Sequence of Random Variables with expectation 0.

Question

Is it possible to construct a sequence of non-negative random variables such that $X_n \rightarrow \infty $ but $\mathbb{E}[X_n] \rightarrow 0$ ? I found one which converges to 0 with expectation infinity but I'm stuck on this one. Any help will be appreciated.

Answer 1

Since $X_n \to \infty$ almost surely, we have

$$M \leq \liminf_{n \to \infty} X_n$$

for any $M>0$. It follows from Fatous lemma that

$$M \leq \mathbb{E} \left( \liminf_{n \to \infty} X_n \right) \leq \liminf_{n \to \infty} \mathbb{E}(X_n).$$

Since this holds true for any $M>0$, this shows

$$\lim_{n \to \infty} \mathbb{E}(X_n) = \infty.$$

Remark: The sequence of random variables $(X_n)_{n \in \mathbb{N}}$ defined by

$$X_n(\omega) := \begin{cases} n, & \omega \in \left[ \frac{1}{n},1 \right], \\ - n \left( n-1 \right), & \omega \in \left[0, \frac{1}{n} \right), \end{cases}$$

on the probability space $([0,1],\mathcal{B}([0,1]))$ (endowed with the Lebesgue measure) satisfies $\mathbb{E}X_n=0$ and $\lim_{n \to \infty} X_n = \infty$ almost surely. This shows that the assumption on the non-negativity is crucial.