Details

Autor(en) / Beteiligte

• Schladitz, K.
• Engelbert, H. J.

Titel

On Probability Density Functions Which are Their Own Characteristic Functions

Ist Teil von

• Theory of probability and its applications, 1996-01, Vol.40 (3), p.577-581

Ort / Verlag

Philadelphia: Society for Industrial and Applied Mathematics

Erscheinungsjahr

1996

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• 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.