Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $\left(X_{n}\right)_{n \geqslant 1}$ be a sequence of random variables with $\mathbb{E}\left(\left|X_{n}\right|^{2}\right) \leqslant 1$ for all $n \geqslant 1$.

(a) Suppose $Z$ is another random variable such that $\mathbb{E}\left(|Z|^{2}\right)<\infty$. Why is $Z X_{n}$ integrable for each $n$ ?

(b) Assume $\mathbb{E}\left(Z X_{n}\right) \underset{n \rightarrow \infty}{\longrightarrow} 0$ for every random variable $Z$ on $(\Omega, \mathcal{F}, \mathbb{P})$ such that $\mathbb{E}\left(|Z|^{2}\right)<\infty$. Show that there is a subsequence $Y_{k}:=X_{n_{k}}, k \geqslant 1$, such that

$$\frac{1}{N} \sum_{k=1}^{N} Y_{k} \underset{N \rightarrow \infty}{\longrightarrow} 0 \text { in } \mathbb{L}^{2}$$

(c) Assume that $X_{n} \rightarrow X$ in probability. Show that $X \in \mathbb{L}^{2}$. Show that $X_{n} \rightarrow X$ in $\mathbb{L}^{1}$. Must it converge also in $\mathbb{L}^{2} ?$ Justify your answer.

(d) Assume that the $\left(X_{n}\right)_{n \geqslant 1}$ are independent. Give a necessary and sufficient condition on the sequence $\left(\mathbb{E}\left(X_{n}\right)_{n \geqslant 1}\right)$ for the sequence

$$Y_{N}=\frac{1}{N} \sum_{k=1}^{N} X_{k}$$

to converge in $\mathbb{L}^{2}$.