Separation problem for a family of Borel and Baire G-powers of shift measures on $ \mathbb{R} $

Abstract

The separation problem for a family of Borel and Baire G-powers of shift measures on $ \mathbb{R} $ is studied for an arbitrary infinite additive group G by using the technique developed in [L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York (1974)], [ A. N. Shiryaev, Probability [in Russian], Nauka, Moscow (1980)], and [G. R. Pantsulaia, Invariant and Quasiinvariant Measures in Infinite-Dimensional Topological Vector Spaces, Nova Sci., New York, 2007]. It is proved that $ {T_n}:{{\mathbb{R}}^n}\to \mathbb{R},\;n\in \mathbb{N} $ , defined by $$ {T_n}\left( {{x_1},\ldots,{x_n}} \right)=-{F^{-1 }}\left( {{n^{-1 }}\#\left( {\left\{ {{x_1},\ldots,{x_n}} \right\}\cap \left( {-\infty; 0} \right]} \right)} \right) $$ for (x 1,…, x n ) ∈ $ {{\mathbb{R}}^n} $ is a consistent estimator of a useful signal θ in the one-dimensional linear stochastic model $$ {\xi_k}=\theta +{\varDelta_k},\quad k\in \mathbb{N}, $$ where #(∙) is a counting measure, ∆ k , k ∈ $ \mathbb{N} $ , is a sequence of independent identically distributed random variables on $ \mathbb{R} $ with a strictly increasing continuous distribution function F, and the expectation of ∆1 does not exist.