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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.

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