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.