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Continuity in probability - Wikipedia

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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]

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

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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]

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