Path and moment criteria from a Lévy triplet
= Path and moment criteria from a Lévy triplet
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For the convention in the <Lévy–Khintchine formula>, a Lévy process has almost surely differentiable paths exactly when $b=0$ and $K=0$, and continuous paths exactly when $K=0$. It is integrable exactly when $\int_{|x|>1}|x|K(dx)<\infty$, and it has a finite second moment exactly when $\int x^2K(dx)<\infty$.