Fisher, Evan On the law of the iterated logarithm for martingales. (English) Zbl 0758.60026 Ann. Probab. 20, No. 2, 675-680 (1992). 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 V. A. Egorov [Theory Probab. Appl. 14, 693-699 (1969); translation from Teor. Veroyatn. Primen. 14, 722-729 (1969; Zbl 0211.489)]. Reviewer: L.Heinrich (Freiberg) Cited in 4 Documents MSC: 60F15 Strong limit theorems 60G42 Martingales with discrete parameter 60G50 Sums of independent random variables; random walks Keywords:law of iterated logarithm; discrete-time martingales Citations:Zbl 0211.489 × Cite Format Result Cite Review PDF Full Text: DOI