How can I prove X = (X1, X2, X3) is independent?

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To prove that X = (X1, X2, X3) is independent, the discussion focuses on the relationships between X1, X2, and X3, which are defined in terms of Z variables derived from a Dirichlet distribution. The initial approach suggests showing independence between pairs, but it is pointed out that independence is not transitive; thus, proving X1 and X2 independent does not guarantee X1 and X3 are independent. A counterexample is provided to illustrate that X3 could be a multiple of X1, undermining the assumption of independence. The conversation emphasizes the need for a different strategy to establish independence in this specific context.
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For (Y1, Y2, Y3, Y4) ~ D4(1,2,3,4;5)
let Xk = [∑(from i=1 to k) Yi] / [∑(from i=1 to k+1) Yi] where k = 1,2,3

How can I prove X = (X1, X2, X3) is independent?
What I did was...

(Y1, Y2, Y3, Y4) ~ D4(1,2,3,4;5) = (Z1, Z2, Z3, Z4) / (Z1+Z2+Z3+Z4+Z5) where Z ~ N(0,1), Z IID G(1/2)

Now, we have
X1 = Z1 / (Z1+Z2)
X2 = (Z1+Z2) / (Z1+Z2+Z3)
X3 = (Z1+Z2+Z3) / (Z1+Z2+Z3+Z4)

I think if i can somehow show X1 and X2 are independent and X2 and X3 are independent then X1 and X3 are independent as well but how? this is a part I don't get T-T
 
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I can't understand your notation, hopefully someone will jump in, but in general, if you're talking about linear independence of vectors, it's not a transitive relation.
 


kkjs said:
I think if i can somehow show X1 and X2 are independent and X2 and X3 are independent then X1 and X3 are independent as well but how? this is a part I don't get T-T
That won't work in general. What if, for example, X3 turned out to be a multiple of X1? If X1 and X2 are independent, then X2 and X3 would also be independent, but X1 and X3 are definitely not independent.

Perhaps you can make it work because of the way the X's are constructed in your specific problem, but I'd guess it's not the right strategy.
 

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