Grinstead-Snell-Probability · Problem 7.1.10

Sums of Random Variables

Mathematics · sums-discrete-random-variables-convolution

Problem

(Lévy) Assume that

n

is an integer, not prime. Show that you can find two distributions

a

and

b

on the nonnegative integers such that the convolution of

a

and

b

is the equiprobable distribution on the set

0, 1, 2, \ldots, n-1

. If

n

is prime this is not possible, but the proof is not so easy. (Assume that neither

a

nor

b

is concentrated at 0.)