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If I define a series of random variables to be $X_n \sim N(0,n)$, does $X_n$ converge almost surely to any limits? Intuitively this is not the case, but I am not sure how to formally show this, would anyone have any hints? thanks.

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Almost sure convergence implies convergence in distribution. Now, assume almost sure convergence and assume that $X_n\rightarrow X$. Since variance approaches $\infty$, for any finite set $A$, $P(X_n\in A)\rightarrow 0$.

Since $X$ is a random variable, we can pick a finite set $A$ for which $P(X\in A)>0$ which yields a contradiction since we have that \begin{equation} 0=\lim_{n\rightarrow\infty}P(X_n\in A)=P(X\in A)>0 \end{equation}

Consequently, $X_n$ does not converge almost surely (or in distribution).

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