On the distribution of the maximum of the Chentsov field on the “steps”

Kruglova, Nataliia V.; Dykhovychny, O.O.; Alekseeva, I. V.; Богданова, Н. В.

doi:10.34121/1028-9763-2020-1-128-139

Cited by 2 publications

(2 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition, considering the limit of the polyline, they have obtained distribution on the boundary of a square. Generalization of these results for a polyline with n vertices was considered by Klesov and Kruglova [10][11][12]. However, the obtained exact distribution of supreremum is given in terms of quite cumbersome integrals direct estimation of which may become problematic.…”

Section: Introductionmentioning

confidence: 93%

�n the distribution of the maximum of a brownian sheet restricted to a lower-dimensional set

Kruglova,

Dykhovychnyi

2021

MMS

View full text Add to dashboard Cite

The paper describes sufficient conditions for determining the distribution of functionals of an -dimensional Brownian sheet (Chentsov field) on a set with the dimension lower than the dimension of the field. The results have been obtained through the generalization of the well-known Doob's Theorem on the identity of the transformation of some Gaussian and Wiener processes in the case of random fields. It is known that the distribution of functionals of a Brownian sheet on the sets with the dimension lower than the dimension of the field and a set of suprema of a piecewise linear curve in particular requires dealing with rather cumbersome integrals that might be hard to estimate. The paper suggests an alternative approach to the problem and considers the probability of suprema of a Brownian sheet being less than a certain drift. The obtained results can significantly simplify the task of determining the distribution of functionals of a Brownian sheet by reducing it to the problem of finding distribution on parallelepipeds with the dimension lower than the dimension of the field. There is also illustrated the validity of the obtained theorem through modeling some Brownian sheets and comparing empirical probabilities with theoretical ones. For the simulations, there have been used the R statistical software. In order to model a Brownian sheet, there has been utilized a special algorithm allowing to model random fields with covariance function of a special form as well as their restrictions on sets of lower dimension (curves, piecewise linear curves). The corresponding R code is provided as well.

show abstract

Section: Introductionmentioning

confidence: 93%

�n the distribution of the maximum of a brownian sheet restricted to a lower-dimensional set

Kruglova,

Dykhovychnyi

2021

MMS

View full text Add to dashboard Cite

show abstract

“…Also, by considering limit of the polyline they have obtained distribution on the boundary of a square. Generalization of these results for a polyline with n vertices was considered by Klesov and Kruglova [13] [12]. However, the obtained exact distribution of supreremum is given in terms of quite cumbersome integrals direct estimation of which may become problematic.…”

Section: Introductionmentioning

confidence: 93%

On the distribution of maximum of a Brownian sheet restricted to a lower-dimensional set

Kruglova,

Dykhovychnyi

2020

Preprint

View full text Add to dashboard Cite

We obtain sufficient conditions of stochastic equivalence of Gaussian random fields with special covariance function. These results generalize Doob's transformation (condition of stochastic equivalence of a Gaussian and a Wiener processes) to the case of random fields (condition of stochastic equivalence of a Gaussian process and a Brownian sheet). We look at the problem of finding the distribution of supremum of a Brownian sheet on a set with a dimension lower than the dimension of the field. We consider the probability of a Brownian sheet with a certain drift to attain zero level. The obtained results can significantly simplify the problem of finding distributions of functionals of a Brownian sheet by reducing it to the problem of finding distributions on parallelepipeds with dimension lower than the dimension of the field. We consider examples that verify validity of the obtained theorem by modeling corresponding fields and comparing empirical probabilities with theoretical ones.

show abstract

On the distribution of the maximum of the Chentsov field on the “steps”

Cited by 2 publications

References 4 publications

�n the distribution of the maximum of a brownian sheet restricted to a lower-dimensional set

�n the distribution of the maximum of a brownian sheet restricted to a lower-dimensional set

On the distribution of maximum of a Brownian sheet restricted to a lower-dimensional set

Contact Info

Product

Resources

About