Nonclassical estimates of precision of normal approximation for martingales (Q1324903)

Let \(X_ 1, X_ 2,\ldots\) be a sequence of random variables (r.v.), \(S_ 0=0\), \(S_ n=X_ 1+\ldots+X_ n\). Let us assume that for any \(i\geq 1\) \[ (1)\quad E\{X_ i\mid S_{i-1}\}=0 \text{ a.s.},\qquad (2)\quad E\{X^ 2_ i\mid S_{i-1}\}=1 \text{ a.s.},\qquad (3)\quad \gamma_ l=\sup_ i E| X_ i|^ l<\infty, \] where \(l>2\). Let \[ S_ n^*=S_ n/\sqrt n;\qquad \delta_ n=\sup_ z\mid P(S_ n^*<z)-\Phi(z)|, \] where \(\Phi(\cdot)\) is the standard normal distribution function (d.f.). Conditions for the convergence of \(\delta_ n\) to zero in terms of the \(\sigma\)-algebras generated by the r.v. \(S_ i\) were previously considered by \textit{A. Dvoretzky} [Proc. 6th Berkeley Symp. math. Stat. Probab., Univ. Calif. 1970, 2, 513-535 (1972; Zbl 0256.60009)]. The present note is concerned with nonclassical estimates of the \(\delta_ n\), i.e., those which take into account the possible proximity of the distribution of the summands to a normal one. The basic goal of the paper here is to get nonclassical analogs of the results of \textit{L. V. Kir'yanova} and \textit{V. I. Rotar'} [Probabilistic problems in discrete mathematics, Interuniv. Collect., Moskva, 136-140 (1988; Zbl 0739.60017)], where (1) and (2) were used and the form of the estimates was determined by specially chosen characteristics of the dependence of the summands.