The Annals of Probability

Large Deviation Local Limit Theorems for Arbitrary Sequences of Random Variables

Narasinga R. Chaganty and J. Sethuraman

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993069

Digital Object Identifier
doi:10.1214/aop/1176993069

Mathematical Reviews number (MathSciNet)
MR770631

Zentralblatt MATH identifier
0559.60030

JSTOR
links.jstor.org

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

Keywords
Local limit theorems large deviations Laplace transform nonparametric inference

Citation

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. https://projecteuclid.org/euclid.aop/1176993069


Export citation