Monthly Archives: February 2017

Portmanteau Theorem

Posted on February 28, 2017 by mg50

Statement: Xn→DX iff E[g(Xn)]→E[g(X)] for any bounded, continuous Lipschitz g. We prove this in two parts: Part 1 Here we show Xn→DX implies E[g(Xn)]→E[g(X)] for any continuous bounded g. Let Fn denote the cdf … Continue reading →

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Uniqueness of Laplace transform

Posted on February 24, 2017 by mg50

Statement: Let Lf(t)=∫0∞f(x)e−txdx. Then if f(x)=o(eax) for some a>0, Lf=Lg implies f=g. Proof: By linearity of the integral, this is equivalent to showing Lf=0 implies $f = … Continue reading →

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Strong Law of Large Numbers

Posted on February 22, 2017 by mg50

Truncation Lemma Statement: Let Xn be a sequence of IID variables with finite mean, and let Yn = XnI|Xn|≤n. Then ∑Var(Yn)n2<∞. Proof: Lemma: If c≥0, then $c^2 \sum_{n=1}^\infty \frac{I_{|c| \leq n}}{n^2} \leq 1 … Continue reading →

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Convergence theorems

Posted on February 21, 2017 by mg50

Monotone Convergence Theorem Statement: Suppose we have a sequence of non-negative functions fn that converges almost everywhere to f on some set D on a measure space. Then lim∫Dfndμ=∫Dfdμ. Proof: Since the sequence of … Continue reading →

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Kolmogorov’s Inequality

Posted on February 18, 2017 by mg50

Statement: Let Xi be independent random variables with mean 0, finite variance and Sn=X1+…+Xn. Then P(max1≤i≤n|Si|≥a)≤1a2Var(Sn). Note that Chebyshev’s inequality already … Continue reading →

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