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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
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References:
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