Subgaussian random vectors and processes. (Russian) Zbl 0627.60049

Buldygin, V. V.; Kozachenko, Yu. V.

Teor. Veroyatn. Mat. Stat., Kiev 36, 10-22 (1987).

In this paper the authors continue their investigation of subgaussian random elements initiated in an earlier paper [Ukr. Math. Zh. 32, 723-729 (1980; Zbl 0459.60002)]. A random vector \(\xi \in R^ n\) is said to be subgaussian if there exists a symmetric positive semidefinite operator \(B:R^ n\to R^ n\) such that \(E\exp\{(u,\xi)\} \leq \exp\{2^{- 1}(Bu,u)\}\), \(u\in R^ n\), where (,) is a scalar product compatible with the natural Euclidean norm in \(R^ n\). Especially, a random vector \(\xi \in R^ n\) is strictly subgaussian if \(E\exp\{(u,\xi)\} \leq \exp\{2^{- 1}(Bu,u)\}\), \(u\in R^ n\), where B is the covariance matrix of \(\xi\).
Let \(\overline{sub}(R^ n)\) denote the class of strictly subgaussian random vectors. A random process \((\xi(t),t\in T)\) is considered as strictly subgaussian if \((\xi (t_ 1),...,\xi (t_ n))\in \overline{sub}(R^ n)\) for all \(t_ 1,...,t_ n\in T\), \(n\geq 1\). Fundamental properties of strictly subgaussian variables, vectors, and processes are formulated in four theorems.

Reviewer: I.Křivý

MSC:

60G15

Gaussian processes

Keywords:

subgaussian random elements; strictly subgaussian variables

Citations:

Zbl 0459.60002