On the separation problem for a family of Borel and Baire G -powers of shift measures on R - Pantsulaia, G.; Saatashvili, G.; Zerakidze, Z. (Ukrains’kyi Matematychnyi Zhurnal, 2013)
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(n^-1 # ({ x_1,…,x_n } (-∞;0])) 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.