Let (X) be a real-valued random variable on ((Omega, mathscr{A}, mathbb{P})) and (mathscr{F} subset mathscr{A}) be (sigma)-algebra.
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Let \(X\) be a real-valued random variable on \((\Omega, \mathscr{A}, \mathbb{P})\) and \(\mathscr{F} \subset \mathscr{A}\) be \(\sigma\)-algebra. Show that
Show that \(\mathbb{E}\left(e^{i \xi X} e^{i \eta \mathbb{1}_{F}}ight)=\mathbb{E}\left(e^{i \xi X}ight) \mathbb{E}\left(e^{i \eta \mathbb{1}_{F}}ight)\) for all \(\xi, \eta \in \mathbb{R}\) and use the characterization of independence by characteristic functions.
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Brownian Motion A Guide To Random Processes And Stochastic Calculus De Gruyter Textbook
ISBN: 9783110741254
3rd Edition
Authors: René L. Schilling, Björn Böttcher
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