A sequence of random variables with bounded first three moments

Question

Suppose $X_n$ is a sequence of random variables such that $$\mathbb{E}(X_n)=0, \mathbb{E}(X_n^2)=1, \mathbb{E}(X_n^3)\leq 1$$ Do we have the following result: for any $\epsilon>0$, $$\mathbb{E}(X_n^21\{X_n>\epsilon\sqrt{n}\})\to0$$ If this is true, is the boundedness condition on the third moment necessary? If this is false, what about when $\mathbb{E}(X_n^3)\to0$?

Answer 1

Let $X_n$ be a random variable such that $\Pr(X_n=\sqrt n)=\Pr(X_n=-\sqrt n)=1/(2n)$ and $\Pr(X_n=0)=1-1/n$. Then $X_n$ and $X_n^3$ have expectation $0$ and $\mathbb E\left(X_n^2\right)=1$. However, for $\varepsilon<1$, $X_n^21\{X_n>\epsilon\sqrt{n}\}=n\{X_n=\sqrt n\}$ hence $$\mathbb{E}\left(X_n^21\{X_n>\epsilon\sqrt{n}\}\right)=\frac 12.$$