For fixed $t$, $(X_n(t))$ is a martingale with independent centered increments and
$$
\sup_n\mathbb E X_n(t)^2=\sum_{i\geq1}c_i(t)^2=t.
$$
The $L^2$ <Martingale convergence theorem> gives convergence both almost surely and in $L^2$ to a random variable $X(t)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 1 / c / ii / Solution

 Solution

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