Open Access
February, 1985 Large Deviation Local Limit Theorems for Arbitrary Sequences of Random Variables
Narasinga R. Chaganty, J. Sethuraman
Ann. Probab. 13(1): 97-114 (February, 1985). DOI: 10.1214/aop/1176993069


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.


Download Citation

Narasinga R. Chaganty. J. Sethuraman. "Large Deviation Local Limit Theorems for Arbitrary Sequences of Random Variables." Ann. Probab. 13 (1) 97 - 114, February, 1985.


Published: February, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0559.60030
MathSciNet: MR770631
Digital Object Identifier: 10.1214/aop/1176993069

Primary: 60F05
Secondary: 60F10

Keywords: Laplace transform , large deviations , Local limit theorems , nonparametric inference

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 1 • February, 1985
Back to Top