Likelihood Ratio and Strong Limit Theorems for the Discrete Random Variable

Abstract

This in virtue of the notion of likelihood ratio and the tool of moment generating function, the limit properties of the sequences of random discrete random variables are studied, and a class of strong deviation theorems which represented by inequalities between random variables and their expectation are obtained. As a result, we obtain some strong deviation theorems for Poisson distribution and binomial distribution.

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W. Li, W. Wang and Z. Liu, "Likelihood Ratio and Strong Limit Theorems for the Discrete Random Variable," Open Journal of Discrete Mathematics, Vol. 2 No. 4, 2012, pp. 169-172. doi: 10.4236/ojdm.2012.24034.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] W. Liu, “Relative Entropy Densities and a Class of Limit Theorems of Sequence of M-Valued Random Variables,” Annals of Probability, Vol. 18, No. 2, 1990, pp. 829-839. doi:10.1214/aop/1176990860 [2] W. Liu, “Strong Deviation Theorems and Analytic Method,” Science Press, Beijing, 2003. [3] W. Liu, “A Kind of Strong Deviation Theorems for the Sequence of Nonnegative Integer-Valued Random Variables,” Statistics & Probability Letters, Vol. 32, No. 4, 1997, pp. 269-276. [4] W. Liu, “Some Limit Properties of the Multivariate Function Sequences of Discrete Random Variables,” Statistics & Probability Letters, Vol. 61, No. 1, 2003, pp. 41-50. doi:10.1016/S0167-7152(02)00304-8 [5] W. G. Yang and X. Yang, “A Note on Strong Limit Theorems for Arbitrary Stochastic Sequences,” Statistics & Probability Letters, Vol. 78, No. 14, 2008, pp. 2018-2023. doi:10.1016/j.spl.2008.01.084 [6] G. R. Li, S. Chen and J. H. Zhang, “A Class of Random Deviation Theorems and the Approach of Laplace Transform,” Statistics & Probability Letters, Vol. 79, No. 2, 2009, pp. 202-210. doi:10.1016/j.spl.2008.07.048 [7] G. R. Li, S. Chen and S. Y. Feng, “A Strong Limit Theorem for Functions of Continuous Random Variables and an Extension of the Shannon-McMillan Theorem,” Journal of Applied Mathematics, Vol. 2008, 2008, pp. 1-10. doi:10.1155/2008/639145 [8] W. G. Yang, “Some Limit Properties for Markov Chains Indexed by a Homogeneous Tree,” Statistics & Probability Letters, Vol. 65, No. 3, 2003, pp. 241-250. doi:10.1016/j.spl.2003.04.001 [9] W. Liu, “Strong Deviation Theorems and Analytic Method (in Chinese) [M],” Science Press, Beijing, 2003. [10] J. L. Doob, “Stochastic Processes,” John Wiley & Sons, New York, 1953.