Revision as of 03:26, 9 June 2024 by Bot (Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> (Lévy<ref group="Notes" >See M. Krasner and B. Ranulae, “Sur une Proprieté des Polynomes de la Division du Circle”; and the following note by J. Hadamard, in ''C.\ R.\ Acad.\ Sci.,'' vol. 204 (1937), pp. 397--399.</ref>) Assume that <math>n...")
BBy Bot
Jun 09'24
Exercise
[math]
\newcommand{\NA}{{\rm NA}}
\newcommand{\mat}[1]{{\bf#1}}
\newcommand{\exref}[1]{\ref{##1}}
\newcommand{\secstoprocess}{\all}
\newcommand{\NA}{{\rm NA}}
\newcommand{\mathds}{\mathbb}[/math]
(Lévy[Notes 1]) Assume
that [math]n[/math] is an integer, not prime. Show that you can find two distributions [math]a[/math] and [math]b[/math] on the nonnegative integers such that the convolution of [math]a[/math] and [math]b[/math] is the equiprobable distribution on the set 0, 1, 2, \dots, [math]n - 1[/math]. If [math]n[/math] is prime this is not possible, but the proof is not so easy. (Assume that neither [math]a[/math] nor [math]b[/math] is concentrated at 0.)
Notes