THE COMPLETE CONVERGENCE FOR PARTIAL SUMS OF INDEPENDENT RANDOM VARIABLES

In this paper, by making use of the equivalenoe between the a. s, eonvergenoe and the convergence in probability for the partial sums of independent random variables, we improved and strengthened a series of Katz and Baum’s results to the case that the random variables need not required to be iid. On the above basis, we got results about the complete convergenoe for the randomly seleoted partial sums of independent random variables. Our main result is the following: Theorem 1. Let X₁,X₂,… be a sequence of independent random variables r＞1, 0<t＜2, and l（x）＞0 be a slowly variable funetion as x→∞. If we have （ⅰ） \sum_k=1^n E\left|X_k-b_k\right|^r^\prime t=O\left(n^1+\alpha\right) for some r’＞1, where b_k=EX_k·I （r’t＞1）+O·I （r’t≤1） and 0≤a＜r’（1+t/2）·I（r’t＞2）+（r’-1）·I（r’t≤2）, （I（A）denotes the indioator function of the set A.） （ⅱ）\sum_n=1^\infty E\left|X_n-b_n\right|^t(r-1) l\left(\left|X_n-b_n\right|^t\right) I\left(X_n-b_n \mid \geqslant \varepsilon \cdot n^1 / t\right)< +\infty for any 8>0, then we obtain that （ⅲ） \sum_n=1^\infty n^r-2 l(n) P\left(\max _1< i< n \mid \sum_i=1^k\left(X_i-b_i\right) \geqslant \varepsilon \cdot n^1 / t\right)<+\infty for any \varepsilon>0 and (iv) \sum_n=1^\infty n^r-2 l(n) P\left(\sup _k >n\left(\left|\sum_i=1^k\left(X_i-b_i\right)\right| / k^1 / t\right) \geqslant 8\right)< +\infty for any 8>0. Conversely, if （ⅲ） or （ⅳ） holds for some sequemee b_n, then （ⅱ） holds.