91Ó°ÊÓ

Let \(\left(X_{i}\right)_{i \in \mathbb{N}}\) be independent, square integrable random variables with \(\mathbf{E}\left[X_{i}\right]=0\) for all \(i \in \mathbb{N}\) (i) Show that \(\sum_{i=1}^{\infty} \operatorname{Var}\left[X_{i}\right]<\infty\) implies that there exists a real random variable \(X\) with \(\sum_{i=1}^{n} X_{i} \stackrel{n \rightarrow \infty}{\longrightarrow} X\) almost surely. (ii) Does the converse implication hold in (i)?

Let \(\mu, \nu, \alpha\) be finite measures on \((\Omega, \mathcal{A})\) with \(\nu \ll \mu \ll \alpha\) (i) Show the chain rule for the Radon-Nikodym derivative: $$ \frac{d \nu}{d \alpha}=\frac{d \nu}{d \mu} \frac{d \mu}{d \alpha} \quad \alpha \text {-a.e. } $$ (ii) Show that \(f:=\frac{d \nu}{d(\mu+\nu)}\) exists and that \(\frac{d y}{d \mu}=\frac{f}{1-f}\) holds \(\mu\)-a.e.