Approximations to inverse moments of double-indexed weighted sums

In this paper, we investigate some approximations to inverse moments of double-indexed weighted sums of random variables. Let {Zn}{Zn} be a sequence of nonnegative independent random variables and {wni}{wni} be a triangular array of nonnegative non-random weights. With the finite first moment, we establish that E(a+∑i=1nwniZi)−α∼(a+∑i=1nwniEZi)−α holds for all a>0a>0 and α>0α>0. By the finite r  -th moment (r>2r>2), the convergence rate of approximation between E(a+∑i=1nwniZi)−α and (a+∑i=1nwniEZi)−α is presented. On the other hand, we obtain some approximations for the estimators of E(∑i=1nwniZia+∑i=1nZi), E(∑i=1nwniZia+∑i=1nZi)2 and Var(∑i=1nwniZia+∑i=1nZi) for all a>0a>0. Finally, some examples and simulations are illustrated.