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

Khusanbaev, Ya. M.; Rakhimov, G. M.

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

Keywords:

conditional variances; characteristic functions; convolution