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Conditions for the weak convergence of distributions of separable statistics. (English. Russian original) Zbl 0624.60040
Math. Notes 40, 928-932 (1986); translation from Mat. Zametki 40, No. 6, 762-769 (1986).
This paper describes the class of weak limits (n\(\to \infty\), \(k_ n\to \infty)\) of the distributions of decomposable statistics of the form \[ \xi_ n'=\sum^{k_ n}_{k=1}f_{nk}(\theta_{nk}), \] where \(f_{nk}\), \(k=1,...,k_ n\) are real Borel functions defined on \({\mathbb{R}}^{\ell +m}\) and the joint distributions of the random quantities \(\theta_{n1},...,\theta_{nk_ n}\) coincide with the conditional distribution of some independent random vectors \((\eta_{nk},\zeta_{nk})\), \(k=1,...,k_ n\), provided \(\eta_ n\equiv \sum^{k_ n}_{k=1}\eta_{nk}=y_ n\), \(\zeta_ n\equiv \sum^{k_ n}_{k=1}\zeta_{nk}=z_ n.\)
Here for all values of n the distribution of the \(\ell\)-dimensional vector \(\eta_ n\) is absolutely continuous in the Lebesgue measure, and the distribution of the vector \(\zeta_ n\) is concentrated on the m- dimensional integer valued lattice.
The above-mentioned weak limits are expressed as an integral of the corresponding limiting conditional joint characteristic function of the vectors \(\eta_ n\) and \(\zeta_ n\). The idea of the Le Cam-Holst method [L. Le Cam, Publ. Inst. Stat. Univ. Paris 7, No.3/4, 7-16 (1959; Zbl 0083.138); L. Holst, Ann. Stat. 7, 551-557 (1979; Zbl 0406.62008) and Ann. Probab. 9, 818-830 (1981; Zbl 0471.60027)] is employed in the proof.

MSC:
60F05 Central limit and other weak theorems
60E10 Characteristic functions; other transforms
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