Conditions for the sample continuity with probability one for square-Gaussian stochastic processes

Kozachenko, Yuriy; Rozora, Iryna

doi:10.1090/tpms/1118

Cited by 3 publications

(6 citation statements)

References 7 publications

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“…The scheme of the proof is analogous to that used in [29], however, with some modifications, since the condition of convergence of another entropy integral is used. Note also that the same method was applied in [18] to prove a similar result for the case of square-Gaussian stochastic processes. In view of this, we present only the main steps.…”

Section: Properties Of Sample Paths Of ϕ-Sub-gaussian Processesmentioning

confidence: 80%

“…(We omit here the details, but note that the derivation of the similar formula for the case of square-Gaussian processes can be found also in [18], see Formula (10) therein.) Then for all λ > 0…”

Section: Properties Of Sample Paths Of ϕ-Sub-gaussian Processesmentioning

confidence: 99%

“…where for the last inequality we used the following estimate: for u, v (2.74) in [29], or Formula (7) in [18], where a quite similar derivation is given for the case of a square-Gaussian process).…”

Section: Properties Of Sample Paths Of ϕ-Sub-gaussian Processesmentioning

confidence: 99%

“…(Omitted intermediary steps can be checked, for example, in [18], see the derivation of Formula (23).) Finally, we obtain:…”

Section: Properties Of Sample Paths Of ϕ-Sub-gaussian Processesmentioning

confidence: 99%

“…The questions of continuity of sample paths are treated in the similar way as in [29], but basing on different conditions. Within a similar approach, estimates for the distribution of increments were derived in [18] for square-Gaussian processes. The results obtained in Section 4 concerning the distribution of supremum for the processes related to the heat equations with ϕ-sub-Gaussian initial conditions provide the generalization and extension of results from papers [13,14] where the cases of Gaussian and sub-Gaussian initial conditions were considered.…”

Section: Introductionmentioning

confidence: 99%

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Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes

Hopkalo¹,

Sakhno²

2021

Modern Stochastics: Theory and Applications

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This paper investigates sample paths properties of ϕ-sub-Gaussian processes by means of entropy methods. Basing on a particular entropy integral, we treat the questions on continuity and the rate of growth of sample paths. The obtained results are then used to investigate the sample paths properties for a particular class of ϕ-sub-Gaussian processes related to the random heat equation. We derive the estimates for the distribution of suprema of such processes and evaluate their rate of growth.

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Section: Properties Of Sample Paths Of ϕ-Sub-gaussian Processesmentioning

confidence: 80%

Section: Properties Of Sample Paths Of ϕ-Sub-gaussian Processesmentioning

confidence: 99%

Section: Properties Of Sample Paths Of ϕ-Sub-gaussian Processesmentioning

confidence: 99%

“…(Omitted intermediary steps can be checked, for example, in [18], see the derivation of Formula (23).) Finally, we obtain:…”

Section: Properties Of Sample Paths Of ϕ-Sub-gaussian Processesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes

Hopkalo¹,

Sakhno²

2021

Modern Stochastics: Theory and Applications

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Sample continuity with probability one for the estimator of impulse response function

Rozora

2020

BKNUPhM

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The problem of estimation of a stochastic linear system has been a matter of active research for the last years. One of the simplest models considers a ‘black box’ with some input and a certain output. The input may be single or multiple and there is the same choice for the output. This generates a great amount of models that can be considered. The sphere of applications of these models is very extensive, ranging from signal processing and automatic control to econometrics (errors-in-variables models). In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function. We assume that impulse function is square-integrable. Input signal is supposed to be Gaussian stationary stochastic process with known spectral density. A sample input–output crosscorrelogram is taken as an estimator of the response function. The conditions on sample continuousness with probability one for impulse response function are investigated.

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Analytical properties of sample paths of some stochastic processes

Rozora

2020

BKNUPhM

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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.

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Conditions for the sample continuity with probability one for square-Gaussian stochastic processes

Cited by 3 publications

References 7 publications

Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes

Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes

Sample continuity with probability one for the estimator of impulse response function

Analytical properties of sample paths of some stochastic processes

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