Approximate local limit theorems for laws outside domains of attraction (Q794053)

Let \((X_ k)\) be a sequence of i.i.d., nondegenerate random variables, and \(S_ n=\sum^{n}_{k=1}X_ k\). Define \(G(x)=P(| X|>x)\), \(K(x)=x^{-2}\int_{| y| \leq x}y^ 2dF(y)\), \(Q(x)=G(x)+K(x)\) for \(x>0\) and \((a_ n)\) by \(Q(a_ n)=n^{-1}\) for large n. It is proved that if \(\lim \sup_{x\to \infty}G(x)/K(x)<\infty\), then there exist \(\epsilon>0\), \(C>0\), such that for each \(M>0\), one can find a sequence \((b_ n)\) and a \(c>0\) for which \(c\leq a_ n P(S_ n\in(x- \epsilon,x+\epsilon))\leq C\) whenever \(| x-b_ n| \leq Ma_ n\) and n is sufficiently large. Other related questions are also discussed.