Let X = (X1, . . ., Xd) be a random vector. We define its distribution function by F(x) = P(X ≤ x). Here x ∈ Rd, and X ≤ x means Xi ≤ xi for i = 1, . . ., d. As in one dimension, F has three obvious properties:
(i) It is nondecreasing, that is, if x ≤ y then F(x) ≤ F(y).
(ii) limx→∞ F(x) = 1, limxi→−∞F(x) = 0.
(iii) F is right continuous, that is, limy↓x F(y) = F(x)
Here x → ∞ means each coordinate xi goes to ∞, xi → −∞ means we let xi → −∞ keeping the other coordinates fixed, and y ↓ x means each coordinate yi ↓ xi
As discussed in Section 1.1, an additional condition is needed to guarantee that F is the distribution function of a probability measure. Let
V = the vertices of the rectangle A. If v ∈ V, let