The Annals of Probability

Large Deviation Local Limit Theorems for Arbitrary Sequences of Random Variables

Narasinga R. Chaganty and J. Sethuraman

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The results of W. Richter (Theory Probab. Appl. (1957) 2 206-219) on sums of independent, identically distributed random variables are generalized to arbitrary sequences of random variables $T_n$. Under simple conditions on the moment generating function of $T_n$, which imply that $T_n/n$ converges to zero, it is shown, for arbitrary sequences $\{m_n\}$, that $k_n(m_n)$, the probability density function of $T_n/n$ at $m_n$, is asymptotic to an expression involving the large deviation rate of $T_n/n$. Analogous results for lattice valued random variables are also given. Applications of these results to statistics appearing in nonparametric inference are presented. Other applications to asymptotic distributions in statistical mechanics are pursued in another paper.

Article information

Ann. Probab., Volume 13, Number 1 (1985), 97-114.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60F05: Central limit and other weak theorems
Secondary: 60F10: Large deviations

Local limit theorems large deviations Laplace transform nonparametric inference


Chaganty, Narasinga R.; Sethuraman, J. Large Deviation Local Limit Theorems for Arbitrary Sequences of Random Variables. Ann. Probab. 13 (1985), no. 1, 97--114. doi:10.1214/aop/1176993069.

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