Strictly $\varphi$-sub-Gaussian quasi shot noise processes
Abstract
In the paper, strictly $\varphi$-sub-Gaussian quasi shot noise processes are considered. There are obtained estimates for distribution of supremum of such a process defined on a compact set and formulated conditions for its sample functions continuity with probability one.References
R.J. Adler, An introduction to continuity, extrema and related topics for general Gaussian processes, Lecture Notes-Monograph Series. Institute of Mathematical Statistics, Hayward, Vol. 12, 1990.
V.V. Buldygin, Sub-Gaussian processes and convergencs of random series in functional spaces, Ukrainian Mathematical Journal, Vol. 29, No. 4, pp. 443–454, 1977.
V. V. Buldygin, and Yu. V. Kozachenko, Metric Characterization of Random Variables and Random Processes, American Mathematical Society, Providence, RI, 2000.
A.V. Bulinski, and A.N. Shiryaev, Theory of Stochastic Processes, Moscow: “Fizmatlit”, 408 p., 2005. (In Russian)
H. Cramer, and M.R. Leadbetter, Stationary and related stochastic processes. Sample function properties and their applications, Wiley, New York–London–Sydney, 348 p., 1967.
I.V. Dariychuk, Yu.V. Kozachenko, and M.M. Perestyuk, Stochastic processes from Orlicz spaces, Chernivtsi: “Zoloti lytavry”, 212p., 2011. (In Ukrainian)
I.V. Dariychuk, and Yu.V. Kozachenko, Some properties of pre-Gaussian shot noise processes, Stochastic Analysis and Random Dynamics. International Conference. Abstracts, Lviv, Ukraine, pp. 57–59, 2009.
I.V. Dariychuk, Yu.V. Kozachenko, The distribution of the supremum of Θ-pre-Gaussian shot noise processes, Theory of Probability and Mathematical Statistics), No. 80, pp. 85–100, 2010.
R. Fukuda, Exponential integrability of sub-Gaussian vectors, Probab. Theory Related Fields, Vol. 85, No. 4, pp. 505–521, 1990.
I.I. Gikhman, and A.V. Skorokhod, Theory of random processes, Vol.1. Moscow: “Nauka”, 1971; English trans., Springer–Verlag, Berlin–Heidelberg–New York, 1974.
R. Giuliano Antonini, Yu. V. Kozachenko, T. Nikitina, Space of φ-sub-Gaussian random variables, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 27, pp. 92–124, 2003.
J.P. Kahane, Propriet ´ es locales des fonctions ´ a series de Fouries al ` eatories ´ , Studia Math., Vol. 19, No. 1, pp. 1–25, 1960.
Yu. Kozachenko, Sufficient conditions for continuity with probability one of sub-Gaussian stochastic processes, Dopovidi AN URSR, No. 2, pp. 113–115, 1968.
Yu. Kozachenko, Random processes in Orlicz spaces. I. Probability Theory and Mathematical Statistics, No. 30, pp.103–117, 1984 Random processes in Orlicz spaces. II. Probability Theory and Mathematical Statistics, No. 31, pp.51–58, 1984.
Yu. Kozachenko, E. Ostrovsky, Banach spaces of random variables of sub-Gaussian type, Theory of Probability and Mathematical Statistics, No. 32, pp.42–53, 1985.
Yu. Kozachenko, M. Perestyuk, O. Vasylyk, On Uniform Convergence of Wavelet Expansions of φ-sub-Gaussian Random Process, Random Operators and Stochastic Equations, Vol.14, No.3, pp. 209–232, 2006.
Yu. Kozachenko, I. Rozora, Accuracy and Reliability of models of stochastic processes of the space Subφ(Ω), Theory of Probability and Mathematical Statistics, No. 71, pp. 105–117, 2005.
Yu. Kozachenko, T. Sottinen, O. Vasilik, Self-similar processes with stationary increments in the spaces SSubφ(Ω), Theory of Probability and Mathematical Statistics, No. 65, pp.77-88, 2002.
Yu. Kozachenko, T. Sottinen, O. Vasylyk, Path space large deviations of a large buffer with Gaussian input traffic, Queueing Systems, No. 42, pp.113-129, 2002.
Yu. Kozachenko, T. Sottinen, O. Vasylyk, Simulation of Weakly Self-Similar Stationary Increment Subφ(Ω)-Processes: A Series Expansion Approach, Methodology and Computing in Applied Probability, Vol. 7, No. 3, pp. 379–400, 2005.
Yu. Kozachenko, O. Vasilik, On the distribution of suprema of Subφ(Ω) random processes, Theory of Stochastic Processes, Vol. 4(20), issue 1–2, pp. 147–160, 1998.
Yu. Kozachenko, O. Vasilik, Stochastic processes of the classes V (φ, ψ), Theory of Probability and Mathematical Statistics, No. 63, pp.109-121, 2001.
Yu. Kozachenko, O. Vasylyk, Sample pathes continuity and estimates of distributions of the increments of separable stochastic processes from the class V (φ, ψ), defined on a compact set, Bulletin of the University of Kiev, Series: Physics and Mathematics, Issue 2, pp.45–50, 2004. (In Ukrainian)
Yu. Kozachenko, R. Yamnenko, O. Vasylyk, Upper estimate of overrunning by Subφ(Ω) random process the level specified by continuous function, Random Oper. Stoch. Equ., Vol. 13, No. 2, pp. 111–128, 2005.
Yu. Kozachenko, R. Yamnenko, O. Vasylyk, φ-sub-Gaussian random process, Kyiv: Vydavnycho-Poligrafichnyi Tsentr, Kyivskyi Universytet (ISBN 978-966-439-051-1), 231 p., 2008. (In Ukrainian).
M.A. Krasnosel’skii, Ya.B. Rutickii, Convex Functions and Orlicz Spaces. Moscow, 1958 (in Russian). English translation: P.Noordhoff Ltd, Groningen, 1961, 249p.
H.J. Landau, L.A. Shepp, On the supremum of Gaussian processes, Sankhya: The Indian Journal of Statistics, Series A, Vol. 32, No. 4, pp. 369–378, 1970.
M.R. Leadbetter, G. Lindgren, H. Rootzen, Extremes and related properties of random sequences and processes, Springer, Berlin, 1983.
M. Ledoux, M. Talagrand, Probability in Banach space, Springer-Verlag, Berlin-New York, 480 p., 1991.
V.A. Liphshits, Gaussian random functions, Kiev: TViMS, 246 p., 1995.
E. I. Ostrovsky, Exponential Bounds for the Distribution of the Maximum of a Non-Gaussian Random Field, Theory of probability and its applications, Vol. 35, No. 3, pp. 487-499, 1991.
A.V. Skorokhod, A Note on Gaussian Measures in a Banach Space, Theory of probability and its applications, Vol. 15, No. 3, pp. 519–520, 1970.
V. Yurinsky, Sums and Gaussian vectors, Lecture Notes in Mathematics, Springer Verlag, Berlin, Heidelberg, Vol. 1617, 1995.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
