On the law of the iterated logarithm for martingales (Q1196931)

Let \(\{U_ n\), \(n\geq 1\}\) be a martingale adapted to an increasing sequence of \(\sigma\)-fields \(\{{\mathcal F}_ n\), \(n\geq 1\}\) and \(s_ n^ 2=\sum_{i=1}^ n E(X_ i^ 2\mid{\mathcal F}_{i-1})\), \(n\geq 1\). The main result establishes (with probability 1) an upper bound of \(\limsup_{n\to\infty}U_ n/s_ n\varphi(s_ n)\) with \(\varphi(x):=(2\log\log(x^ 2\vee e^ 2))\) which generalizes the corresponding estimate for sums of i.i.d. r.v.'s derived by \textit{V. A. Egorov} [Theory Probab. Appl. 14, 693-699 (1969); translation from Teor. Veroyatn. Primen. 14, 722-729 (1969; Zbl 0211.489)].