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.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

From WIkipedia

the central limit theorem states that the sums Sn scaled by the factor $1/\sqrt{n}$ converge in distribution to a standard normal distribution. Combined with Kolmogorov's zero-one law, this implies that these quantities converge neither in probability nor almost surely: $$ \frac{S_n}{\sqrt n} \ \stackrel{p}{\nrightarrow}\ \forall, \qquad \frac{S_n}{\sqrt n} \ \stackrel{a.s.}{\nrightarrow}\ \forall, \qquad \text{as}\ \ n\to\infty. $$

How do Kolmogorov 0-1 law and CLT imply normalized sample mean doesn't converge in probability nor a.s.?

Thanks!

share|cite|improve this question
    
I wrote up the argument here: math.stackexchange.com/questions/210131/… – Chris Janjigian Oct 18 '12 at 13:42
    
Thanks, @Chris! – Tim Oct 18 '12 at 13:49

Your Answer

 
discard

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

Browse other questions tagged or ask your own question.