Approximate Local Limit Theorems for Laws Outside Domains of Attraction
Approximate Local Limit Theorems for Laws Outside Domains of Attraction
Let $\{X_k\}$ be a sequence of independent, identically distributed, nondegenerate random variables and $S_n = X_1 + \cdots + X_n$. Define $G(x) = P\{|X| > x\}, K(x) = x^{-2} \int_{|y| \leq x}y^2 dF(y), Q(x) = G(x) + K(x)$ for $x > 0$, and $\{a_n\}$ by $Q(a_n) = n^{-1}$ for large …