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$$T_n(x_1,…,x_n) = -F^{-1}\left(n^{-1 } # (\{ x_1,…,x_n \} \bigcap (-∞;0])\right)$$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.