theorems of exponential type
cites work: Local times and related sample path properties of certain self-similar processes
cites work: Local nondeterminism and local times for stable processes
cites work: Regularly varying functions
full work available at URL: s0304-4149(99)00011-3
OpenAlex ID: W2012240275
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description: scientific article; zbMATH DE number 1792509