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Some limit theorems in log density. (English) Zbl 0785.60014
Let $$(X_ n, n\geq 1)$$ be independent r.v.’s, $$S_ n=\sum^ n_{i=1} X_ i$$, $$Y_ n=(S_ n- b_ n)/a_ n$$ and $Ef(| Y_ n|)\leq(\log\log n)^{-1-\varepsilon} f\bigl(\exp((\log n)^{1- \varepsilon})\bigr),\;n\geq n_ 0,\quad\text{for some } \varepsilon>0,\tag{1}$ where functions $$f(x)>0$$, $$x/f(x)$$ are nondecreasing and the right-hand side of (1) is nondecreasing; $$a_ n>0$$, $$b_ n\in R$$ and $a_ i/a_ k\geq \mathbb{C}(i/k)^ \gamma,\;i\geq k,\quad\text{for some } \mathbb{C}>0,\;\gamma>0.\tag{2}$ Then Theorem 1 asserts that for any distribution function $$G$$ and any Borel set $$A\subset R$$ with $$G(\partial A)=0$$ the relation $\lim_{N\to\infty} (1/\log N) \sum_{k\leq N} (1/k)\;I\{Y_ k\in A\}= G(A)\quad\text{a.s.}$ is fulfilled if and only if $\lim_{N\to \infty}(1/\log N) \sum_{k\leq N} (1/k)\;P\{Y_ k\in A\}= G(A).$ A functional version of this result on $$D[0,1]$$ also holds (Theorem 2). The log density $$\mu(H)$$ of $$H\subset N$$ is defined by $\mu(H)= \lim_{N\to\infty} (1/\log N) \sum_{k\in H, k\leq N} (1/k).$ $$\xi_ n@> P>>\xi$$ (log) means that there exists a set $$H$$ with $$\mu(H)=1$$ such that $$\xi_ n@> P>>\xi$$ as $$n\to \infty$$, $$n\in H$$. $$\xi_ n\to \xi$$ a.s. (log) means that for a.e. $$\omega$$ there exists a set $$H_ \omega$$ with $$\mu(H_ \omega)=1$$ such that $$\xi_ n(\omega)\to\xi(\omega)$$ as $$n\to\infty$$, $$n\in H_ \omega$$. Theorem 3 asserts that the relations $$S_ n/a_ n\to 0$$ a.s. (log) and $$S_ n/a_ n@>P>>0$$ (log) are equivalent, if (1), (2) hold with $$b_ n=0$$. A.s. invariance principles in log density are proved to exist, too.

MSC:
 60F05 Central limit and other weak theorems 60F17 Functional limit theorems; invariance principles 60F15 Strong limit theorems
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