The paper investigates Lp convergence and Marcinkiewicz-Zygmund strong laws of large numbers for random elements in a Banach space under the condition that the Banach space is of Rademacher type p, 1 〈 p 〈 2. The paper also discusses Lτ convergence and Lτ bound for random elements without any geometric restriction condition on the Banach space.
In the paper we extend and generalize some results of complete moment convergence results （or the refinement of complete convergence） obtained by Chow [On the rate of moment complete convergence of sample sums and extremes. Bull. Inst. Math. Academia Sinica, 16, 177-201 （1988）] and Li ＆ Spataru [Refinement of convergence rates for tail probabilities. J. Theor. Probab., 18, 933-947 （2005）] to sequences of identically distributed φ-mixing random variables.
The present paper first shows that, without any dependent structure assumptions for a sequence of random variables, the refined results of the complete convergence for the sequence is equivalent to the corresponding complete moment convergence of the sequence. Then this paper investigates the convergence rates and refined convergence rates （or complete moment convergence） for probabilities of moderate deviations of moving average processes. The results in this paper extend and generalize some well-known results.
