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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?

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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.

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