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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

   

 

Bounds for the distribution of some functionals of processes with $ \varphi$-sub-Gaussian increments


Author: R. E. Yamnenko
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 85 (2011).
Journal: Theor. Probability and Math. Statist. 85 (2012), 181-197
MSC (2000): Primary 60G07; Secondary 60K25
Published electronically: January 14, 2013
MathSciNet review: 2933713
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Abstract: Bounds for the distribution of some functionals of a stochastic process $ \{X(t),t\in T\}$ belonging to the class $ V(\varphi ,\psi )$ are obtained. An example of the functionals studied in the paper is given by

$\displaystyle \mathsf {F}\left \{\sup _{s\le t; s,t \in B}(X(t)-X(s)-(f(t)-f(s)))>x\right \}, $

where $ f(t)$ is a continuous function that can be viewed as a service output rate of a queue formed by the process $ X(t)$. For the latter interpretation, the bounds can be viewed as upper estimates for the buffer overflow probabilities with buffer size $ x>0$. The results obtained in the paper apply to Gaussian stochastic processes. As an example, we show an application for the generalized fractional Brownian motion defined on a finite interval.

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Additional Information

R. E. Yamnenko
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 2, Kiev 03127, Ukraine
Email: yamnenko@univ.kiev.ua

DOI: http://dx.doi.org/10.1090/S0094-9000-2013-00884-9
Keywords: Generalized fractional Brownian motion, metric entropy, queue, bounds for the distribution, sub-Gaussian process
Received by editor(s): June 11, 2011
Published electronically: January 14, 2013
Article copyright: © Copyright 2013 American Mathematical Society