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The law of the iterated logarithm for weakly dependent Hilbert space valued random variables. (Russian, English) Zbl 1046.60023

Sib. Mat. Zh. 44, No. 6, 1407-1424 (2003); translation in Sib. Math. J. 44, No. 6, 1111-1126 (2003).
Let \(X_1, X_2, \dots\) be a sequence of centered identically distributed random vectors in a separable Banach space \(B\) with norm \(\| \cdot\| \) and dual \(B^*\). Say that \(\{X_n\}\) satisfies the bounded law of the iterated logarithm if \[ \limsup_{n\to\infty}\frac{\| X_{1}+\cdots+X_{n}\| }{\sqrt{2n\log\log n}}<\infty \;\text{ a.s.} \] Let \(\varphi(n)\) be the mixing coefficient \[ \varphi(n)=\sup\{| P(A/C)-P(A)| : k\in\mathbb N, \;C\in F_1^k,\;A\in F_{n+k}^\infty, \;P(C)>0\}, \] where \(F_a^b\) is the \(\sigma\)-algebra generated by the random variables \(X_a,\dots,X_b\). The author considers the sequence of random vectors \(\{X_n\}\) which satisfies the following conditions: \[ \begin{gathered} Ef(X_1)=0,\;Ef^2(X_1)<\infty\;\text{ for all }\;f\in B^*, \\ \sum_{k=1}^\infty\varphi^{1/\theta}(2^k)<\infty \;\text{ for some }\;\theta>3, \quad \lim_{n\to\infty}\frac{Ef^2(S_n)}{n}<\infty \;\text{ for all }\;f\in B^*. \end{gathered} \] Under these conditions, the author proves that \(\{X_n\}\) satisfies the bounded law of the iterated logarithm if and only if \( E\| X_1\| ^2/ \log\log\| X_1\| )<\infty\). Necessary and sufficient conditions are also given for the compact law of the iterated logarithm to be satisfied.

MSC:

60F05 Central limit and other weak theorems
60F20 Zero-one laws
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