Sign up ×
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.

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

1 Answer 1

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


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.