Article proofs | Statistical randomness
This article is supplemental for “Convergence of random variables” and provides proofs for selected results. Several results will be established using the portmanteau lemma: A sequence {Xn} converges in distribution to X if and only if any of the following conditions are met: 1. * E[f(Xn)] → E[f(X)] for all bounded, continuous functions f; 2. * E[f(Xn)] → E[f(X)] for all bounded, Lipschitz functions f; 3. * limsup{Pr(Xn ∈ C)} ≤ Pr(X ∈ C) for all closed sets C; (Wikipedia).
How to Prove a Sequence with Two Components Converges
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From playlist Advanced Calculus
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We prove the limit law for the difference of convergent series. If two series converge to a and b, then the series whose terms are the differences of the terms of the original two series is the difference of the limits: a-b. We'll prove this using the limit law for the difference of conver
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From playlist Real Analysis
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From playlist MIT 6.262 Discrete Stochastic Processes, Spring 2011
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MIT 6.262 Discrete Stochastic Processes, Spring 2011 View the complete course: http://ocw.mit.edu/6-262S11 Instructor: Robert Gallager License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
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