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

Author:
R. E. Yamnenko

Translated by:
S. Kvasko

Journal:
Theor. Probability and Math. Statist. **85** (2012), 181-197

MSC (2000):
Primary 60G07; Secondary 60K25

DOI:
https://doi.org/10.1090/S0094-9000-2013-00884-9

Published electronically:
January 14, 2013

MathSciNet review:
2933713

Full-text PDF Free Access

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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 \[ \mathsf {F}\left \{\sup _{s\le t; s,t \in B}(X(t)-X(s)-(f(t)-f(s)))>x\right \}, \] 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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References
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*Most probable paths and performance formulae for buffers with Gaussian input traffic*, Eur. Trans. Telecommun. **13(3)** (2002), 183–196.
- V. V. Buldygin and Yu. V. Kozachenko,
*Metric Characterization of Random Variables and Random Processes*, “TBiMC”, Kiev, 1998; English transl., American Mathematical Society, Providence, RI, 2000. MR **1743716 (2001g:60089)**
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*On the use of fractional Brownian motions in the theory of connectionless networks*, IEEE Journal on Selected Areas in Communications **13** (1995), no. 6, 953–962.
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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

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