By Kai Lai Chung, Ruth J. Williams
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Extra resources for Introduction to stochastic integration, Second Edition
This is manifest in the following two results: a variant of Slutsky’s theorem and the so-called delta method. The former deals with limit behavior whereas the latter deals with an extension of the central limit theorem. 4 (Slutsky’s Theorem) Denote by Xi , Yi sequences of random variables with Xi → X and Yi → c for c ∈ R in probability. Moreover, denote by g(x, y) a function which is continuous for all (x, c). In this case the random variable g(Xi , Yi ) converges in probability to g(X, c). g. [Bil68].
1 nearest neighbor classifier. Depending on whether the query point x is closest to the star, diamond or triangles, it uses one of the three labels for it. Fig. 18. k-Nearest neighbor classifiers using Euclidean distances. Left: decision boundaries obtained from a 1-nearest neighbor classifier. Middle: color-coded sets of where the number of red / blue points ranges between 7 and 0. Right: decision boundary determining where the blue or red dots are in the majority. flexible. For instance, we could use string edit distances to compare two documents or information theory based measures.
G. [Bil68]. 4 is often referred to as the continuous mapping theorem (Slutsky only proved the result for affine functions). It means that for functions of random variables it is possible to pull the limiting procedure into the function. Such a device is useful when trying to prove asymptotic normality and in order to obtain characterizations of the limiting distribution. 5 (Delta Method) Assume that Xn ∈ Rd is asymptotically 2 normal with a−2 n (Xn − b) → N(0, Σ) for an → 0. Moreover, assume that d l g : R → R is a mapping which is continuously differentiable at b.