Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 217 / 2 / Solution

Letbe the event that the partial sums first cross level $t$ at time $k$. On $A_k$, write $R_k=S_n-S_k$. Conditional on $X_1,\dots ,X_k$, the random variable $R_k$ is independent and symmetric. Sinceat least one of the two norms on the right exceeds $t$. Symmetry of $R_k$ consequently giveson $A_k$. The events $A_k$ are disjoint, so summation proves the Lévy maximal inequality

For the Gaussian series, put $S_m={\sum }_{j=1}^m{g}_j{u}_j$. Apply the inequality to the symmetric independent increments from $n+1$ through $m$, followed by Markov inequality in squared norm:Here the cross terms vanish by orthogonality of independent centered Hilbert-space random variables. Letting $m\to \infty$ and then $n\to \infty$ showsThe convergence criterion in the question now shows that ${\sum }_j{g}_j{u}_j$ converges almost surely in $V$.

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