Continuity in probability

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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

^ ^a ^b Applebaum, D. "Lectures on Lévy processes and Stochastic calculus, Braunschweig; Lecture 2: Lévy processes" (PDF). University of Sheffield. pp. 37–53.
^ ^a ^b ^c 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.

[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.

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