This paper discusses a new methodology for modeling non-Gaussian time series with long-range dependence. The class of models proposed admits continuous or discrete data and considers the conditional variance as a function of the conditional mean. These types of models are motivated by empirical properties exhibited by some time series. The proposed methodology is illustrated with the analysis of two real-life persistent time series. The first application is concerned with the modeling of stock market daily trading volumes, whereas the second application consists of a study of mineral deposit measurements. Copyright (C) 2010 John Wiley & Sons, Ltd.