Continuity in probability explained

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} 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.

References

^{[1] [2] [3]}

Notes and References

1. Book: Kallenberg . Olav . Olav Kallenberg . 2002 . Foundations of Modern Probability. New York . Springer . 2nd. 286.
2. Web site: Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes. Applebaum, D.. 37–53. University of Sheffield.
3. Book: Kallenberg . Olav . Olav Kallenberg . 2002 . Foundations of Modern Probability. New York . Springer . 2nd. 290.