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

Suppose there is a sequence $(X_n)_n$ of independent random variables, $X_i \sim Poisson(\lambda)$.

I order to almost surely compute $\lim\limits_{n\to \infty} \sqrt[n]{X_1X_2\dots X_n}$, I thought of using the law of large numbers for $\ln(X_1) + \ln(X_2) + \dots + \ln(X_n)\over n$.

However, the natural logarithm is not defined for $0$, which is one value the random variables could take. I suppose that makes applying the law incorrect. Any other thoughts?

share|cite|improve this question
up vote 6 down vote accepted

Since ${\Bbb P}(X_i=0)>0$, a.s. infinitely many $X_i$ equal $0$, so the limit is a.s. $0$.

share|cite|improve this answer

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.