Author Archives: mg50

Portmanteau Theorem

Posted on February 28, 2017 by mg50

Statement: $X_n \overset{D}{\to} X$ iff $E[g(X_n)] \to E[g(X)]$ for any bounded, continuous Lipschitz g. We prove this in two parts: Part 1 Here we show $X_n \overset{D}{\to} X$ implies $E[g(X_n)] \to E[g(X)]$ for any continuous bounded g. Let $F_n$ denote the cdf … Continue reading →

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Uniqueness of Laplace transform

Posted on February 24, 2017 by mg50

Statement: Let $L_f(t) = \int_0^\infty f(x)e^{-tx} dx$. Then if $f(x) = o(e^{ax})$ for some $a > 0$, $L_f = L_g$ implies $f = g$. Proof: By linearity of the integral, this is equivalent to showing $L_f = 0$ implies $f = … Continue reading →

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Strong Law of Large Numbers

Posted on February 22, 2017 by mg50

Truncation Lemma Statement: Let $X_n$ be a sequence of IID variables with finite mean, and let $Y_n$ = $X_n I_{|X_n| \leq n}$. Then $\sum \frac{Var(Y_n)}{n^2} < \infty$. Proof: Lemma: If $c \geq 0$, 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 $f_n$ that converges almost everywhere to $f$ on some set $D$ on a measure space. Then $\lim \int_D f_n d\mu = \int_D f d\mu$. Proof: Since the sequence of … Continue reading →

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Kolmogorov’s Inequality

Posted on February 18, 2017 by mg50

Statement: Let $X_i$ be independent random variables with mean 0, finite variance and $S_n = X_1 + \ldots + X_n$. Then $P\left( \displaystyle \max_{1 \leq i \leq n} |S_i| \geq a \right) \leq \frac{1}{a^2} Var(S_n)$. Note that Chebyshev’s inequality already … Continue reading →

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Mahalanobis Distance

Posted on February 23, 2016 by mg50

Let $\mathbf x$ be a vector, and assume we have a distribution with mean $\mathbf \mu$ and covariance matrix $S$. The Mahalanobis distance with respect to the distribution is defined as $$D(\mathbf x) = \sqrt{(\mathbf x – \mathbf \mu)^TS^{-1}(\mathbf x … Continue reading →

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