# Expected value of a log Poisson distribution

Suppose there is a sequence $(X_n)_n$ of independent random variables, $X_i \sim Poisson(\lambda)$.

I order to almost surely compute $\lim\limits_{n\to \infty} \sqrt[n]{X_1X_2\dots X_n}$, I thought of using the law of large numbers for $\ln(X_1) + \ln(X_2) + \dots + \ln(X_n)\over n$.

However, the natural logarithm is not defined for $0$, which is one value the random variables could take. I suppose that makes applying the law incorrect. Any other thoughts?

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Since ${\Bbb P}(X_i=0)>0$, a.s. infinitely many $X_i$ equal $0$, so the limit is a.s. $0$.