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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
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: distribution functions, Bernoulli distributions, singular functions, Schilling equation, Vieta's product
      Subjects: 39-xx Finite differences and functional equations > 39Bxx Functional equations and inequalities
      62-xx Statistics > 62Exx Distribution theory
      60-xx Probability theory and stochastic processes > 60Exx Distribution theory
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 16 Nov 2003
      Last Modified: 07 Sep 2014 21:27
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/90

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