  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; Sankhy\={a} Ser. A 60 (1998) 171-175; Sankhy\={a} Ser.
A 63 (2001) 128-132; J. Statist. Plann. Inference 92 (2001) 1-5; Bernoulli 17
(2011) 1044-1053).
