4
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ Hint: If you assume $\limsup X_n$ is a finite number ... can you show this leads to a contradiction? $\endgroup$ – WetlabStudent May 18 '15 at 18:48
1
$\begingroup$

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 $b < X_m \leq mc$.

Note: This is just my attempt. Please feel free to continue OP.

$\endgroup$
  • 1
    $\begingroup$ I'm not sure if this is meant as a hint or a proof. Hints are preferred for homework-style questions. That being said, your last sentence needs motivation: perhaps the OP can supply this. $\endgroup$ – ekvall May 19 '15 at 0:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.