Let (F: mathbb{R} ightarrow[0,1]) be a distribution function. a) Show that there exists a probability space ((Omega,
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Let \(F: \mathbb{R} ightarrow[0,1]\) be a distribution function.
a) Show that there exists a probability space \((\Omega, \mathscr{A}, \mathbb{P})\) and a random variable \(X\) such that \(F(x)=\mathbb{P}(X \leqslant x)\).
b) Show that there exists a probability space \((\Omega, \mathscr{A}, \mathbb{P})\) and an iid sequence of random variables \(X_{n}\) such that \(F(x)=\mathbb{P}\left(X_{n} \leqslant xight)\).
c) State and prove the corresponding assertions for the \(d\)-dimensional case.
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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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