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

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Ukrainian Mathematical Journal Aims and scope

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.

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References

  1. I. Sh. Ibramkhallilov and A. V. Skorokhod, Consistent Estimates of the Parameters of Stochastic Processes [in Russian], Kiev (1980).

  2. L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, etc. (1974).

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 4, pp. 470–485, April, 2013.

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Zerakidze, Z., Pantsulaia, G. & Saatashvili, G. Separation problem for a family of Borel and Baire G-powers of shift measures on \( \mathbb{R} \) . Ukr Math J 65, 513–530 (2013). https://doi.org/10.1007/s11253-013-0792-8

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  • DOI: https://doi.org/10.1007/s11253-013-0792-8

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