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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
DOI: https://doi.org/10.1090/S0094-9000-2013-00884-9
Published electronically: January 14, 2013
MathSciNet review: 2933713
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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

$\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.

References [Enhancements On Off] (What's this?)

  • 1. 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.
  • 2. 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)
  • 3. Yu. V. Kozachenko, T. Sottinen, and O. I. Vasylyk, Self-similar processes with stationary increments in the spaces $ \mathrm {SSub}_\varphi (\Omega )$, Teor. Imovir. Mat. Stat. 65 (2001), 67-78; English transl. in Theory Probab. Math. Statist. 65 (2002), 77-88. MR 1936131 (2004a:60078)
  • 4. Yu. Kozachenko, O. Vasylyk, and R. Yamnenko, Upper estimate of overrunning by $ \mathrm {Sub}_\varphi (\Omega )$ random process the level specified by continuous function, Random Oper. Stoch. Equ. 13 (2005), no. 2, 111-128. MR 2152102 (2006b:60207)
  • 5. 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.
  • 6. R. Yamnenko, Ruin probability for generalized $ \varphi $-sub-Gaussian fractional Brownian motion, Theory Stoch. Process. 12(28) (2006), no. 1-2, 261-275. MR 2316577 (2008g:60109)
  • 7. R. Yamnenko and O. Vasylyk, Random process from the class $ V(\varphi ,\psi )$: exceeding a curve, Theory Stoch. Process. 13(29) (2007), no. 4, 219-232. MR 2482262 (2010a:60125)
  • 8. Yu. Kozachenko, O. I. Vasylyk, and R. E. Yamnenko, $ \varphi $-sub-Gaussian Random Processes, ``Kyiv University'', Kyiv, 2008. (Ukrainian)
  • 9. R. E. Yamnenko and O. S. Shramko, On the distribution of storage processes from the class $ V(\varphi ,\psi )$, Teor. Imovir. Mat. Stat. 83 (2010), 163-176; English transl. in Theory Probab. Math. Statist. 83 (2011), 191-206. MR 2768858 (2011j:60268)

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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: https://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

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