Analytical properties of sample paths of some stochastic processes
DOI:
https://doi.org/10.17721/1812-5409.2020/4.1Abstract
The study of the analytical properties of random processes and their functionals, without a doubt, was and remains the relevant topic of the theory of random processes. The first result from which the study of the local properties of random processes began is Kolmogorov’s theorem on sample continuity with probability one. The classic result for Gaussian random processes is Dudley’s theorem. This paper is devoted to the study of local properties of sample paths of random processes that can be represented as a sum of squares of Gaussian random processes. Such processes are called square-Gaussian. We investigate the sufficient conditions of sample continuity with probability 1 for square-Gaussian processes based on the convergence of entropy Dudley type integrals. The estimation of the distribution of the continuity module is studied for square-Gaussian random processes. It is considered in detail an example with an estimator (correlogram) of the covariance function of a Gaussian stationary random process. The conditions on continuity of correlogram’s trajectories with probability one are found and the distribution of the continuity module is also estimated.
Key words: Gaussian process, square-Gaussian process, correlogram, sample continuity.
Pages of the article in the issue: 11 - 15
Language of the article: Ukrainian
References
DUDLEY R. (1973) Sample functions of the Gaussian processes Annals of Probability. 1, № 1, 3–68.
BULDYGIN, V., KOZACHENKO, YU. (2000) Metric characterization of random variables and random processes, Amer. Math. Soc., Providence, RI.
GIKHMAN, I.I. and SKOROKHOD, A.V. (1971), Teoriya sluchainyh processov, v.1, M.: Nauka, 664 p.
KOZACHENKO YU., ROZORA I. (2019) Conditions of sample continuity with probability one for Square-Gaussian stochastic processes. Theor. Probability and Math. Statist., 101, 134-146.
LEADBETTER M., LINDGREN G., ROOTZEN H. (1983) Extremes and related properties of random sequences and processes, Springer, Berlin.
LEDOUX M., TALAGRAND M. (2013) Probability in Banach Spaces: Isoperimetry and Processes, Springer Science and Business Media.
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