On a Local Limit Theorem Concerning Variables in the Domain of Normal Attraction of a Stable Law of Index $\alpha, 1 < \alpha < 2$

Abstract

Let $\{X_n\}$ be a sequence of independent and identically distributed random variables with $EX_1 = 0$. Suppose that there exists a constant $a > 0$, such that $Z_n = (an^r)^{-1}(X_1 + X_2 + \cdots + X_n)$ converges in law to a stable distribution function (df) $V(x)$ as $n \rightarrow \infty$. If, in addition, we assume that the characteristic function of $X_1$ is absolutely integrable in $m$th power for some integer $m \geqq 1$, then for all large $n$, the df $F_n$ of $Z_n$ is absolutely continuous with a probability density function (pdf) $f_n$ such that the relation $$\lim_{n\rightarrow\infty}|x\|f_n(x) - \nu(x)| = 0$$ holds uniformly in $x, -\infty < x < \infty$, where $v$ is the pdf of $V$.