Relationship between moments

Question

Suppose a random variable $\mathbf{X}$ with some statistical distribution. Let's say the $E[\cdot]$ is the expectation operator. Is it possible to establish a relation between $E[ \mathbf{X}^n]$ and $E[ \mathbf{X} ]^n$? I would like to see something like $E[ \mathbf{X}^n ]= f(\omega, n) \times E[ \mathbf{X} ]^n$ for any existing distribution. Can someone give me a hint on how I can design this?

Answer 1

As stated, this is not possible. For example, we can let $X$ be a standard normal $N(0,1)$ random variable. Then $\mathbb{E}[X^2] = 1$ but $\mathbb{E}[X]^2 = 0$ and we cannot have $1 = f(\omega, 2) \cdot 0 = 0$ for any function $f$. Even if we replace $\mathbb{E}[X]^n$ with $\mathbb{E}[|X|]^n$ this can't work because there are distributions where $\mathbb{E}[|X|] < \infty$ but $\mathbb{E}[X^2] = \infty$. The only relation you could get with certainty is that $\mathbb{E}[|X|^n] \ge \mathbb{E}[|X|]^n$ by Holder's inequality.

Answer 2

I'm afraid you can find centered random variable with arbitrary moment of order $n$, if $n$ is fixed.

What is the $\omega$ in your $f(\omega, n)$ ?