Summary
Kolmogorov's law of the iterated logarithm has been sharpened by Strassen who proved a more refined theorem by using tools from functional analysis. The present paper gives a “classical” proof of Strassen's theorem, using a method along the lines of Kolmogorov's original approach. At the same time the result proved here is more general since a) the random variables involved need not have the same distributions, b) the condition of independence is weakened and c) instead of Kolmogorov's growth condition on the random variables, only a mild restriction on their moments of order l≧3 is needed.
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References
Kolmogorov, A.: Das Gesetz des iterierten Logarithmus. Math. Ann. 101, 126–135 (1929)
Rényi, Wahrscheinlichkeitsrechnung. Berlin: Deutscher Verlag der Wiss. 1962
Strassen, V.: An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie verw. Gebiete 3, 211–226 (1964)
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Szüsz, P., Volkmann, B. On Strassen's law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie verw Gebiete 61, 453–458 (1982). https://doi.org/10.1007/BF00531616
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DOI: https://doi.org/10.1007/BF00531616