Solution
= Solution
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 $\varepsilon>0$,
$$
\mathbb P\!\left(\sup_{0\leq t\leq T}|X_t^n-X_t|>\varepsilon\right)\longrightarrow0.
$$