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
Keywords:Chentsov random field, Distribution of the maximum, Wiener process, Doob’s transformation theorem
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
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.
LicenseCopyright (c) 2017 NTUU KPI Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under CC BY 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work