## The Annals of Probability

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

### Large Deviation Local Limit Theorems for Arbitrary Sequences of Random Variables

Narasinga R. Chaganty and J. Sethuraman

#### 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