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
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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.
 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), 183196.
 2.
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
(2001g:60089)
 3.
Yu.
Kozachenko, T.
Sottīnen, and O.
Vasilik, Selfsimilar processes with stationary increments in the
spaces 𝑆𝑆𝑢𝑏ᵩ(Ω), Teor.
Ĭmovīr. Mat. Stat. 65 (2001), 67–78
(Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math.
Statist. 65 (2002), 77–88. MR 1936131
(2004a:60078)
 4.
Olga
Vasylyk, Yuriy
Kozachenko, and Rostyslav
Yamnenko, Upper estimate of overrunning by
𝑆𝑢𝑏ᵩ(Ω) random process the level
specified by continuous function, Random Oper. Stochastic Equations
13 (2005), no. 2, 111–128. MR 2152102
(2006b:60207), http://dx.doi.org/10.1163/156939705323383832
 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, 953962.
 6.
Rostyslav
Yamnenko, Ruin probability for generalized 𝜑subGaussian
fractional Brownian motion, Theory Stoch. Process. 12
(2006), no. 34, 261–275. MR 2316577
(2008g:60109)
 7.
Rostyslav
Yamnenko and Olga
Vasylyk, Random process from the class
𝑉(𝜑,𝜓): exceeding a curve, Theory Stoch.
Process. 13 (2007), no. 4, 219–232. MR 2482262
(2010a:60125)
 8.
Yu. Kozachenko, O. I. Vasylyk, and R. E. Yamnenko, subGaussian Random Processes, ``Kyiv University'', Kyiv, 2008. (Ukrainian)
 9.
R.
Ē. Yamnenko and O.
S. Shramko, On the distribution of storage processes from the class
𝑉(𝜑,𝜓), 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
(2011j:60268)
 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), 183196.
 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, 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)
 4.
 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)
 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, 953962.
 6.
 R. Yamnenko, Ruin probability for generalized subGaussian fractional Brownian motion, Theory Stoch. Process. 12(28) (2006), no. 12, 261275. MR 2316577 (2008g:60109)
 7.
 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)
 8.
 Yu. Kozachenko, O. I. Vasylyk, and R. E. Yamnenko, subGaussian Random Processes, ``Kyiv University'', Kyiv, 2008. (Ukrainian)
 9.
 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
PII:
S 00949000(2013)008849
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
