The Annals of Probability

Density Versions of the Univariate Central Limit Theorem

Sujit K. Basu

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F05: Central limit and other weak theorems
Secondary: 60E05: Distributions: general theory 62E15: Exact distribution theory

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


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

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