Existence of Moments in Distributions of the Form Tan(X)

Abstract

In this work, we consider the existence of the moments of functions of random variables supported on a bounded interval. Our approach begins by working with an arbitrary diffeomorphism, but later we restrict attention to the tan function–the corresponding distribution is a generalization of the Cauchy distribution, which is derived when one applies tan to a uniformly distributed variable. For a continuous random variable X, we derive a necessary and sufficient condition for the existence of a moment of a given order of the distribution of tan(X) in terms of the behaviour of the probability density of X near the points ± \(\frac{\pi}{2}\). As a consequence, we obtain classes of examples, somewhere the moments exist and somewhere they do not at all.