MathDB

Let

X

and

Y

be independent random variables with "Saint-Petersburg" distribution, i.e. for any

k=1,2,\ldots

their value is

2^k

with probability

\frac{1}{2^k}

. Show that

X

and

Y

can be realized on a sufficiently big probability space such that there exists another pair of independent "Saint-Petersburg" random variables

(X', Y')

on this space with the property that

X+Y=2X'+Y'I(Y'\le X')

almost surely (here

I(A)

denotes the indicator function of the event

A

).
(translated by L. Erdős)

Problem Statement