## On approximation of sums of conditionally independent random variables.(English. Russian original)Zbl 0856.60051

Theory Probab. Math. Stat. 50, 133-142 (1995); translation from Teor. Jmovirn. Mat. Stat. 50, 129-137 (1994).
For every fixed $$n\in \mathbb{N}$$ let $$Y_n(k)$$, $$k= 1, 2,\dots$$, be a sequence of random variables with values in the measure space $$(X, {\mathfrak B})$$ and $$X_n(k, x)$$, $$k= 1, 2,\dots$$, a sequence of independent real random variables depending on the parameter $$x\in X$$. The sequences $$X_n(k, x)$$ and $$Y_n(k)$$ are independent. Put $$Z_n= \sum^n_{k= 1} X_n(k, Y_n(k))$$, $$b^2_n(k, x)= \text{Var } X_n(k, x)< \infty$$, $$b^2_n= \sum^n_{k= 1} Eb^2_n(k, Y_n(k))$$, $$A_n= EZ_n$$, $$B^2_n= \text{Var } Z_n$$ and $$S_n= E(Z_n\mid Y_n(1), Y_n(2),\dots)$$. The authors assume a condition on conditional variances and on the second-order behaviour at zero of the characteristic functions of the $$X_n(k, x)$$ and then prove that $E\exp\{itB^{- 1}_n(Z_n- A_n)\}- \exp(-\textstyle{{1\over 2}} t^2 b^2_n B^{- 2}_n) E\exp\{itB^{- 1}_n(S_n- A_n)\}\to 0$ as $$n\to \infty$$. So a limiting distribution could be the convolution of the usual normal one and a second one arising from the $$S_n$$. Conditions on second moments decide which limit occurs. Under further conditions, including bounds on absolute third moments, the authors derive bounds on $$\sup\{|F_n(y)- H(y)|: y{\i}\mathbb{R}\}$$, where $$F_n$$ is the distribution function of $$B^{- 1}_n(Z_n- A_n)$$ and $$H$$ the limiting one.
Reviewer: A.J.Stam (Winsum)

### MSC:

 60G50 Sums of independent random variables; random walks 60E10 Characteristic functions; other transforms