Local limit theorem in the case of convergence to a law of class L (Q1106537)

Suppose \(\{\xi_ 1,\xi_ 2,...\}\) is a sequence of integer-valued independent random variables. Let \(S_ n\) denote its partial sum of first n elements. Assume \(\{B_ n\}\), \(\{a_ n\}\) are two sequences of constants such that \(S_ n/B_ n-a_ n\) tends in distribution to an L- class distribution with density p. Then the local limit theorem for the sequence \(\{\xi_ n\}\) is defined as \[ \sup_{m}| B_ nP(S_ n=m)-p(m/B_ n-a_ n)| \to 0,\quad as\quad n\to \infty. \] Certain sufficient conditions are given for guaranteeing the local limit theorem.