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. ^[1][2]

Definition

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 ${\displaystyle s}$.^[2]

Examples and Applications

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.^[1] Any process that is continuous in probability and has independent increments has a version that is càdlàg.^[2] As a result, some authors immediately define Lévy process as being càdlàg and having independent increments.^[3]