Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of (real-valued, Borel-measurable) random variables on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$.

(a) Let $\left(A_{n}\right)_{n \in \mathbb{N}}$ be a sequence of events in $\mathcal{A}$.

What does it mean for the events $\left(A_{n}\right)_{n \in \mathbb{N}}$ to be independent?

What does it mean for the random variables $\left(X_{n}\right)_{n \in \mathbb{N}}$ to be independent?

(b) Define the tail $\sigma$-algebra $\mathcal{T}$ for a sequence $\left(X_{n}\right)_{n \in \mathbb{N}}$ and state Kolmogorov's $0-1$ law.

(c) Consider the following events in $\mathcal{A}$,

$$\begin{gathered} \left\{X_{n} \leqslant 0 \text { eventually }\right\} \\ \left\{\lim _{n \rightarrow \infty} X_{1}+\ldots+X_{n} \text { exists }\right\} \\ \left\{X_{1}+\ldots+X_{n} \leqslant 0 \text { infinitely often }\right\} \end{gathered}$$

Which of them are tail events for $\left(X_{n}\right)_{n \in \mathbb{N}}$ ? Justify your answers.

(d) Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be independent random variables with

$$\mathbb{P}\left(X_{n}=0\right)=\mathbb{P}\left(X_{n}=1\right)=\frac{1}{2} \quad \text { for all } n \in \mathbb{N},$$

and define $U_{n}=X_{1} X_{2}+X_{2} X_{3}+\ldots+X_{2 n} X_{2 n+1}$.

Show that $U_{n} / n \rightarrow c$ a.s. for some $c \in \mathbb{R}$, and determine $c$.

[Standard results may be used without proof, but should be clearly stated.]