Let $p\in(0,1)$. For each $n\in\mathbb{N}$, consider a random variable $X_n$, $X_n$ has binomial distribution with parameters $n$ and $p$. The $X_n$ are independent. Compute, for each $\epsilon>0$
\begin{equation} \displaystyle \mathrm{P}(\limsup_{n\longrightarrow \infty} X_n\geq n^{1-\epsilon}) \end{equation}
My attempt:
- First, I think in this property. For each event $A_n$ \begin{equation} \displaystyle \mathrm{P}(\limsup A_n)\geq \limsup\mathrm{P}( A_n) \end{equation} From my point of view, this is useful only if $\limsup_{n\longrightarrow \infty}\mathrm{P}( X_n\geq n^{1-\epsilon})=1$. But I don't know how to prove it.
2.(I know this point gives a total different result). I tried to use The First Borel-Cantelli Lemma, i.e., if $\sum_{n=1}^\infty P(A_n)<\infty\Rightarrow P(A_n i.o.)=0$. But I don't know how to define the summands such that the series converges.
My problem is I don't know how to control $n^{1-\epsilon}$, because all the exercises I've done before $n^{1-\epsilon}$ is replaced by only one variable, namely, $\epsilon$, the bound is fixed, but in this case the lower bound is changing.