Weird submartingale sequence, $X_n \to -\infty$ but $E(X_n) \to +\infty$.

Question

could you give an example of a sequence $(X_n)$ being a submartingale such that $X_n \to -\infty$ but $E(X_n) \to +\infty$?

Thanks a lot!

Answer 1

Let $Y_i$ be independent RVs with $P(Y_i=2^i)=1/i^2$ and then $P(Y_i=-1) = 1-(1/i^2)$. Note that $E[Y_i] >0, $ and $E[Y_i]$ is increasing in $i$, thus $S_n := \sum_{i=1}^n Y_i$ is a submartingale relative to $\mathcal{F}_n:= \sigma(Y_i,1\leq i \leq n)$. (It is an extremely biased random walk.)

Clearly $E[S_n] > n E[Y_1] \to \infty$ since $E[Y_1]>0$.

However, $P[Y_i=2^i, \mathrm{i.o.}] = 0$ by Borel-Cantelli so that $Y_i = -1$ eventually with probability 1. Thus $S_n \to -\infty$ a.s.