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Let $X_n$ a sequence of random variables such that $X_n\sim \text{Poisson}(\frac 1n)$. Study the almost-sure convergence of $X_n$.

Since $X_n$ is integer-valued and $P(X_n=0) = \exp(-\frac 1n)$ it is easy to prove that $X_n$ converges to $0$ in probability.

Note that $P(X_n\geq 1) = 1-\exp(-\frac 1n)\sim \frac 1n$, hence $\sum_n P(X_n \geq 1) = \infty$. If the $X_n$ are independent, Borel-Cantelli lemma yields $P\left(\limsup_n \left(X_n\geq 1\right)\right)=1$, hence $X_n$ does not converge to $0$ almost surely.

What can be said when the $X_n$ are not independent ?

If the events $(X_n\geq 1)$ are negatively correlated, a stronger version of Borel-Cantelli (derived from Kochen-Stone lemma) still yields $P\left(\limsup_n \left(X_n\geq 1\right)\right)=1$ (see this).

If $X_n\to 0$ a.s, then $P(\liminf_n (X_n=0))=1$ but I haven't been able to get anything useful out of this.

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Here is an example where the almost sure convergence to zero holds. Let $\left(\xi_i\right)_{i\geqslant 1}$ be an independent sequence of random variables, where $\xi_i$ has a Poisson distribution of parameter $i^{-1}-(i+1)^{-1}$. Define $X_n:=\sum_{i\geqslant n}\xi_i$ (one can check that these random variables are well-defined and that $X_n$ has a Poisson distribution of parameter $1/n$). Then $X_n\to 0$ almost surely.

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