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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
60F05 Central limit and other weak theorems
41A25 Rate of convergence, degree of approximation
60G42 Martingales with discrete parameter
Full Text: EuDML
[1] A. K. Basu: On the rate of approximation in the Central limit theorem for dependent random variables and random vectors. J. Multivariate Anal. 10, (1980), 565-578. · Zbl 0452.60027 · doi:10.1016/0047-259X(80)90070-6
[2] P. L. Butzer L. Hahn W. Westphal: On the rate of approximation in the Central limit theorem. J. Approx. Theory 13 (1975), 327-340. · Zbl 0298.60014 · doi:10.1016/0021-9045(75)90042-8
[3] M. Mamatov I. Nematov: On a limit theorem for sums of a random number of independent random variables. (Russian). Izv. Akad. Nauk, USSR Ser. Fiz. Mat. Nauk, 3 (1971), 18-24. · Zbl 0226.60037
[4] H. Robbins: The asymptotic distribution of the Sum of a random number of random variables. Bull. Amer. Math. Soc. 54 (1948), 1151-1161. · Zbl 0034.22503 · doi:10.1090/S0002-9904-1948-09142-X
[5] Z. Rychlík D. Szynal: On the limit behavior of Sum of a random number of independent random variables. Coll. Math. 28 (1973), 147-159. · Zbl 0238.60015
[6] Z. Rychlík D. Szynal: On the rate of approximation in the random C-L.T. Theory of probability and Appl. 24 (1979), 620-625.
[7] E. Rychlík Z. Rychlík: The generalized Anscombe Condition and its applications in random sum limit theorems. Lecture Notes in Math. Probability in Banach spaces I Springer-Verlag 828 (1980), 244-250.
[8] V. Sakalauskas: An estimate in the Multidimensional Central Limit Theorem. Lithuanian Math. Jour. (Eng. Trans.) 17, 4 (1977), 567-572. · Zbl 0404.60030 · doi:10.1007/BF00972282
[9] S. Kh. Sirazhdinov G. Orazov: Generalization of a theorem of Robbins. (Russian), In Limit theorems and Statistical Inferences, Tashkent 1960, 154-162.
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