Some inequalities of linear combinations of independent random variables: II

Abstract

Linear combinations of independent random variables have been extensively studied in the literature. However, most of the work is based on some specific distribution assumptions. In this paper, a companion of (J. Appl. Probab. 48 (2011) 1179–1188), we unify the study of linear combinations of independent nonnegative random variables under the general setup by using some monotone transforms. The results are further generalized to the case of independent but not necessarily identically distributed nonnegative random variables. The main results complement and generalize the results in the literature including (In Studies in Econometrics, Time Series, and Multivariate Statistics (1983) 465–489 Academic Press; Sankhyā Ser. A 60 (1998) 171–175; Sankhyā Ser. A 63 (2001) 128–132; J. Statist. Plann. Inference 92 (2001) 1–5; Bernoulli 17 (2011) 1044–1053).