The convergence of the Fourier integral of a unimodal distribution

Abstract

Abstract. We show that the characteristic function of a unimodal probability distribution function can be inverted by the Fourier transform a.e. if and only if the distribution is absolutely continuous. The result complements Khintchine's criterion for unimodal distributions.The distribution function F (x) = Pr(X ≤ x) of a random variable X is unimodal with vertex at the origin if F (x) is convex for x < 0 and concave for x > 0. Some examples of unimodal distributions are: i) the standard normal, exponential and Cauchy probability densities; ii) the class of stable distributions; iii) the class of symmetric infinitely divisible distributions. According to Khintchine's criterion ( for 0 < x ≤ t < ∞, −t −1 for − ∞ < t ≤ x < 0, 0 otherwise.