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 $$ \R^n $$. 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.