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I have a task: $(X_n)_{n>=1}$ are independent. $P(X_n=0)=1/n$ and $P(X_n=2n)=1-1/n$. Check the weak convergence $\frac{X_1+X_2+X_3+....+X_n}{n}-n$. I tried use the Lapunow theorem or Linderberg theorem to $X_{n,k}=\frac{X_k}{n}-1$. But the variance converges to infinity.

Thanks in advance.

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