On Probability Density Functions Which are Their Own Characteristic Functions

Abstract

Let p be the probability density of a probability distribution P on the real line ${\bf R}$ with respect to the Lebesgue measure. The characteristic function $\widehat p$ of p is defined as \[ \widehat p(x): = \int_{\bf R} {e^{ixy} p(y)} dy,\quad x \in {\bf R}. \] We consider probability densities p which are their own characteristic functions, that means \[ (1)\qquad \widehat p(x) = \frac{1}{{p(0)}}p(x),\quad x \in {\bf R}. \] By linear combination of Hermitian functions we find a family of probability densities which are solutions of this integral equation. These solutions are entire functions of order 2 and type $\tfrac{1}{2}$. This is contradictory to Corollary 3 in [J. L. Teugels, Bull. Soc. Math. Belg., 23 (1971), pp. 236–262.].Furthermore, we characterize the general solution of the integral equation (1) within the convex cone of probability density functions.