The <Martingale convergence theorem> states that a discrete-time martingale $(X_n)$ with <uniform integrability> has an integrable random variable $X_\infty$ such that
$$
X_n\longrightarrow X_\infty
$$
almost surely and in $L^1$. Moreover, the martingale is closed by its limit:
$$
X_n=\mathbb E[X_\infty\mid\mathcal F_n].
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 201 / 6 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-201/6/a/solution