A class of random deviation theorems for sums of nonnegative stochastic sequence and strong law of large numbers

In this paper, the notion of limit log-likelihood ratio of random sequences, as a measure of dissimilarity   between the true density pn(x1,…,xn)(n=1,2,…) and the product of their marginals ∏i=1npi(xi), is introduced. Establish a.s. convergence supermartingale by means of constructing new probability density functions and under suitable restrict conditions, some random deviation theorems for arbitrary stochastically dominated continuous random variables and some strong law of large numbers are obtained.