Summary: This article studies the convolutions of Pascal random variables, i.e., negative binomial distributions with integer parameters. We show that the probability distributions of such convolutions can be expressed as a generalized mixture (i.e., mixture with negative and positive coefficients) of a finite number of Pascal distribution functions. Based on this result, further study on the limiting behavior of the failure rate function of the convolution is presented. A sufficient condition is given to establish the likelihood ratio order between two convolutions of Pascal random variables. Similar results are obtained for the convolutions of Erlang random variables.