Extreme events: blending PCA and ICA [8]

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Bullet points include: We can subdivide joint ‘fat-tailed-ness’ into two parts: How fat-tailed each series is in isolation, i.e. each marginal distribution, and How fat-tailed is their co-movement, i.e. their (joint) copula function. Sklar’s theorem: Suppose that X1, X2, ..., XN are random variables. With marginal distribution functions, i.e. individual cumulative probability distribution functions, say, F1(x1), F2(x2), ..., FN(xN). And a joint distribution function F(x1, x2, ..., xN). Then F can always be characterised by the N marginal distributions and an N-dimensional copula, C, i.e. a function that maps a vector of N numbers each between 0 and 1 onto some value in the range 0 to 1, using:

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