basic correlation linearity question

[supermarco]

(St) = SUM (t=1 to T){X(St)}
(Pt) = SUM (t=1 to T){-Y(St) x constant_1 + constant_2}

CORRELATION [(St), (Pt)] = CORRELATION [SUM (t=1 to T){X(St)} , - SUM (t=1 to T){Y(St)}]

 

[stapel]

(St) = SUM (t=1 to T){X(St)}
(Pt) = SUM (t=1 to T){-Y(St) x constant_1 + constant_2}

CORRELATION [(St), (Pt)] = CORRELATION [SUM (t=1 to T){X(St)} , - SUM (t=1 to T){Y(St)}]

Okay; you've posted some definitions and equations. Did you have a question?

 

[supermarco]

Hello, thank you! I'm sorry, i'm not sure i was clear. Let me do it again to see if you agree with all steps:
X_(n) = ∑_(t=1)^(n) { x_t }
Z_(n) = ∑_(t=1)^(n) { z_t - (z_t - z_(t-1) ) × c } = ∑_(t=1)^(n) {z_t - (z_n - z_0 ) × c}
With 2 constants (z_0,c). Then, we should have:
correlation[ X_(n) , Z_(n) ] = correlation[ ∑_(t=1)^(n) {x_t} , ∑_(t=1)^(n) {z_t} ] + correlation( ∑_(t=1)^(n) {x_(t )} ,∑_(t=1)^(n) {-(z_n - z_0 ) c }
= correlation[ X_(n) , Z_(n) ] = correlation[ ∑_(t=1)^(n) {x_t} , ∑_(t=1)^(n) {z_t} ] + correlation( ∑_(t=1)^(n) {x_(t )} ,∑_(t=1)^(n) {-(z_n ) }
= correlation[ X_(n) , Z_(n) ] = correlation[ ∑_(t=1)^(n) {x_t} , ∑_(t=1)^(n) {z_t} ] + 0

 

[stapel]

Hello, thank you! I'm sorry, i'm not sure i was clear. Let me do it again to see if you agree with all steps:
X_(n) = ∑_(t=1)^(n) { x_t }
Z_(n) = ∑_(t=1)^(n) { z_t - (z_t - z_(t-1) ) × c } = ∑_(t=1)^(n) {z_t - (z_n - z_0 ) × c}
With 2 constants (z_0,c). Then, we should have:
correlation[ X_(n) , Z_(n) ] = correlation[ ∑_(t=1)^(n) {x_t} , ∑_(t=1)^(n) {z_t} ] + correlation( ∑_(t=1)^(n) {x_(t )} ,∑_(t=1)^(n) {-(z_n - z_0 ) c }
= correlation[ X_(n) , Z_(n) ] = correlation[ ∑_(t=1)^(n) {x_t} , ∑_(t=1)^(n) {z_t} ] + correlation( ∑_(t=1)^(n) {x_(t )} ,∑_(t=1)^(n) {-(z_n ) }
= correlation[ X_(n) , Z_(n) ] = correlation[ ∑_(t=1)^(n) {x_t} , ∑_(t=1)^(n) {z_t} ] + 0

Okay; now you've posted "Starting with (this), we have two (those), so then we have (these others)". But what is your *question*? Are you asking if your steps/conclusion/etc are correct? Was there an original exercise for which what you've posted is part of the solution?

 

[supermarco]

Hello, thank you for your response, no it's not a problem, just a question. My question is to know if you agree that the second term of correlation is 0.