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Probability, Fourth Edition by Rick Durrett

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3.9 Limit Theorems in Rd

Let X = (X1, . . ., Xd) be a random vector. We define its distribution function by F(x) = P(Xx). Here xRd, and Xx means Xixi for i = 1, . . ., d. As in one dimension, F has three obvious properties:

 (i)   It is nondecreasing, that is, if xy then F(x) ≤ F(y).

 (ii)   limx→∞ F(x) = 1, limxi→−∞F(x) = 0.

(iii)   F is right continuous, that is, limyx F(y) = F(x)

Here x → ∞ means each coordinate xi goes to ∞, xi → −∞ means we let xi → −∞ keeping the other coordinates fixed, and yx means each coordinate yixi

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 vV, let

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