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Martingale methods for proving central limit theorems for dependent variables. (English) Zbl 0741.60015

Statistics and control of stochastic processes. Vol. 2, Pap. Steklov Semin., Moscow/USSR 1985-86, Transl. Ser. Math. Eng., 41-67 (1989).
[For the entire collection see Zbl 0732.00014.]
Let \(\{X_ n,\;-\infty<n<+\infty\}\) be a strictly stationary sequence of random variables with \(E X_ 0=0\), \(E X^ 2_ 0<\infty\). Define \({\mathcal F}_ n\) to be the sigma-field generated by \(\{X_ i:\;i\leq n\}\), and let \({\mathcal T}\) be a shift- invariant sigma-field such that \({\mathcal T}\subset{\mathcal F}_ 0\). Conditions are given which ensure that, for \(\alpha>1/2\), \[ E(\exp(itn^{-\alpha-1/2}\sum_{k=1}^ n k^ \alpha X_ k) \mid {\mathcal T})\to \exp\{-t^ 2 \sigma^ 2/2\}, \] where \(\sigma^ 2=\sum^ \infty_{n=-\infty}E(X_ n X_ 0 \mid {\mathcal T})\). In particular, this convergence is shown to hold for certain stationary mixing sequences. These results are derived from a more general theorem which establishes a similar convergence for weighted sums \(\sum^ n_{k=1}a_ k X_ k\), where \(a_ k=k^ \alpha f(k)\) and \(\{f(k)\}\) is slowly oscillating.

MSC:

60F05 Central limit and other weak theorems
60G42 Martingales with discrete parameter
60G10 Stationary stochastic processes

Citations:

Zbl 0732.00014
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