An inequality of Anderson and unimodal measures. (English. Russian original) Zbl 0634.60007

Buldygin, V. V.; Kharazishvili, A. B.

Theory Probab. Math. Stat. 35, 13-26 (1987); translation from Teor. Veroyatn. Mat. Stat. 35, 13-27 (1986).

The paper under review studies unimodal distributions on topological vector spaces F in the sense of Anderson which are defined by the inequality \(\mu (D+ty)\geq \mu (D+y)\) for \(y\in F\), \(0\leq t\leq 1\), and convex symmetric Borel set D. It turns out that in one dimension the concept coincides with the usual unimodality of Khintchine for symmetric measures.
The main results show that a linear transformation of a product measure with unimodal components remains unimodal. The result can be applied to Gaussian measures on the infinite product of the reals. It should be mentioned that the inequalities of Anderson are also important in statistics in connection with critical regions.

Reviewer: A.Janssen

MSC:

60B11	Probability theory on linear topological spaces
60E15	Inequalities; stochastic orderings
60E07	Infinitely divisible distributions; stable distributions
28A35	Measures and integrals in product spaces

Keywords:

unimodal distributions on topological vector spaces; symmetric measures; unimodal components; Gaussian measures; inequalities; critical regions