Why is $H_j=-\infty$ almost surely when $E(H_1)\in [-\infty, 0)$

Question

If we have a random walk $(H_j)_{j \ge 0}$ such that $E(H_1) \in [-\infty, 0)$ why can we conclude that $\lim_{j\to\infty}H_j = -\infty$ almost surely?

Thanks in advance!

Answer 1

This is not true. Let $X_i$ i.i.d. s.t. $$\mathbb P\{X_i=\pm 1\}=\frac{1}{2}.$$ Let $$H_n=-1+X_1+...+X_n.$$ Then $\mathbb E[H_n]=-1<0$ for all $n\in\mathbb N$ but $$\lim_{n\to \infty }H_n=-\infty $$ doesn't hold.