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

[kkjs]

For (Y₁, Y₂, Y₃, Y₄) ~ D₄(1,2,3,4;5)
let X_k = [∑(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

 

[radou]

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.

 

[vela]

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.