Generalized strictly pre-Gaussian stochastic processes. (English. Ukrainian original) Zbl 0940.60048

Kozachenko, Yu. V.; Stus, O. V.

Theory Probab. Math. Stat. 58, 67-81 (1999); translation from Teor. Jmovirn. Mat. Stat. 58, 61-75 (1998).

Spaces of generalized strong pre-Gaussian random variables \((G,S,P)\) and random processes from these spaces are investigated. Spaces of square-Gaussian random variables SG\(_{\Sigma}(\Omega)\), i.e. a closure in \(L_2(\Omega)\) of quadratic forms from a family of jointly Gaussian random variables, are a particular case of \((G,S,P).\) An inequality for the distributions of quadratic forms on random variables \(\xi_{k}\in (G,S,P)\), \(k=1,\ldots,n\), is proved. This inequality allows to construct confidence ellipsoids for estimators of covariance functions of jointly Gaussian stationary random processes. The authors find estimates of the distributions of the supremum of quadratic forms on random processes \(\xi_{k}= \{\xi_{k}(t),t\in T\}\), \(\xi_{k}\in (G,S,P)\), \(k=1,\ldots,n.\) These estimates for square-Gaussian random processes allow to construct confidence ellipsoids for uniform (in some set) estimators of covariance functions of jointly Gaussian stationary random processes. For example, the following assertion is proved:
Let \(\vec\xi=(\xi_1,\ldots, \xi_{d})^{\top}\), \(\xi_{k}\in\) GSP\((\Delta,f,a)\), where \(a>0\), \(f=\{ f(s),|s|<1\}\) is an even function such that \(f(0)\geq 1\) and \(f(t)\) increases as \(s>0\). Let \(A\) be a symmetric positive semi-definite matrix. Then for all \(0\leq s\leq a^{-1}\) the following inequality holds true \[ EG\biggl({s^2(A{\vec\xi},{\vec\xi})\over \text{Sp}(A\cdot B({\vec\eta}(\vec\xi)))}\biggr)\leq f(as), \quad G(x)={\sin h\sqrt{x}\over\sqrt{x}},\quad x>0. \] For square-Gaussian random vectors \(\vec\xi\) we have \(B({\vec\eta}(\vec\xi))=\text{cov}(\vec\xi)\), \(a=\sqrt 2\), \(f(s)= \exp\{{1/8}-{|s|/2}\} (1-|s|)^{-1/2}.\)

Reviewer: M.P.Moklyachuk (Kyïv)

MSC:

60G07	General theory of stochastic processes
60E15	Inequalities; stochastic orderings

Keywords:

pre-Gaussian random process; square-Gaussian random process; quadratic form; Banach space; covariance function