Geodesic
On probability density functions which are their own characteristic functions
Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 3, pp. 694-698

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Let $p$ be the probability density of a probability distribution $P$ on the real line $\mathbf R$ with respect to the Lebesgue measure. The characteristic function $\widehat p$ of $p$ is defined as $$ \widehat p(x):=\int_{\mathbf{R}}e^{ixy}p(y)\,dy,\qquad x\in\mathbf{R}. $$ We consider probability densities $p$ which are their own characteristic functions, that means \begin{equation} \widehat p(x)=\frac1{p(0)}p(x),\qquad x\in\mathbf{R}. \tag{1} \end{equation} 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 $\frac12$. 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.
Keywords: probability density functions, characteristic function, positive definite functions, Hermitian functions
Mots-clés : Fourier transform. 
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     author = {K. Schladitz and H. J. Engelbert},
     title = {On probability density functions which are their own characteristic functions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     url = {http://geodesic.mathdoc.fr/item/TVP_1995_40_3_a21/}
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K. Schladitz; H. J. Engelbert. On probability density functions which are their own characteristic functions. Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 3, pp. 694-698. http://geodesic.mathdoc.fr/item/TVP_1995_40_3_a21/