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aakinshin.net | ||
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djalil.chafai.net
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| | | | | Convergence in law to a constant. Let \( {{(X_n)}_{n\geq1}} \) be a sequence of random variables defined on a common probability space \( {(\Omega,\mathcal{A},\mathbb{P})} \), and taking their values in a metric space \( {(E,d)} \) equipped with its Borel sigma-field. It is well known that if \( {{(X_n)}_{n\geq1}} \) converges in law as \( {n\rightarrow\infty} \) to some Dirac... | |
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gregorygundersen.com
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mkatkov.wordpress.com
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| | | | | For probability space $latex (\Omega, \mathcal{F}, \mathbb{P})$ with $latex A \in \mathcal{F}$ the indicator random variable $latex {\bf 1}_A : \Omega \rightarrow \mathbb{R} = \left\{ \begin{array}{cc} 1, & \omega \in A \\ 0, & \omega \notin A \end{array} \right.$ Than expected value of the indicator variable is the probability of the event $latex \omega \in... | |
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jmanton.wordpress.com
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| | | If $latex Y$ is a $latex \sigma(X)$-measurable random variable then there exists a Borel-measurable function $latex f \colon \mathbb{R} \rightarrow \mathbb{R}$ such that $latex Y = f(X)$. The standard proof of this fact leaves several questions unanswered. This note explains what goes wrong when attempting a "direct" proof. It also explains how the standard proof... | ||
