LIL behavior for B-valued strong mixing random variables
The authors prove a law of the iterated logarithm for \(\phi^*\)-mixing Banach space-valued random variables, by this means generalizing an earlier results of \textit{U. Einmahl} and \textit{D. Li} [Trans. Am. Math. Soc. 360, No.~12, 6677--6693 (2008; Zbl 1181.60010)] and \textit{O. Sh. Sharipov} [Theory Probab. Appl. 49, No.~3, 444--457 (2005); translation from Teor. Veroyatn. Primen. 49, No.~3, 522--537 (2004; Zbl 1095.60002)]. Subject to several technical conditions, they prove \[ \limsup_{n \to \infty} \frac{\|X_1 + \dots + X_n \|}{c_n} = \alpha_0 \qquad \text{a.s.}, \] where \((c_n)_{n \geq 1}\) is an appropriate increasing sequence, and \[ \alpha_0 = \sup \left\{ \alpha: ~\sum_{n=1}^\infty \frac{1}{n} \exp \left(-\frac{\alpha^2 c_n^2}{2 H_n}\right) = \infty \right\} < \infty, \] where \[ H_n = \sup_{f \in B_1^*} \text{E} f^2 \left( \sum_{i=1}^n X_i ~\mathbf{I}\left\{\|X_i\| \leq \delta c_n\right\} - \text{E} X_i ~\mathbf{I} \left\{\|X_i\| \leq \delta c_n\right\} \right), \] where \(\delta>0\) and \(B_1^*\) is the unit ball of the topological dual \(B^*\) of the Banach space \(B\).
scientific article; zbMATH DE number 5916823