Concerning the Moment Convergence Result for Properly Normalised Delayed Sums

Abstract

Let {X_n, n\(\ge\)1} be a sequence of independent and identically distributed random variables with a common distribution function F. Let (S_n) be the partial sum sequence. Set $$T_n=S_{n+a_n}-S_n=\sum_{k=n+1}^{n+a_n} X_k$$. The sum T_n is referred to as a (forward) delayed sum. We derive a moment convergence result for the delayed sums when the random variables are within the domain of normal attraction of a stable law with an index \(\alpha\), 1 < \(\alpha\) < 2 The results can be used to obtain a density version of a local limit theorem.