Limit theorems in \(R^ k\). III
[For part I see ibid. 22, No.2, 51-68 (1982; Zbl 0494.60024) and for part II see the foregoing entry, Zbl 0567.60024.] Let \(\{\xi_ j^{(n)}\), \(j=1,...,n\), \(n=1,2,...\}\) be a series of k- dimensional random vectors independent and identically distributed in each series with zero mean and a nonsingular covariance matrix. Suppose that m of the first coordinates of the vector \(\xi_ j^{(n)}\) are lattice in \(\{v_ mH_ m^{(n)'}\}\), and (l-m) of the next coordinates are absolutely continuous, where \(0\leq m\leq l\leq k\). Thus, the distribution of the random vector \(\xi_ j^{(n)}\), \(j=1,...,n\) is of the form \[ P\{\xi_ j^{(n)}\in A\}= \] \[ \sum_{v_ mH_ m^{(n)'}\in A_ m}P\{\xi_{jm}^{(n)}=v_ mH_ m^{(n)'}\}\int_{A_{l-m}(v_ m)}p^{(n)}(v_ m\times x_{l-m}; A_{k-l}(v_ m\times x_{l-m}))dx_{l-m}. \] For any Borel set \(A_{k-l}\in R^{k-l}\) the function \(p^{(n)}(v_ m\times x_{l-m}; A_{k-l})\) will be called the ''conditional density of the (l-m)-th order'' of the random vector \(\xi^{(n)}_{j(k-m)}\). Denote \(Z_ n=\sum^{n}_{j=1}\xi_ j^{(n)}\), \(f(x_{k-l})\) is a measurable function in \(R^{k-l}\) and \(p_ n(v_ m\times x_{l-m}; A_{k-l})\) is the ''conditional density of the (l-m)-th order'' of the random vector \(n^{-1/2}Z_{n(k-m)}.\) The estimation of the remainder term and necessary and sufficient conditions are obtained for the convergence rate in the approximation of the function \[ n^{m/2}\int f(x_{k-l})P\{Z_{nm}=v_ mH_ m^{(n)'}\}p_ n(v_ m\times x_{l-m}; dx_{k-l})/| \det H_ m^{(n)}| \] by the limit distribution and by the corresponding asymptotic expansion.
scientific article; zbMATH DE number 3905544