Averaged deviations of Orlicz processes and majorizing measures

Rostyslav Yamnenko

Abstract


This paper is devoted to investigation of supremum of averaged deviations $| X(t) - f(t) - \int_{\mathbb{T}} (X(u) - f(u)) \,\mathrm{d}\mu(u) / \mu(\mathbb{T})|$ 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.

Keywords


Orlicz space; Orlicz process; supremum distribution; method of majorizing measures; Ornstein–Uhlenbeck process

Full Text:

PDF

References


[1] Buldygin, V.V., Kozachenko, Y.V.: Metric Characterization of Random Variables and Random Processes. AMS, Providence, RI (2000). MR1743716

[2] Dudley, R.M.: Sample functions of the Gaussian process. Ann. Probab. 1, 66–103 (1973). MR0346884

[3] Fernique, X.: Régularité de processus gaussiens. Invent. Math. 12, 304–320 (1971). MR0286166

[4] 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

[5] Kôno, N.: Sample path properties of stochastic processes. J. Math. Kyoto Univ. 20, 295–313 (1980). MR0582169

[6] Kozachenko, Y.: Random processes in Orlicz spaces. I. Theory Probab. Math. Stat. 30, 103–117 (1985). MR0800835

[7] 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

[8] 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

[9] 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

[10] Kozachenko, Y., Vasylyk, O., Yamnenko, R.: Upper estimate of overrunning by Sub φ(Ω) random process the level specified by continuous function. Random Oper. Stoch. Equ. 13(2), 111–128 (2005). MR2152102. doi:10.1163/156939705323383832

[11] Krasnoselskii, M.A., Rutitskii, Y.B.: Convex Functions and Orlicz Spaces. P. Noordhoff Ltd., Groningen (1961). MR0126722

[12] 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

[13] Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer, Berlin, New York (1991). MR1102015. doi:10.1007/978-3-642-20212-4

[14] Nanopoulos, C., Nobelis, P.: Régularité et propriétés limites des fonctions aléatoires. Lect. Notes Math. 649, 567–690 (1978). MR0520031

[15] Talagrand, M.: Regularity of Gaussian processes. Acta Math. 159, 99–149 (1987). MR0906527. doi:10.1007/BF02392556

[16] Talagrand, M.: Majorizing measures: The generic chaining. Ann. Probab. 24, 1049–1103 (1996). MR1411488. doi:10.1214/aop/1065725175

[17] 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

[18] Yamnenko, R.: A bound for norms in Lp(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




DOI: http://dx.doi.org/10.15559/16-VMSTA64

Refbacks

  • There are currently no refbacks.


Copyright (c) 2016 Rostyslav Yamnenko

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.