Since $X_1-X_s$ is centered Gaussian with variance $1-s$,
$$
\mathbb E\int_0^1\frac{|X_1-X_s|}{1-s}\,ds
=C\int_0^1(1-s)^{-1/2}\,ds<\infty.
$$
<Fubini's theorem> therefore shows that the defining integral for $A_1$ is absolutely finite almost surely.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 201 / 4 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-201/4/b/solution