Suppose you have a sequence of independent random variables {$X_i, i\geq1$}, such that $$\Bbb P(X_i=i^2 -1)=i^{-2},$$ $$ \Bbb P(X_i=-1)=1-i^{-2}.$$ Then, $\frac1n \sum_{i=1}^n X_i$ converges almost surely to a constant c. Find the value of c.
For this question, I thought of applying the strong law of large numbers which says: $$ \frac1n \sum_{i=1}^n X_i$$ converges almost surely to $\mu$, where {$X_i, i\geq1$} is a sequence of IID random variables whose mean $\mu$ exists.
However, for the question above, $\mu=0$. But c is not equal to $0$. c is, in fact, equal to -1.
Any clarification on this would greatly be appreciated, as I'm not sure how $c=-1$.
Use the Borel-Cantelli lemma. Since $\sum_{n \geq 1} \mathbb{P}(X_n \neq -1) = \sum_{n \geq 1} n^{-2} < \infty$, we know that almost surely, $X_n = -1$ for all $n$ sufficiently large. Then the desired result follows easily: if $(a_n)$ is a sequence that is eventually constant, it's straightforward to verify that $\frac{1}{N} \sum_{n=1}^{N} a_n$ converges to that constant as $N \to \infty$.