# Prove that limsup of a sum of iid random variables(with 0 expectation) is infinity

Let $Y_1, Y_2, Y_3,\ldots,Y_n,\ldots$ be iid and bounded random variables with $E[Y_1]=0$. Define $X_n = Y_1+Y_2+ \cdots + Y_n$.

If $\Pr(Y_1 \neq 0) \gt 0$, then $\limsup X_n = \infty$ with probability $1$.

The conclusion seems intuitive, but how would I approach it rigorously?

• Hint: If you assume $\limsup X_n$ is a finite number ... can you show this leads to a contradiction? – WetlabStudent May 18 '15 at 18:48

Suppose that $\limsup X_n = b$ for some $b \in \mathbb{R}$. We know that $|Y_i| \leq c$ for some $c \in \mathbb{R}^{+}$. So $X_n \leq nc$. Now for some large $N$, $n \geq N$ means that $X_n \leq b$. But we can choose a large enough $n$ (say $m$) such that $b < X_m \leq mc$.