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Assume that X, Y are two bounded random variables. If for any integers $m, n \geq 0$, $E[X^m Y^n] = E[X^m]E[Y^n]$, then X and Y are independent.

I've worked out that, for sure, if $E[f(X)g(Y)] = E[f(X)]E[g(Y)]$ for any functions $f(X)$, $g(Y)$, then they must be independent (which can be proven by assuming $P(X = x$ and $Y = y) \neq P(X = x)P(Y = y)$, taking f, g to be characteristic functions, and reaching a contradiction). However, I can't think of any way to weaken the condition to just whole-number powers, either directly or indirectly (through the "any function" case). I suppose I could try using a Taylor series expansion, but that seems way off-base for this problem, and like it wouldn't even cover most functions, since that really only applies to differentiable ones.

Xindaris
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1 Answers1

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Just notice that you can approximate the characteristic function of (of points) by polynomials. Also notice that if $f,g$ are polynomials, then the above condition gives $E(f(X)g(Y))=E(f(X))E(g(Y))$

voldemort
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