Are the Hermite polynomials independent?

Question

I believe they are not independent as obviously, $$X := He_1(x) = x \; \text{and} \; Y:=He_2(x) = x^2-1$$ then $$P(Y\in A | X \in A) \neq P(Y \in A) \; \; \text{for some set } A \subset \mathbb{R}.$$

This is a counter-example (assuming the x's are distributed with Gaussian measure.)

But this seems too trivial, and so I suspect I may be wrong. Could anyone please verify my proof?

Answer 1

If $X$ is a random variable with a continuous distribution and $f$ is any non-constant polynomial, $X$ and $f(X)$ are never independent. Indeed, the conditional distribution of $f(X)$ given $X=x$ is the unit mass at $f(x)$, and this depends on $x$.