On the separation problem for a family of Borel and Baire <i>G</i> -powers of shift measures on <i>R</i>

Abstract

The separation problem for a family of Borel and Baire G-powers of shift measures on 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 $T_n: R^n → R,\;n∈N$, defined by $$T_n(x_1,…,x_n) = -F^{-1}\left(n^{-1 } \# (\{ x_1,…,x_n \} \bigcap (-\infty;0])\right)$$ for $(x_1,…, x_n) ∈ R^n$ is a consistent estimator of a useful signal $θ$ in the one-dimensional linear stochastic model $$ξ_k = θ + ∆_k,\; k ∈ N,$$ where $\#(·)$ is a counting measure, $∆_k,\; k ∈ N$, is a sequence of independent identically distributed random variables on $R$ with a strictly increasing continuous distribution function $F$, and the expectation of $∆_1$ does not exist.