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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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##### References:
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