On the complete convergence of sequences of random elements in Banach spaces

Abstract

For a sequence $${\{X_{n}, {n \geqslant 1}\}}$$ of independent random elements taking values in a Rademacher type p Banach space with the k-th partial sum $${S_{k} (k \geqslant 1)}$$ , we provide necessary and sufficient conditions for the convergence of $${\sum_{n=1}^{\infty} \frac{1}{n}\,\mathbb{P} ({\rm max}_{1 \leqslant{k}\leqslant{n}} \|S_{k}\| > \varepsilon{n}^{\alpha})}$$ and $${\sum_{n=1}^{\infty} \frac{{\rm log} n}{n} \, \mathbb{P} (\max_{1\leqslant{k}\leqslant{n}} \|S_{k}\| > \varepsilon{n}^{\alpha})}$$ for every $${\varepsilon > 0}$$ .