Strong Law Of Large Numbers And Complete Convergence For Arbitrary Stochastic Sequences

The first part of this paper, we study the strong law of large numbers for the sequences of arbitrary random variables. Before we study, we introduce a functionφ(y).Letφ(y) be a function which satisfies the follow conditions:(1) For some d≥0,φis strictly increasing on [d,∞) with range [0,∞);(2) There exist c and a positive integer k₀, such that (φ(y+1))/(φ(y))≤c,y≥k₀;(3) There exist constant a and b such thatφ²(s) integral from n=s to +∞1/(φ²(x))dx≤as+b, s＞d.The character of this function is that it can not only connect with several regular summability methods which we are familiar with, but also relate to nonregular summability methods which has more practical valuable. Then we proof the strong law of large numbers for the sequences of arbitrary random variables by a moment conditions which is between the moment conditions of the SLLN of Marcinkiewicz and the moment conditions of the SLLN of Kolmogrov. The next part of this paper, we add suitable moment conditions to the conditions of the first part of the theorem, then the results extend to the complete convergence of arbitrary random variables. The complete convergence of several special sequences that we often discuss such as NA sequences, PA sequences, martingale sequence, martingale difference sequence are the particular cases of the results of the theorem. Furthermore, because these special sequences have lots of prior properties, their convergence are stronger than the convergence of arbitrary random variables. Finally, in order to generalize the conclusions of the paper, the author discuss the conditions of the complete convergence fbr the sequences of arbitrary random variables.