Large deviation local limit theorems for ratio statistics |
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Authors: | Narasinga Rao Chaganty Sanjeev Sabnis |
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Affiliation: | 1. Department of Mathematics and Statistics , Old Dominion University , Norfolk, 23529, Virginia;2. Department of Mathematics , Indian Institute of Technology , Bombay, 400076, India |
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Abstract: | Let {Tn, n ≥ 1} be an arbitrary sequence of nonlattice random variables and let {Sn, n ≥ 1} be another sequence of positive random variables. Assume that the sequences are independent. In this paper we obtain asymptotic expression for the density function of the ratio statistic Rn = Tn/Sn based on simple conditions on the moment generating functions of Tn and Sn. When Sn = re, our main result reduces to that of Chaganty and Sethura-man[Ann. Probab. 13(1985):97-114]. We also obtain analogous results when Tn and Sn are both lattice random variables. We call our theorems large deviation local limit theorems for Rn, since the conditions of our theorems imply that Rn → c in probability for some constant c. We present some examples to illustrate our theorems. |
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Keywords: | Large Deviations Local Limit Theorems Saddle Point |
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