## Strong deviation theorems for the sequence of arbitrary random variables with respect to product distribution in random selection system

**Wang Kangkang1,Li Fang2(1.School of Mathematics and Physics,Jiangsu University of Science and Technology,Zhenjiang Jiangsu,212003,China)(2.School of Mathematics and Computer Science,An′hui Normal University,Wuhu An′hui,241000,China)**

As a measure of deviation between a sequence of the integer-valued random variables and a sequence of independent random variables with the biometric distribution,the notion of the limit logarithmic likelihood ratio was introduced in this paper.A subset of the sample space was given by restricting the likelihood ratio;and then,a class of limit theorems,which was represented by inequalities,was obtained based on this subset for the sequence of arbitrary integer-valued random variables on the gambling system.As corollaries,strong deviation theorems for arbitrary stochastic sequence on product biometric distribution were gotten.Moreover,a class of strong laws for sequences of independent random variables with biometric distributions was obtained.

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