# Functional Equations and Distribution Functions

Borwein, Jonathan M. and Girgensohn, Roland (1994) Functional Equations and Distribution Functions. Results in Mathematics, 26 (3-4). pp. 229-237.

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## Abstract

We consider the functional equation $\ds f(t) = {1 \over b} \sum_{\nu=0}^{b-1} f\left({t-\beta_\nu \over a}\right) \quad \mbox{for all } t \in \R,$ where $0<a<1$, $b \in \N \setminus \{1\}$ and $-1 = \beta_0 \le \beta_1 \le \dots \le \beta_{b-1}=1$ are given parameters, $f:\R \to \R$ is the unknown. We show that there is a unique bounded function $f$ which solves (F) and satisfies $f(t)=0$ for $t<-1/(1-a)$, $f(t)=1$ for $t>1/(1-a)$. This solution can be interpreted as the distribution function of a certain random series. It is known to be either singular or absolutely continuous, but the problem for which parameters it is absolutely continuous is largely open. We collect some previously established partial answers and generalize them. We also point out an interesting connection to the so-called Schilling equation.

Item Type: Article pubdom FALSE distribution functions, Bernoulli distributions, singular functions, Schilling equation, Vieta's product 39-xx Finite differences and functional equations > 39Bxx Functional equations and inequalities62-xx Statistics > 62Exx Distribution theory60-xx Probability theory and stochastic processes > 60Exx Distribution theory UNSPECIFIED Users 1 not found. 16 Nov 2003 07 Sep 2014 21:27 https://docserver.carma.newcastle.edu.au/id/eprint/90

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