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On the rate of approximation in the random sum CLT for dependent variables. (English) Zbl 0631.60024
Let $$\{X_ n,n\geq 1\}$$ be a sequence of real r.v. defined on a probability space ($$\Omega$$,$${\mathcal A},P)$$, and $${\mathcal F}_ 0\subset {\mathcal F}_ 1\subset...\subset {\mathcal F}_ n$$ be a sequence of $$\sigma$$- algebras such that $$X_ n$$ is $${\mathcal F}_ n$$-measurable. Denote by $$N_{\lambda}^ a$$positive integer-valued r.v. for $$\lambda >0$$. Assume that $$N_{\lambda}$$ and $$\{X_ n,n\geq 1\}$$ are independent and $$E(X_ k| {\mathcal F}_{k-1})=0$$, $$k=1,...,n$$. The author proves “large-$${\mathcal O}''$$ and “little-$$o''$$ approximation theorems for the expression $| E[f(S_{N_{\lambda}}/M_{N_{\lambda}})]-\int f(x)d\phi (x)|$ for smooth functions f, where $$\phi$$ denotes the standard normal d.f. and $$S_{N_{\lambda}}=\sum^{N_{\lambda}}_{k=1}X_ k$$, $$M^ 2_{N_{\lambda}}=\sum^{N_{\lambda}}_{k=1}\sigma^ 2_ k$$, $$\sigma^ 2_ k=E(X^ 2_ k| {\mathcal F}_{k-1})$$. Finally, these results are extended to the multi-dimensional case.
Reviewer: L.Hahn
##### MSC:
 60F05 Central limit and other weak theorems 41A25 Rate of convergence, degree of approximation 60G42 Martingales with discrete parameter
##### Keywords:
random sums; central limit theorem; approximation theorems
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##### References:
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