Asymptotic behavior of the distribution of the maximum of a Chentsov field on polygonal lines. (Ukrainian, English) Zbl 1224.60075

Kruglova, N. V.

Teor. Jmovirn. Mat. Stat. 81, 88-101 (2009); translation in Theory Probab. Math. Stat. 81, 101-115 (2010).

The author deals with the two-parameter Chentsov field \(X(s, t)\) (also known as the Brownian sheet) which is a real-valued separable Gaussian random field \(\{X(s, t):(s, t)\in D=[0,1]\times[0,1]\}\) such that (1) \(X(0, t) = X(s, 0) = 0\) for all \(s, t\in [0, 1]\); (2) \(E[X(s, t)] = 0\) for all \((s, t)\in D\); (3) \(E[X(s, t)X(s_1, t_1)] =\min\{s, s_1\}\min\{t, t_1\}\) for all \((s, t)\in D\) and \((s_1, t_1)\in D\) [N. N. Chentsov, Dokl. Akad. Nauk SSSR 106, 607–609 (1956; Zbl 0074.12503); J. Yeh, Trans. Am. Math. Soc. 95, 433–450 (1960; Zbl 0201.49402)]. The closed form of the distribution of functionals such as \(\max_{(s,t)\in D} X(s, t)\) is still unknown for Chentsov fields. The distribution of the supremum of \(X(s, t)\) on the boundary of the unit square was found by S. R. Paranjape and C. Park [J. Appl. Probab. 10, 875–880 (1973; Zbl 0281.60081)]. I. I. Klesov [Theory Probab. Math. Stat. 51, 63–67 (1995); translation from Teor. Jmovirn. Mat. Stat. 51, 62–66 (1994; Zbl 0939.60042)] found an explicit form of the probability \[ P(L,g)=P\left\{ \sup_{(s,t)\in L} (X(s,t)-g(s,t))<0 \right\}, \] where \(X(s,t)\) is a Chentsov field on \(D\), \(L\) is a polygonal line with two linear sections, and \(g(s,t)\) is a linear function. In this article, the asymptotic behaviour of the tail of the distribution of the maximum of the field \(X(s, t)\) \[ P(L,\lambda)=P\left\{ \sup_{(s,t)\in L} X(s,t)>\lambda \right\} \] is investigated as \(\lambda\to\infty\) in the case where \(L\) is a polygonal line with several linear sections.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60G15	Gaussian processes
60G60	Random fields

Keywords:

Chentsov field; asymptotic behaviour; distribution of maximum; polygonal line

Citations:

Zbl 0074.12503; Zbl 0201.49402; Zbl 0281.60081; Zbl 0939.60042