= Closure of Lévy processes under locally controlled convergence in probability
Suppose <Lévy processes> $X^n$ converge to $X$ in probability at every fixed time, and
$$
\lim_{n\to\infty}\limsup_{t\downarrow0}\mathbb P(|X_t^n-X_t|>\varepsilon)=0\qquad(\varepsilon>0).
$$
Then $X$ starts at zero, has <independent increments> and <stationary increments>, and has <stochastic continuity>. Finite increment vectors inherit their independent laws by <convergence in distribution>; the near-zero estimate supplies continuity at zero. Hence $X$ has a <càdlàg modification> that is a <Lévy process>. The hypotheses alone cannot assert that the original version has <càdlàg> paths.
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