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.

Definition

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

Examples and Applications

Feller processes are continuous in probability at $t=0$. 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.