We present a computationally efficient method to generate random variables from a univariate conditional probability density function (PDF) derived from a multivariate α-sub-Gaussian (αSG) distribution. The approach may be used to sequentially generate variates for sliding-window models that constrain immediately adjacent samples to be αSG random vectors. We initially derive and establish various properties of the conditional PDF and show it to be equivalent to a Student's t-distribution in an asymptotic sense. As the αSG PDF does not exist in closed form, we use these insights to develop a method based on the rejection sampling (accept-reject) algorithm that allows generating random variates with computational ease. HighlightsAn efficient method to generate random variates for a-sub-Gaussian processes with memory is presented.Properties of the univariate conditional α-sub-Gaussian distribution are investigated.Convergence of the aforementioned distribution to a Student's t-distribution is proven in an asymptotic sense.Using the above properties and tabulation of a heavy-tailed function, rejection sampling is used to generate realizations.The method may be used in simulation-based performance analysis of systems operating in colored α-sub-Gaussian noise.