Averaged deviations of Orlicz processes and majorizing measures
Volume 3, Issue 3 (2016), pp. 249–268
Pub. online: 11 November 2016
Type: Research Article
Open Access
Open Access
Received
7 September 2016
7 September 2016
Revised
28 October 2016
28 October 2016
Accepted
28 October 2016
28 October 2016
Published
11 November 2016
11 November 2016
Abstract
This paper is devoted to investigation of supremum of averaged deviations $|X(t)-f(t)-\int _{\mathbb{T}}(X(u)-f(u))\hspace{0.1667em}\mathrm{d}\mu (u)/\mu (\mathbb{T})|$ of a stochastic process from Orlicz space of random variables using the method of majorizing measures. An estimate of distribution of supremum of deviations $|X(t)-f(t)|$ is derived. A special case of the $L_{q}$ space is considered. As an example, the obtained results are applied to stochastic processes from the $L_{2}$ space with known covariance functions.
References
Buldygin, V.V., Kozachenko, Y.V.: Metric Characterization of Random Variables and Random Processes. AMS, Providence, RI (2000). MR1743716
Dudley, R.M.: Sample functions of the Gaussian process. Ann. Probab. 1, 66–103 (1973). MR0346884
Fernique, X.: Régularité de processus gaussiens. Invent. Math. 12, 304–320 (1971). MR0286166
Fernique, X.: Régularité des trajectoires des fonctions aléatoires gaussiennes. In: Ecole d’Eté de Probabilités de Saint-Flour. IV, 1974. Lect. Notes Math. vol. 480, pp. 1–96 (1971). MR0413238
Kôno, N.: Sample path properties of stochastic processes. J. Math. Kyoto Univ. 20, 295–313 (1980). MR0582169
Kozachenko, Y.: Random processes in Orlicz spaces. I. Theory Probab. Math. Stat. 30, 103–117 (1985). MR0800835
Kozachenko, Y., Moklyachuk, O.: Large deviation probabilities in terms of majorizing measures. Random Oper. Stoch. Equ. 11(1), 1–20 (2003). MR1969189. doi:10.1163/156939703322003953
Kozachenko, Y., Ryazantseva, V.: Conditions for boundedness and continuity in terms of majorizing measures of random processes in certain Orlicz space. Theory Probab. Math. Stat. 44, 77–83 (1992). MR1168430
Kozachenko, Y., Sergiienko, M.: The criterion of hypothesis testing on the covariance function of a Gaussian stochastic process. Monte Carlo Methods Appl. 20(1), 137–145 (2014). MR3213591. doi:10.1515/mcma-2013-0023
Kozachenko, Y., Vasylyk, O., Yamnenko, R.: Upper estimate of overrunning by $\operatorname{Sub}_{\varphi }(\varOmega )$ random process the level specified by continuous function. Random Oper. Stoch. Equ. 13(2), 111–128 (2005). MR2152102. doi:10.1163/156939705323383832
Krasnoselskii, M.A., Rutitskii, Y.B.: Convex Functions and Orlicz Spaces. P. Noordhoff Ltd., Groningen (1961). MR0126722
Ledoux, M.: Isoperimetry and Gaussian analysis. In: Lectures on probability theory and statistics, Ecole d’Eté de Probabilités de Saint-Flour, Lect. Notes Math., vol. 1648, pp. 165–294 (1996). MR1600888. doi:10.1007/BFb0095676
Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer, Berlin, New York (1991). MR1102015. doi:10.1007/978-3-642-20212-4
Nanopoulos, C., Nobelis, P.: Régularité et propriétés limites des fonctions aléatoires. Lect. Notes Math. 649, 567–690 (1978). MR0520031
Talagrand, M.: Regularity of Gaussian processes. Acta Math. 159, 99–149 (1987). MR0906527. doi:10.1007/BF02392556
Talagrand, M.: Majorizing measures: The generic chaining. Ann. Probab. 24, 1049–1103 (1996). MR1411488. doi:10.1214/aop/1065725175
Yamnenko, R.: An estimate of the probability that the queue length exceeds the maximum for a queue that is a generalized Ornstein–Uhlenbeck stochastic process. Theory Probab. Math. Stat. 73, 181–194 (2006). MR2213851. doi:10.1090/S0094-9000-07-00691-6
Yamnenko, R.: A bound for norms in $L_{p}(T)$ of deviations of φ-sub-Gaussian stochastic processes. Lith. Math. J. 55(2), 291–300 (2015). MR3347597. doi:10.1007/s10986-015-9281-0
