This paper is motivated by the problem of estimating the marginal distribution or quantiles of a linear process. For one dimensional transformations of Gaussian processes, direct estimation of the marginal probability distribution function proceeds from Hermite expansion of centered indicator functions, whereas for linear processes, one uses Appell polynomial expansions. In contrast to the empirical distribution function, in the goodness-of-fit literature, the empirical moment generating function is often used. In particular, statistics based on derivatives of this smooth function are plotted to reveal various distributional features. In this paper, we derive some asymptotic results concerning the empirical moment generating function of a stationary linear process with short and long memory serial correlations.