Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Another homework problem that is supposed to be "easy" but I can't see any way around. Any direction at all would be appreciated.

Let $X_{n}$ be iid with $\mathbb{E}[X] = 0$. Define $S_n = \sum\limits_{k=1}^{n} X_k$. Show that $\frac{S_n}{n^{1/p}} \rightarrow 0$ almost surely implies $\mathbb{E}[|X|^{p}] < \infty.$

My initial thoughts were around Jensen's Inequality and convexity, but that only applies to $f(x) = x^p$ for $p \geq 1$. I really have no idea where to go from here.

share|improve this question

2 Answers 2

up vote 5 down vote accepted

Here are some indications.

Fact 1: If $S_n/n^{1/p}$ converges to a finite limit, then $X_n/n^{1/p}$ converges to zero.

Fact 2: $\mathrm E(|X|^p)$ is finite if and only if $\sum\limits_n\mathrm P(|X|\ge n^{1/p})=\sum\limits_n\mathrm P(|X_n|\ge n^{1/p})$ converges.

Fact 3: For any sequence $(Y_n)$ of independent random variables, $Y_n\to0$ almost surely if and only if $\sum\limits_n\mathrm P(|Y_n|\ge y)$ converges, for every positive $y$.

For the proof of the problem, use fact 1, then the direct implication in fact 2, and finally the reverse implication in fact 3. This gives a stronger version of the result where one does not assume that $\mathrm E(X)=0$ nor that $S_n/n^{1/p}$ converges almost surely to $0$ but only that it converges almost surely to a finite limit.

Hint for the proof of fact 1: $X_n=S_n-S_{n-1}$.

Hint for the proof of fact 2: Start by recalling or reproving the classical equivalence that $\mathrm E(|X|)$ is finite if and only if $\sum\limits_n\mathrm P(|X|\ge n)$ converges.

Hint for the proof of fact 3: Borel-Cantelli lemma with independence for the direct implication, Borel-Cantelli lemma without independence for the reverse implication.

share|improve this answer
    
Got it! Though I didn't follow your order for the facts, I still used all of them to prove what I wanted. –  duckworthd Sep 19 '11 at 18:59

Let $\phi_X(t)$ be a characteristic function of $X$. Then $\phi_X(t) \sim 1 + o(t)$.

Now $\phi_{\frac{S_n}{n^{1/p}}}\left(t\right) = \left( \phi_X\left(\frac{t}{n^{1/p}}\right) \right)^n$. If $n^{-1/p} S_n \to 0$ a.s. then $\phi_{\frac{S_n}{n^{1/p}}}\left(t\right) \to 1$.

This is only possible when $\phi_X(t) \sim 1 + o(t^p)$.

share|improve this answer
    
This may be true, but I have yet to encounter characteristic functions of random variables in my coursework. Perhaps there is an alternative proof? –  duckworthd Sep 18 '11 at 6:40

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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