A characterization of the Gaussian distribution in a Banach space. (English. Russian original) Zbl 0628.60007

Dorogovtsev, A. A.

Theory Probab. Math. Stat. 33, 21-25 (1986); translation from Teor. Veroyatn. Mat. Stat. 33, 20-25 (1985).

The paper deals with the following problem. Let \(\xi_ 1,...,\xi_ n\) be independent identically distributed random variables with values in a separable Banach space X. The problem is to find conditions under which the following random variables \[ T_ B=\sum^{n}_{i,j=1}a_{ij}B(\xi_ i,\xi_ j)\quad and\quad L=\sum^{n}_{i=1}\xi_ i \] are independent for any continuous bilinear form B on \(X\times X\). The following statement is proved.
Theorem. If \(\sum^{n}_{i=1}a_{ii}\neq 0\), \(\sum^{n}_{i,j=1}a_{ij}=0\), E \(\xi\) \({}_ 1=0\), E \(\| \xi_ 1\|^ 2<\infty\), then \(T_ B\) and L are independent for any continuous bilinear form B on \(X\times X\) if and only if \(\xi_ 1\) is Gaussian and \(\sum^{n}_{j=1}a_{ij}=0\), \(i=1,...,n.\)
In the scalar case the analogous result has been proved earlier by R. G. Laha [Vest. Leningr. Univ. 11, No.1, 25-32 (1956; Zbl 0075.144)].

Reviewer: S.A.Chobanjan

MSC:

60B11	Probability theory on linear topological spaces
60G15	Gaussian processes

Keywords:

linear and bilinear forms of Gaussian random elements; tensor product of Gaussian random elements; Banach space; continuous bilinear form

Citations:

Zbl 0075.144