On Moment Convergence Result for Properly Normalized Delayed Sums
Kokkada Vidyalaxmi
Department of Studies in Statistics, Manasagangothri, University of Mysore, Mysuru, 570 006, Karnataka, India.
Gooty Divanji *
Department of Studies in Statistics, Manasagangothri, University of Mysore, Mysuru, 570 006, Karnataka, India.
*Author to whom correspondence should be addressed.
Abstract
Let \(\left\{X_{n}, n \geq 1\right\}\) 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 called a (forward) delayed sum. The present work aims to obtain a moment convergence result for the delayed sums when the random variables are in the domain of normal attraction of a stable law with an index \(\alpha ,1, < \alpha < 2 \). This result plays a vital role in studying a local limit theorem.
Keywords: Domain of normal attraction, stable law, delayed sum, moment convergence, local limit theorem