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On conditional limit distributions of sums of random vectors. (Russian) Zbl 0571.62008
Let $$(\xi_{nk},\eta_{nk},\zeta_{nk})$$, $$k=1,...,k_ n$$, be a sequence of independent $$(\ell +m+1)$$-dimensional random vectors. Let $$\xi_ n=\sum^{k_ n}_{k=1}\xi_{nk}$$ and $$\eta_ n$$, $$\zeta_ n$$ be analogically defined. The distribution of $$\eta_ n$$ is absolutely continuous with respect to the Lebesgue measure for every n. The distribution of $$\zeta_ n$$ is a lattice distribution, which is concentrated on the m-dimensional lattice with integer coordinates.
The class of conditional limit distributions $$(n\to \infty$$, $$k_ n\to \infty)$$ of $$\xi_ n$$, if $$\eta_ n$$ and $$\zeta_ n$$ are fixed is studied. It is proved, that the sequence $$\{\hat P_ n(\cdot | y_ n,z_ n)\}$$ converges weakly to the continuous version $$\hat P(\cdot | y,z)$$ of a certain regular conditional distribution. Examples of special distributions are studied and applications to goodness-of-fit and homogeneity testing are given.
Reviewer: P.Froněk

##### MSC:
 62E20 Asymptotic distribution theory in statistics 60F05 Central limit and other weak theorems 62F03 Parametric hypothesis testing 62G10 Nonparametric hypothesis testing