Connection Between the Distribution of the Maximum of the Wiener Process with a Linear Shift and Distribution of the Maximum of the Chentsov Random Field on Polygonal Lines

Наталія Володимирівна Прохоренко

Abstract


Background. The probability distributions of functionals of Chentsov random field X(s, t) like supremum on the unit square are not yet known. Some trivial probability distribution theory for X(s, t) can be obtained by using known results about the standard Wiener process. O. Klesov and N. Kruglova expressed probability that Wiener process crossing a step-wise linear barriers in terms of n-tuple integral of a function involving exponents and the standard Gaussian density. Also they used this result for obtaining the distribution of the maximum of the Chentsov field on polygonal lines. We consider the problem of finding the distribution of the maximum of the Chentsov random field on polygonal lines with several points of break and the problem of finding zero-crossing probabilities of the Wiener process with a step-wise linear shift in this paper.

Objective. The purpose of this paper is to find the conditions for a correspondence between these problems.

Methods. We used Doob’s Transformation Theorem in the proof of the main results.

Results. Conditions for a correspondence between the problem of finding the distribution of the maximum of the Chentsov random field on polygonal lines with several points of break and the problem of finding zero-crossing  probabilities of the Wiener process with a step-wise linear shift are found. It is proved that at the certain conditions a polygonal line with n points of break one-to-one corresponds to a step-wise linear shift.

Conclusions. Under certain conditions there is a two-way correspondence between the considered problems

Keywords


Chentsov random field; Distribution of the maximum; Wiener process; Doob’s transformation theorem

References


J.L. Doob, “Heuristic approach to Kolmogorov-Smirnov theorems”, Ann. Math. Statist, vol. 20, pp. 393–403, 1949.

S. Malmquist, “On certain confidence contours for distribution functions”, Ann. Math. Statist, vol. 25, pp. 523–533, 1954.

S.R. Paranjape and C. Park, “Distribution of the supremum of the two-parameter Yeh-Wiener process on the boundary”, J. Appl. Probab., vol. 10, no. 4, pp. 875–880, 1973.

R. Paranjape and C. Park, “Probabilities of Wiener paths crossing differentiable curves”, Pacific J. Math, vol. 53, pp. 579–583, 1974.

O.I. Klesov and N.V. Kruglova, “The distribution of a functional of theWiener process and its application to the Brownian sheet”, Statistics, vol. 45, no. 1, pp. 19–26, 2011.

I.I. Klesov “On the probability of attainment of a curvilinear level by a Wiener field”, Teoriya Ymovirnostey ta Matematychna Statystyka, no. 51, pp. 63–67, 1995 (in Ukrainian).

O.I. Klesov and N.V. Kruglova “The distribution of a functional of the maximum type for the two-parameter Chentsov random field”, Naukovi Visti NTUU KPI, no. 4, pp. 136–141, 2007 (in Ukrainian).

N.V. Kruglova, “Distribution of the maximum of the Chentsov random field”, Theory Stoch. Proc., no. 1, pp. 76–81, 2008.


GOST Style Citations


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DOI: https://doi.org/10.20535/1810-0546.2015.4.50363

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