Let *X* = (*X*_{1}, . . ., *X*_{d}) be a random vector. We define its **distribution function** by *F*(*x*) = *P*(*X* ≤ *x*). Here *x* ∈ **R**^{d}, and *X* ≤ *x* means *X*_{i} ≤ *x*_{i} 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)* lim_{x→∞} *F*(*x*) = 1, lim_{xi→−∞}*F*(*x*) = 0.

*(iii)* *F* is right continuous, that is, lim_{y↓x} *F*(*y*) = *F*(*x*)

Here *x* → ∞ means each coordinate *x*_{i} goes to ∞, *x*_{i} → −∞ means we let *x*_{i} → −∞ keeping the other coordinates fixed, and *y* ↓ *x* means each coordinate *y*_{i} ↓ *x*_{i}

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

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