A functional equation from probability theory

Baker, John A.

doi:10.1090/S0002-9939-1994-1186127-5

A functional equation from probability theory
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by John A. Baker PDF

Proc. Amer. Math. Soc. 121 (1994), 767-773 Request permission

Abstract:

The functional equation \begin{equation}\tag {$(1)$} f(x) = \prod \limits _{j = 1}^N {{{[f({\beta _j}x)]}^{{\gamma _j}}}}\end{equation} has been used by Laha and Lukacs (Aequationes Math. 16 (1977), 259-274) to characterize normal distributions. The aim of the present paper is to study (1) under somewhat different assumptions than those assumed by Laha and Lukacs by using techniques which, in the author’s opinion, are simpler than those employed by the afore-mentioned authors. We will prove, for example, that if $0 < {\beta _j} < 1$ and ${\gamma _j} > 0$ for $1 \leq j \leq N, \sum {_{j = 1}^N\beta _j^k{\gamma _j} = 1}$, where k is a natural number, $f:\mathbb {R} \to [0, + \infty )$, (1) holds for $x \in \mathbb {R}$ and ${f^{(k)}}(0)$ exists then either $f \equiv 0$ or there exists a real constant c such that $f(x) = \exp (c{x^k})$ for all $x \in \mathbb {R}$.

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