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Let d> 2. Let X = (X₁,..., Xd) be an Rª-valued random variable with infinitely divisible distribution. Let (A, v, y) be its generating triplet Show the following. (i) X₁,..., Xd are independent if and only if A is diagonal and v is supported on the union of the coordinate axes. (ii) If X₁,..., Xd are pairwise independent, that is, if X, and X are independent for every choice of j and k with j k, then X₁,..., Xd are independent.

Let d> 2. Let X = (X₁,..., Xd) be an Rª-valued random variable with infinitely divisible distribution. Let (A, v, y) be its generating triplet Show the following. (i) X₁,..., Xd are independent if and only if A is diagonal and v is supported on the union of the coordinate axes. (ii) If X₁,..., Xd are pairwise independent, that is, if X, and X are independent for every choice of j and k with j k, then X₁,..., Xd are independent.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Let d> 2. Let X = (X₁,..., Xd) be an Rd-valued random variable with infinitely divisible distribution. Let (A, v, y) be its generating triplet Show the following. (i) X₁,..., Xd are independent if and only if A is diagonal and is supported on the union of the coordinate axes. (ii) If X₁,..., Xd are pairwise independent, that is, if X, and X are independent for every choice of j and k with j k, then X₁,..., Xd are independent.
Transcribed Image Text:Let d> 2. Let X = (X₁,..., Xd) be an Rd-valued random variable with infinitely divisible distribution. Let (A, v, y) be its generating triplet Show the following. (i) X₁,..., Xd are independent if and only if A is diagonal and is supported on the union of the coordinate axes. (ii) If X₁,..., Xd are pairwise independent, that is, if X, and X are independent for every choice of j and k with j k, then X₁,..., Xd are independent.
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