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  be a stochastic process in . The process  is continuous in probability when  converges in probability to  whenever  converges to . Examples and Applications 
Feller processes are continuous in probability at . 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. References 

Stochastic processes