Precise rate in the law of iterated logarithm for ρ-mixing sequence

Abstract

Let {X, X n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set $$S_n = \sum\nolimits_{k = 1}^n X _k ,M_n = max_{k \leqslant n} \left| {S_k } \right|,n \geqslant 1$$ . Suppose lim n→∞ $$ES_n^2 /n = :\sigma ^2 > 0$$ and $$\sum\limits_{n = 1}^\infty {\rho ^{2/d} \left( {2^n } \right)} < \infty $$ , where d=2, if −1 2(b+1), if b≥0. It is proved that, for any b>−1, $$\begin{gathered} \mathop {lim}\limits_{\varepsilon \searrow 0} \varepsilon ^{2\left( {b + 1} \right)} \sum\limits_{n = 1}^\infty {\frac{{\left( {log logn} \right)^b }}{{nlogn}}} P\left\{ {M_n \geqslant \varepsilon \sigma \sqrt {2nlog logn} } \right\} = \hfill \\ \frac{2}{{\left( {b + 1} \right)\sqrt \pi }}\Gamma \left( {b + 3/2} \right)\sum\limits_{k = 0}^\infty {\frac{{\left( { - 1} \right)^k }}{{\left( {2k + 1} \right)^{2b + 2} }}} \hfill \\ \end{gathered} $$ , where Γ(•) is a Gamma function.