## The Annals of Probability

- Ann. Probab.
- Volume 2, Number 2 (1974), 270-276.

### Density Versions of the Univariate Central Limit Theorem

#### Abstract

Let $\{X_n\}$ be a sequence of independent random variables each with a finite expectation and a finite variance. Write $Z_n$ for the standardized sum of $X_1, X_2, \cdots, X_n$ and suppose that for all large $n, Z_n$ has a probability density function which we denote by $h_n(x)$. It is well known that the usual assumptions of the Central Limit Theorem do not necessarily imply the convergence of $h_n(x)$ to the standard normal density $\phi(x)$. In this study, we find a set of sufficient conditions under which the relation $$\lim_{n\rightarrow\infty} |x|^k|h_n(x) - \phi(x)| = 0$$ holds uniformly with respect to $x\in (-\infty, +\infty), k$ being an integer greater than or equal to 2.

#### Article information

**Source**

Ann. Probab., Volume 2, Number 2 (1974), 270-276.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996708

**Digital Object Identifier**

doi:10.1214/aop/1176996708

**Mathematical Reviews number (MathSciNet)**

MR356176

**Zentralblatt MATH identifier**

0304.60012

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60E05: Distributions: general theory 62E15: Exact distribution theory

**Keywords**

Central Limit Theorem smoothing subsequence Lindeberg condition of order $k$

#### Citation

Basu, Sujit K. Density Versions of the Univariate Central Limit Theorem. Ann. Probab. 2 (1974), no. 2, 270--276. doi:10.1214/aop/1176996708. https://projecteuclid.org/euclid.aop/1176996708