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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1980, Volume 25, Issue 3, Pages 502–512 (Mi tvp1090)

A new variant of the functional law of the iterated logarithm

A. V. Bulinskiĭ

Moscow

Abstract: The full description of the set of limit points of the sequence (3) is given, where $W(t)$ is a $d$-dimensional Brownian motion consisting of $d$ independent Brownian motions and $\varphi( \cdot )$ is arbitrary function such that $\varphi(t)\uparrow\infty$ ($t\uparrow\infty$). We show that with probability one this set coincides with the set $K_{R(\varphi)}$ specified in theorems 1–3. The sequences of the form (18) are also considered. The result of V. Strassen is a special case when $\varphi(t)=\sqrt{2\ln\ln t}$. The generalization of Hartman–Wintner's theorem is obtained. Theorems 4, 5 are valid for all sequences satisfying the almost sure invariance principles (martingale-differences, sequences with mixing etc.).

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English version:
Theory of Probability and its Applications, 1981, 25:3, 493–503

Bibliographic databases:

Citation: A. V. Bulinskiǐ, “A new variant of the functional law of the iterated logarithm”, Teor. Veroyatnost. i Primenen., 25:3 (1980), 502–512; Theory Probab. Appl., 25:3 (1981), 493–503

Citation in format AMSBIB
\Bibitem{Bul80} \by A.~V.~Bulinski{\v\i} \paper A new variant of the functional law of the iterated logarithm \jour Teor. Veroyatnost. i Primenen. \yr 1980 \vol 25 \issue 3 \pages 502--512 \mathnet{http://mi.mathnet.ru/tvp1090} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=582580} \zmath{https://zbmath.org/?q=an:0462.60035|0436.60031} \transl \jour Theory Probab. Appl. \yr 1981 \vol 25 \issue 3 \pages 493--503 \crossref{https://doi.org/10.1137/1125061} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1980MB70100005} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. V. Bulinski, M. A. Lifshits, “Rate of convergence in the functional law of the iterated logarithm with non-standard normalizing factors”, Russian Math. Surveys, 50:5 (1995), 925–944
2. A. P. Shashkin, “Generalization of the Law of the Iterated Logarithm for Associated Random Fields”, Math. Notes, 98:5 (2015), 831–842