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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

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