Abstract
A continuous random variable is expanded as a sum of a sequence of uncorrelated random variables. These variables are principal dimensions in continuous scaling on a distance function, as an extension of classic scaling on a distance matrix. For a particular distance, these dimensions are principal components. Then some properties are studied and an inequality is obtained. Diagonal expansions are considered from the same continuous scaling point of view, by means of the chi-square distance. The geometric dimension of a bivariate distribution is defined and illustrated with copulas. It is shown that the dimension can have the power of continuum.
Highlights
Let X be a random variable on a probability space (Ω, P) with rang= e I [a,b] ⊂, absolutely continuous cdf F and density f w.r.t. the Lebesgue measure
N≥1 where ( X n ) is a sequence of uncorrelated random variables, which can be seen as a decomposition of the so-called geometric variability Vδ ( X ), defined below, a dispersion measure of X in relation with a suitable distance function δ ( x, x′), x, x′∈ I
Orthogonal expansions and series appear in the theory of stochastic processes, in martingales in the wide sense ([4], Chap. 4; [5], Chap. 10), in non-parametric statistics [6], in goodness-of-fit tests [7,8], in testing independence [9] and in characterizing distributions [10]
Summary
Let X be a random variable on a probability space (Ω, , P) with rang= e I [a,b] ⊂ , absolutely continuous cdf F and density f w.r.t. the Lebesgue measure. N≥1 where ( X n ) is a sequence of uncorrelated random variables, which can be seen as a decomposition of the so-called geometric variability Vδ ( X ), defined below, a dispersion measure of X in relation with a suitable distance function δ ( x, x′), x, x′∈ I. Expansion (1) is obtained following a similar procedure, except that we have a sequence of uncorrelated rather than independent random variables. Orthogonal expansions and series appear in the theory of stochastic processes, in martingales in the wide sense CUADRAS [15,16]
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