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 $X=(X_{t})_{t\in T}$ 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 $s$ .[2]

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.[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]

References

Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.
Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 286.
Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 290.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[applebaum-1] Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.

[KallebergFoundations286-2] Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 286.

[KallebergFoundations290-3] Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 290.