Continuity in probability

In probability theory, a stochastic process is said to be continuous in probability or stochastically continuous if its distributions converge whenever the values in the index set converge.

Let ${\displaystyle X=(X_{t})_{t\in T}}$be a stochastic process in ${\displaystyle \mathbb {R} ^{n}}$. The process ${\displaystyle X}$is continuous in probability when ${\displaystyle X_{r}}$converges in probability to ${\displaystyle X_{s}}$whenever ${\displaystyle r}$converges to ${\displaystyle s}$.

Feller processes are continuous in probability at ${\displaystyle t=0}$. Continuity in probability is a sometimes used as one of the defining property for Lévy process. Any process that is continuous in probability and has independent increments has a version that is càdlàg. As a result, some authors immediately define Lévy process as being càdlàg and having independent increments.