Student's t-process is a parametrized generalization of the Gaussian process, which is widely used for the analysis of heavy-tailed data. Although these processes share several principal properties, an important difference can be observed once their Renyi entropy rates are analyzed instead of the commonly considered Shannon entropy rate. In this talk, we show that stationary Student's t-processes have, in general, a nonfinite Renyi entropy rate, while a specific class of nonstationary Student's t-processes has a finite Renyi entropy rate, which is fully opposed to the Shannon case. After that, we extend considerations to the generalized statistical complexity rates, which represent an interplay between order and disorder levels of a random process. We show that the statistical complexity rates are invariant with respect to the stationarity of the Student's t-process while keeping information about its shape parameter.