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738 CHAPTER 27. ANALYTICAL CONSIDERATIONS

Thus, using this in the above,∫Rp1

∫Rp2

∫Rp3

XE (x)XF (y)XG (z)dλZdλY dλX

=∫Rp1

∫Rp2

∫Rp3

XE (x)XF (y)XG (z)dλZ|xydλY dλX

and also it reduces to∫Rp1×Rp2

∫Rp3

XE (x)XF (y)XG (z)dλZdλ (X,Y )

=∫Rp1×Rp2

∫Rp3

XE (x)XF (y)XG (z)dλZ|xydλ (X,Y )

Now by uniqueness of the slicing measures again, for λ (X,Y ) a.e. (x,y) , it follows that

λZ = λZ|xy

Similar conclusions hold for λX ,λY . In each case, off a set of measure zero the distribu-tion measures equal the slicing measures.

Conversely, if the distribution measures equal the slicing measures off sets of measurezero as described above, then it is obvious that the random variables are independent. Thesame reasoning applies for any number of random variables.

Thus this gives a different and more analytical way to think of independence of finitelymany random variables. Clearly, the argument given above will apply to any finite set ofrandom variables.

Proposition 27.2.3 Equations 27.2 and 27.1 hold with XE replaced by any nonnega-tive Borel measurable function and for any bounded continuous function or for any functionin L1.

Proof: The two equations hold for simple functions in place of XE and so an appli-cation of the monotone convergence theorem applied to an increasing sequence of simplefunctions converging pointwise to a given nonnegative Borel measurable function yields theconclusion of the proposition in the case of the nonnegative Borel function. For a boundedcontinuous function or one in L1, one can apply the result just established to the positiveand negative parts of the real and imaginary parts of the function.

Lemma 27.2.4 LetX1, · · · ,Xn be random vectors with values in Rp1 , · · · ,Rpn respec-tively and let

g : Rp1 ×·· ·×Rpn → Rk

be Borel measurable. Then g (X1, · · · ,Xn) is a random vector with values in Rk and ifh : Rk→ [0,∞), then ∫

Rkh(y)dλg(X1,··· ,Xn) (y) =∫

Rp1×···×Rpnh(g (x1, · · · ,xn))dλ (X1,··· ,Xn). (27.3)

If X i is a random vector with values in Rpi , i = 1,2, · · · and if gi : Rpi → Rki , where gi isBorel measurable, then the random vectors gi (X i) are also independent whenever theX iare independent.

738 CHAPTER 27. ANALYTICAL CONSIDERATIONSThus, using this in the above,Lo bon [eos 28 (@) 2% (y) Roz) dr zddyddxJRPL JIRP2 JIRP3= [ | Lz (w) Xe (y) Xo (Z) dA gay dd xJet JIRP2 JIRP3and also it reduces toi p(w) Xe (y) Ro (z) dd zdd x,y)R?1 xR?2 JR?3_ [ vrcana Jam 2E (@) 2 (Y) Xo (2) dd zhoydh(x,¥)Now by uniqueness of the slicing measures again, for Ai X,Y) ae. (x,y), it follows thatSimilar conclusions hold for Ax ,Ay. In each case, off a set of measure zero the distribu-tion measures equal the slicing measures.Conversely, if the distribution measures equal the slicing measures off sets of measurezero as described above, then it is obvious that the random variables are independent. Thesame reasoning applies for any number of random variables.Thus this gives a different and more analytical way to think of independence of finitelymany random variables. Clearly, the argument given above will apply to any finite set ofrandom variables.Proposition 27.2.3. Equations 27.2 and 27.1 hold with 2% replaced by any nonnega-tive Borel measurable function and for any bounded continuous function or for any functionin L,Proof: The two equations hold for simple functions in place of 2g and so an appli-cation of the monotone convergence theorem applied to an increasing sequence of simplefunctions converging pointwise to a given nonnegative Borel measurable function yields theconclusion of the proposition in the case of the nonnegative Borel function. For a boundedcontinuous function or one in L', one can apply the result just established to the positiveand negative parts of the real and imaginary parts of the function.Lemma 27.2.4 Let X\,---,X,, be random vectors with values in R?!,--- IR?" respec-tively and letg 2 RP! x ++ x Rn + REbe Borel measurable. Then g(X\,-++,Xn) is a random vector with values in R* and ifh: R* > 0,00), then[Bde xi.) 0) =[ A(g(@1,-+* @n)) dA (x, Xn): (27.3)JRP1 X--x RenIf X; is a random vector with values in R?',i = 1,2,--+ and if g; : R?' +> R", where g; isBorel measurable, then the random vectors g; (X;) are also independent whenever the X ;are independent.