On the distribution of the maximum of a brownian sheet restricted to a lower-dimensional set
DOI:
https://doi.org/10.34121/1028-9763-2021-3-71-80Keywords:
Wiener process, Chentsov random field, Brownian sheet, distribution of the supremum of a Brownian sheet, Doob’s Transformation Theorem, вінерівський процес, поле Ченцова, броунівський лист, розподіл супремума броунівського листа, теорема Дуба про трансформаціюAbstract
The paper describes sufficient conditions for determining the distribution of functionals of an n-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.References
1. Chentsov N.N. Wiener random fields depending on several parameters. Doklady AN SSSR. 1956. Vol. 46, N 4. P. 607–609 (in Russian).
2. Yeh J. Wiener Measure in a Space of Functions of Two Variables. Transactions of the American Mathematical Society. 1960. N 95. P. 433–450.
3. Allen E.J. Modeling with Ito Stochastic Differential Equations. Dordrecht: Springer, 2007. 228 p. doi: 10.1007/978-1-4020-5953-7.
4. Bovkun V.A. On models that lead to an infinite-dimensional stochastic Cauchy problem. Teor. Veroyatnost. i Primenen. 2017. Vol. 62, N 4. P. 803–804 (in Russian). doi: 10.4213/tvp5151.
5. Davar Khoshnevisan. Five Lectures on Brownian Sheet. 2001. URL: http://www.math.utah.edu/?davar.
6. Men'shikov M.V., Molchanov S.A., Sidorenko A.F. Percolation theory and some applications. J Math Sci. 1988. N 42. P. 1766–1810. URL: https://doi.org/10.1007/BF01095508.
7. Paranjape S. R., Park C. Distribution of the supremum of the two-parameter Yeh-Wiener process on the boundary. J. Appl. Probab. 1973. Vol. 10, N 4. P. 875–880.
8. Park C., Schuurmann F. J. Evaluations of barrier-crossing probabilities of Wiener paths. J. Appl. Prob. 1976. N 13. P. 267–275.
9. Park C. Representations of Gaussian processes by Wiener processes. Pacific Journal of Mathematics. 1981. Vol. 94, N 2. P. 407–415.
10. Kruglova N.V. Distribution of the maximum of the Chentsov random field. Theor. Stoch. Process. 2008. N 1. P. 76–81.
11. Klesov O.I., Kruglova N.V. Distribution of the functionals of the Chentsov R3 random field. Naukovi visti NTUU-KPI. 2007. N 6. P. 145–150 (in Ukrainian).
12. Prokhorenko N.V. Stochastic Equivalence of Gaussian Process to the Wiener Process, Brownian Bridge, Ornstein-Uhlenbeck Process. Naukovi visti NTUU-KPI. 2016. N 4. Р. 85–93 (in Ukrainian).
13. Dykhovychnyi O.O., Kruglova N.V., Virstiuk O.I. Methods for identification of probability distribution of random variables from data samples with R statistical computing language. Mathematics in Modern Technical University. 2018. N 1. Р. 91–100 (in Ukrainian).
14. Doob J.L. Heuristic approach to Kolmogorov-Smirnov theorems. Ann. Math. Statist. 1949. N 20. P. 393–403.
15. Kruglova N., Dykhovychnyi O. On the distribution of the maximum of a Brownian sheet restricted to a lower-dimensional set. Preprint arXiv: 2006.06243. 2020. 8 p.
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