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

We have a sequence of independent random variables defined as $Y_n= a+n$ with probability $\frac{1}{n}$ and $Y_n=a$ with probability $1-\frac{1}{n}$.

  1. Does the sequence converge in probability?
  2. Does the sequence converge in mean square sense?

If it is convergent in either sense, what's the limit?

share|cite|improve this question

Hint: let $X_n=Y_n-a$. Then $P(|X_n|\geq 1)=\frac 1n$, so what can you deduce about convergence in probability?

We have $E[X_n^2]=(a+n)^2/n+a^2(1-1/n)$, so $E[X_n^2]$ is not bounded.

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.