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], page 265; or see e.g. Breiman [9], page 27, for analogous properties if distribution functions are defined, alternatively, by F (x) = P (X < x).) It follows from the above that Fd is a subclass of the collection BVH(R), the class of all functions of bounded variation in the sense of Hardy-Krause (see Clarkson and Adams [13], page 825, and Hildebrandt [20], chapter 3, for the case d = 2). [Most of the current literature on functions of bounded variation on Rd seems to concern the classes BV (Rd) ≡ BVT (Rd) of bounded variation in the sense of Tonelli (see Clarkson and Adams [13] and Ziemer [30], chapter 5, definition 5.5.1, page 220, and his historical note 5.1, page 280.