Transformations and Hardy-Krause Variation

A generalization of Hoeffding’s lemma, and a new class of covariance inequalities

MSC: primary 62H05 60E15 secondary 62M10 a b s t r a c t We generalize Hoeffding's lemma to apply to covariances between functions of several random variables. Our generalization leads to a new class of covariance inequalities involving the Vitali or Hardy–Krause variation. These inequalities are relevant to the study of weakly dependent processes. Published by Elsevier B.V.

Covering numbers, dyadic chaining and discrepancy

is called the star-discrepancy of (z1, . . . , zN ). Here and in the sequel λ denotes the sdimensional Lebesgue measure. The Koksma-Hlawka inequality states that the difference between the integral of a function f over the s-dimensional unit cube and the arithmetic mean of the function values f(z1), . . . , f(zN ) is bounded by the product of the total variation of f (in the sense of Hardy and ...

How Many Distribution Functions Are There? Bracketing Entropy Bounds for High Dimensional Distribution Functions

This means that every F ∈ Fd satisfies: (i) (non-negativity). For finite intervals I = (a1, b1] × · · · × (ad, bd] ≡ (a, b], with a, b ∈ Rd, F (I) = ∆dF (a, b] ≥ 0 where ∆d denotes the d−dimensional difference operator. (ii) (continuity from above). If y ↓ x, then F (y) ↓ F (x). (iii) (normalization). If x1∧. . .∧xd → −∞, then F (x) → 0; if x1∧. . .∧xd → +∞, then F (x) → 1. (See Billingsley [4]...

Polymorphous Transformations in Alkaline-Earth Silicates

QMC integration of improper integrals1

In financial applications of QMC methods, often integrals appear with a singular integrand on the integration boundary. As the variation in the sense of Hardy and Krause is not bounded for such functions, one has to use other theorems than the Koksma-Hlawka inequality to show convergence of QMC methods and estimate the convergence rate. Recently, a lot of research effort has been spent on this ...