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.

Strong Law of Large Numbers demonstrates if $X_1,\,X_2,\,\ldots$ are i.i.d. random variables with $\mathbb{E}|X|<\infty$, $S_n:=\sum_{k=1}^n X_k$, then $$\frac{S_n}{n}\to \mathbb{E} X\quad\text{a.s.}$$ My question is can you give an example such that $\frac{S_n}{n}\not\to \mathbb{E} X$ in $L^1$?

share|improve this question
add comment

1 Answer 1

up vote 1 down vote accepted

No you cannot find such an example since $$ \frac{S_n}{n}\to E[X]\quad \text{in }L^1. $$ It follows from the fact that it converges almost surely and that $$ \left\{\frac{S_n}{n}\;\bigg|\; n\geq 1\right\} $$ is uniformly integrable.


Hint on how to show the uniform integrability property:

  1. Using the iid assumption, show that $\{X_n\mid n\geq 1\}$ is uniform integrable.

  2. Now show that $\left\{\frac{S_n}{n}\mid n\geq 1\right\}$ is uniformly integrable using for example the equivalent formulation of uniform integrability $(*)$.

$(*)$ A sequence $(X_n)_{n\geq 1}$ is uniformly integrable if and only if $$ \sup_{n\geq 1}\int |X_n|\,\mathrm dP<\infty $$ and for all $\varepsilon>0$ there exists a $\delta>0$ such that if $P(A)\leq \delta$ then $$ \sup_{n\geq 1}\int_A |X_n|\,\mathrm dP\leq \varepsilon. $$

share|improve this answer
    
Perhaps you could add some hint how to prove the uniform integrability. –  saz Feb 26 '13 at 8:46
    
@Stephan Hensan Thanks indeed. I'm not sure if the second condition in your hint is easily applied. But perhaps a better way is to realize that $\frac{S_n}{n}$ is a backward martingale. Another way is to check directly by definition that $\frac{S_n}{n}$ is $U.I.$ Thank you a second time. –  Coiacy Feb 28 '13 at 2:55
add comment

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.