The Study Of Convergences And Inequalities For Some Random Variables

Borel obtained the strong law of large numbers through studing the Bernoulli trails, and an extended Borel strong law of large numbers was mentioned in some references. In this paper We improved the extended Borel strong law of large numbers, whose condition dn=O(1/n) was weakened to dn=O(1/nα), a>0or dn=O(Cn), Cn≥0,(?)m∈N+, m≥2such that (?)Cknm<∞. Furthermore, we generalize those results to the case of the bounded random variables and get their Borel strong law of large numbers. If{Xn,n≥1} is an arbitrary sequence of random variables, some convergence results for the partial sums of arbitrary sequence of random variables are obtained, which generalize the known results for independent sequences, NA sequences p-mixing sequences and φ-mixing sequences, etc. In addition, If{Xn,n≥1} is a M-Z-type sequence of random variables and Sn(a)=(?)Xi, n≥1, a≥0, where{Xn,n≥1} was defined in Lp, maximal inequalities and large deviations for M-Z-type sequence of random variables were discussed:for p>2,μ(|Sn (a)|>n)≤cn-p/2, and for1<p≤2,μ(|Sn(a)|>n)≤cn1-At last, we obtain and prove the large deviations for maximal sequence max Sk (a) and the partial sums of mixing sequence Sn (a).