The closure property of the up shifted likelihood ratio order under convolutions was first proved by Shanthikumar and Yao (1986) by establishing a stochastic monotonicity property of birth-death processes. Lillo, Nanda and Shaked (2000) made a slight extension of this closure property for any random variables with interval supports by using the result of Shanthikumar and Yao (1986). A new analytic proof of the closure property is given, and the method is applied to establish another result involving the up shifted hazard rate and reversed hazard rate orders.