Bounds for the distribution of some functionals of processes with subGaussian 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), 181197
MSC (2000):
Primary 60G07; Secondary 60K25
Published electronically:
January 14, 2013
MathSciNet review:
2933713
Fulltext PDF
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Abstract: Bounds for the distribution of some functionals of a stochastic process belonging to the class are obtained. An example of the functionals studied in the paper is given by where is a continuous function that can be viewed as a service output rate of a queue formed by the process . For the latter interpretation, the bounds can be viewed as upper estimates for the buffer overflow probabilities with buffer size . 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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 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), 183196.
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 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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 Yu. V. Kozachenko, T. Sottinen, and O. I. Vasylyk, Selfsimilar processes with stationary increments in the spaces , Teor. Imovir. Mat. Stat. 65 (2001), 6778; English transl. in Theory Probab. Math. Statist. 65 (2002), 7788. MR 1936131 (2004a:60078)
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 Yu. Kozachenko, O. Vasylyk, and R. Yamnenko, Upper estimate of overrunning by random process the level specified by continuous function, Random Oper. Stoch. Equ. 13 (2005), no. 2, 111128. MR 2152102 (2006b:60207)
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 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, 953962.
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 R. Yamnenko, Ruin probability for generalized subGaussian fractional Brownian motion, Theory Stoch. Process. 12(28) (2006), no. 12, 261275. MR 2316577 (2008g:60109)
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 R. Yamnenko and O. Vasylyk, Random process from the class : exceeding a curve, Theory Stoch. Process. 13(29) (2007), no. 4, 219232. MR 2482262 (2010a:60125)
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 Yu. Kozachenko, O. I. Vasylyk, and R. E. Yamnenko, subGaussian Random Processes, ``Kyiv University'', Kyiv, 2008. (Ukrainian)
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 R. E. Yamnenko and O. S. Shramko, On the distribution of storage processes from the class , Teor. Imovir. Mat. Stat. 83 (2010), 163176; English transl. in Theory Probab. Math. Statist. 83 (2011), 191206. 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:
http://dx.doi.org/10.1090/S009490002013008849
Keywords:
Generalized fractional Brownian motion,
metric entropy,
queue,
bounds for the distribution,
subGaussian process
Received by editor(s):
June 11, 2011
Published electronically:
January 14, 2013
Article copyright:
© Copyright 2013
American Mathematical Society
