$X_n$ is a sequence of independent random variables with $\mathbb{P}[X_n=n^2-1]=n^{-2},\mathbb{P}[X_n=-1]=1-n^{-2}$ and $Var[X_n]$ is unbounded. Set $S_n=X_1+…+X_n$. Prove that $\frac{S_n}{n} \rightarrow -1$ in probability.
Till now, I have proved that $\mathbb{E}[X_n]=0$. But then I am confused, no matter by weak law of large numbers or strong law of large numbers, this doesn't make sense to me that $\frac{S_n}{n}$ converges to -1. Can anyone explain to me what happened? Thanks!
Best Answer
The series $\sum\limits_nP[X_n\ne-1]$ converges hence $X_n=-1$ for every $n$ large enough, almost surely, say for every $n\geqslant N$. Thus, $S_n=S_N+N-n$ for every $n\geqslant N$, in particular $S_n/n\to-1$ almost surely (hence also in probability). The independence hypothesis is not needed.