# Almost sure convergence of $\text{Poisson}(\frac 1n)$ to $0$

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.

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.