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April, 1974 Density Versions of the Univariate Central Limit Theorem
Sujit K. Basu
Ann. Probab. 2(2): 270-276 (April, 1974). DOI: 10.1214/aop/1176996708


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.


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Sujit K. Basu. "Density Versions of the Univariate Central Limit Theorem." Ann. Probab. 2 (2) 270 - 276, April, 1974.


Published: April, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0304.60012
MathSciNet: MR356176
Digital Object Identifier: 10.1214/aop/1176996708

Primary: 60F05
Secondary: 60E05 , 62E15

Keywords: central limit theorem , Lindeberg condition of order $k$ , smoothing subsequence

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 2 • April, 1974
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