A <càdlàg process> $A$ is a <finite-variation process> when, almost surely, for every $T<\infty$,
$$
\sup_\pi\sum_{k}|A_{t_{k+1}}-A_{t_k}|<\infty,
$$
where the <supremum> is over every finite <partition of an interval> $0=t_0<\cdots<t_n=T$.

<Uniform convergence on compacts in probability> of $X^n$ to $X$ means that, for every $T<\infty$ and $\varepsilon>0$,
$$
\mathbb P\!\left(\sup_{0\leq t\leq T}|X_t^n-X_t|>\varepsilon\right)\longrightarrow0.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 2 / a / Solution

A càdlàg process has sample paths that are <càdlàg functions>: they are right-continuous and have a finite left limit at every positive time.

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