Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathcal{A}=\left(A_{i}: i=1,2, \ldots\right)$ be a sequence of events.

(a) What is meant by saying that $\mathcal{A}$ is a family of independent events?

(b) Define the events $\left\{A_{n}\right.$ infinitely often $\}$ and $\left\{A_{n}\right.$ eventually $\}$. State and prove the two Borel-Cantelli lemmas for $\mathcal{A}$.

(c) Let $X_{1}, X_{2}, \ldots$ be the outcomes of a sequence of independent flips of a fair coin,

$$\mathbb{P}\left(X_{i}=0\right)=\mathbb{P}\left(X_{i}=1\right)=\frac{1}{2} \quad \text { for } i \geqslant 1$$

Let $L_{n}$ be the length of the run beginning at the $n^{\text {th }}$flip. For example, if the first fourteen outcomes are 01110010000110 , then $L_{1}=1, L_{2}=3, L_{3}=2$, etc.

Show that

$$\mathbb{P}\left(\limsup _{n \rightarrow \infty} \frac{L_{n}}{\log _{2} n}>1\right)=0$$

and furthermore that

$$\mathbb{P}\left(\limsup _{n \rightarrow \infty} \frac{L_{n}}{\log _{2} n}=1\right)=1$$