Kolmogorov's law of the iterated logarithm for noncommutative martingales
The law of the iterated logarithm is a main subject in probability theory. It is a strong almost sure theorem which gives a very precise bound on the almost sure fluctuations of the sums of random variables normalized as in the central limit theorem. The first studied case was for sums of independent identically distributed random variables, namely, Bernoulli random variables (\(\pm 1\) with \(\mathbb{P} = 1/2\)), this was done by Khintchine. This result was extended to independent but not identically distributed random variables by Kolmogorov when the variables are individually bounded but the bound grows at a controlled rate. \textit{P. Hartman} and \textit{A. Wintner} [Am. J. Math. 63, 169--176 (1941; Zbl 0024.15802)] proved that the finiteness of the second moment suffices for the law of the iterated logarithm in the independent identically distributed case. The case of martingales was first studied in 1937 by Lévy, later by \textit{V. Strassen} [in: Proc. 5th Berkeley Symp. Math. Stat. Probab., Univ. Calif. 1965/66, 2, Part 1, 315--343 (1967; Zbl 0201.49903)] and \textit{W. F. Stout} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 15, 279--290 (1970; Zbl 0209.49004)] under some boundedness conditions similar to those of Kolmogorov. In the present work, the author proves Kolomogorov's law of the iterated logarithm for noncommutative martingales.
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