On strong attraction of stationary sequences to a normal law (Q2711157)

The paper deals with stationary sequences satisfying the uniform strong mixing condition with mixing coefficients \(\varphi (n)\) such that \(\varphi (1) <1\) (the author calls this \(\varphi _1\)-mixing). Let \({\mathcal N}_0\) denote the domain of the strong attraction of the normal law, that is the set of all distributions \(L(\xi)\) such that any stationary sequence \(\{\xi _n\}\) meeting \(\varphi _1\)-mixing and such that \(L(\xi _1)\in {\mathcal N}_0\) is attracted to the normal law. Let, further, \({\mathcal N}_1\) be the set of all distributions \(L(\xi)\) such that any stationary sequence \(\{\xi _n\}\) satisfying the \(\varphi _1\)-mixing condition and such that \(L(\xi _1)=L(\xi)\) obeys the central limit theorem. Clearly, \({\mathcal N}_1\subset {\mathcal N}_0\). The author gives conditions under which \(L(\xi)\) belongs to \({\mathcal N}_0\) or \({\mathcal N}_1\) and which are more general than those known before. His main result looks as follows: Let \(\hat{\xi}_1,\ldots,\hat{\xi}_n\) be a sequence of independent random variables such that \(\hat{\xi}_k\buildrel d \over =\xi_k, k=1,2,\ldots\), and \(c_n(p)=\{E|\max_{1\leq k\leq n}\hat{\xi_k}|^p\}^{1/p}\). Then NEWLINENEWLINENEWLINE1) if, for some \(p>0\) NEWLINE\[NEWLINE \lim_{N\to\infty}\lim \sup_{n\to\infty}n\int_{N}^{\infty}P(\xi ^2\geq xc_n^2(p))dx=0, NEWLINE\]NEWLINE then \(L(\xi)\in {\mathcal N}_1\); NEWLINENEWLINENEWLINE2) if, for some \(p>0\) NEWLINE\[NEWLINE \lim_{\varepsilon\to 0}\lim \inf_{n\to\infty}\int_{\varepsilon}^{\infty} P(\xi ^2\geq xc_n^2(p)) dx=\infty, NEWLINE\]NEWLINE then \(L(\xi)\in {\mathcal N}_0\).