(i) Let $(E, \mathcal{E}, \mu)$ be a measure space and let $1 \leqslant p<\infty$. For a measurable function $f$, let $\|f\|_{p}=\left(\int|f|^{p} d \mu\right)^{1 / p}$. Give the definition of the space $L^{p}$. Prove that $\left(L^{p},\|\cdot\|_{p}\right)$ forms a Banach space.

[You may assume that $L^{p}$ is a normed vector space. You may also use in your proof any other result from the course provided that it is clearly stated.]

(ii) Show that convergence in probability implies convergence in distribution.

[Hint: Show the pointwise convergence of the characteristic function, using without proof the inequality $\left|e^{i y}-e^{i x}\right| \leqslant|x-y|$ for $x, y \in \mathbb{R}$.]

(iii) Let $\left(\alpha_{j}\right)_{j \geqslant 1}$ be a given real-valued sequence such that $\sum_{j=1}^{\infty} \alpha_{j}^{2}=\sigma^{2}<\infty$. Let $\left(X_{j}\right)_{j \geqslant 1}$ be a sequence of independent standard Gaussian random variables defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Let

$$Y_{n}=\sum_{j=1}^{n} \alpha_{j} X_{j}$$

Prove that there exists a random variable $Y$ such that $Y_{n} \rightarrow Y$ in $L^{2}$.

(iv) Specify the distribution of the random variable $Y$ defined in part (iii), justifying carefully your answer.