The Hàjek-Rènyi inequlity for Banach space valued random variable sequences and its application

Abstract

In this paper we prove the following Hàjek-Rènyi inequality: Let 0≤p≤1, then for any Banach spaceB, anyL ^p integrableB valued random variable sequence {D _n , n≥1}, any real number sequence {b _n , n≥1} with 0<b _n, ↑ ∞, any integern≥1, there exist a constantC=C _p>0 (only depending onp) such that

$$P\mathop {(\sup }\limits_{j \geqslant n} \frac{{||\sum\limits_{i = 1}^j {D_1 ||} }}{{b_1 }} \geqslant \in ) \leqslant C \in ^{ - p} \sum\limits_{j = n + 1}^\infty {\frac{{E||D_j ||^p }}{{b_j^p }}} + \sum\limits_{j = 1}^n {\frac{{E||D_j ||^p }}{{b_n^p }}} )$$

In the other direction, we prove some strong laws of large numbers and the integrability of the maximal functions forB valued random variable sequences by using this inequality and the Hàjeck-Rènyi inequality we have obtained recently. Some known results are extended and improved.