Hello


Here is a little problem for which I have no clue, and I don't even know if it is difficult.


Does there exist a measurable (!) function $\psi:[0,1]^2\mapsto [0,1]$ such that if $(X\_i)\_i$ is a sequence of iid uniform variables on $[0,1]$, then the $\psi(X\_i,X\_j), i< j$ are indepdendant (and of course identically distributed) variables?


The setting of the problem ($[0,1]$, uniform law, ...) can be changed, the only requirement is that the support of the $X\_i$ and the arrival space have at least two values.


For info, the examples 1) $X\_i$ are Bernoulli and $\psi(x,y)=x y$ and 2) $X\_i$ are uniform and $\psi(x,y)$ is the congruence of $x+y$ modulo $1$ do not work.


Any remark is welcome too...



Transfer independance from $\mathbb{N}$ to $\mathbb{N}^2$
If the sequence $X\_1,X\_2,\ldots$ has finite range, then this is impossible except in the trivial way mentioned by Emil.


The "input" random variables (e.g. $X\_1$) all have equal and finite entropy $a$ and the "output" random variables (e.g. $\phi(X\_1,X\_2)$) all have equal entropy $b$. By independence the entropy of $(X\_1,\ldots, X\_n)$ is $na$. There are $\binom{n}{2}$ independent output random variables defined deterministically in terms of these $n$ inputs, so their total entropy can be no greater: $\binom{n}{2}b \leq na$. Letting $n$ go to infinity we obtain $b=0$, i.e., the outputs are constant almost surely.


