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

Abstract | References | Similar Articles | Additional Information

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.

- R. Addie, P. Mannersalo, and I. Norros,
*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*, Translations of Mathematical Monographs, vol. 188, American Mathematical Society, Providence, RI, 2000. Translated from the 1998 Russian original by V. Zaiats. MR**1743716** - Yu. Kozachenko, T. Sottīnen, and O. Vasilik,
*Self-similar processes with stationary increments in the spaces ${\rm SSub}_\phi (\Omega )$*, Teor. Ĭmovīr. Mat. Stat.**65**(2001), 67–78 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**65**(2002), 77–88. MR**1936131** - Olga Vasylyk, Yuriy Kozachenko, and Rostyslav Yamnenko,
*Upper estimate of overrunning by ${\rm Sub}_\phi (\Omega )$ random process the level specified by continuous function*, Random Oper. Stochastic Equations**13**(2005), no. 2, 111–128. MR**2152102**, DOI https://doi.org/10.1163/156939705323383832 - I. Norros,
*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. - Rostyslav Yamnenko,
*Ruin probability for generalized $\phi $-sub-Gaussian fractional Brownian motion*, Theory Stoch. Process.**12**(2006), no. 3-4, 261–275. MR**2316577** - Rostyslav Yamnenko and Olga Vasylyk,
*Random process from the class $V(\phi ,\psi )$: exceeding a curve*, Theory Stoch. Process.**13**(2007), no. 4, 219–232. MR**2482262** - Yu. Kozachenko, O. I. Vasylyk, and R. E. Yamnenko,
*$\varphi$-sub-Gaussian Random Processes*, “Kyiv University”, Kyiv, 2008. (Ukrainian) - R. Ē. Yamnenko and O. S. Shramko,
*On the distribution of storage processes from the class $V(\phi ,\psi )$*, Teor. Ĭmovīr. Mat. Stat.**83**(2010), 163–176 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist.**83**(2011), 191–206. MR**2768858**, DOI https://doi.org/10.1090/S0094-9000-2012-00851-X

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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