Upper bound on expected value $E[X^2]$

Question

Assume $X_n$ is a sequence of positive random variables such that $E[X_n]=\mathcal{O}(f(n))$ as ${n \to \infty}$, where $f$ is a function. Is the following true? If yes, prove it: $$E[X_n^2]=\mathcal{O}(f^2(n))\text{ as } {n \to \infty}$$ If it is not generally true, is there a set of assumptions under which the above equation is true?

Using Jensen's inequality: $E[X_n^2]\geq E[X_n]^2$ which does not help. Any idea how to solve this?

Answer 1

About all you can say is this. If $0 \le X \le b$ a.s., then $\mathbb E[X^2] \le b\; \mathbb E[X]$. This is best possible, in that equality holds when the only possible values of $X$ are $0$ and $b$.

Answer 2

It's false. For example, setting $X_n$ to be $n$ with probability $\frac1n$ and $0$ otherwise gives us $E[X_n] = 1$ for all $n$, but $E[X_n^2] = n$.