Joint characterisitc function

[lolypop]

1. The problem statement

X1,X2,X3...Xn are RVs with the same mean and covariance function as follow :
COV(Xi,Xj)=ρ (σ)^|i-j|

if Sn=X1+X2+...Xm , Sm=X1+X2+...Xm where m<n
find the joint characterisitic function of Sn and Sm
2. Homework Equations .
the general equation of the characterisitic function

3. The Attempt at a Solution

I tried using the joint characterisitic function for Sn and Sm but got stuck since they didn't specified the distribution of Xi . but there was a hint in the question I couldn't use that is to condition on Sm but how to use it since Xi are dependent (since the Cov(Xi,Xj)not= zero (so they are uncorrelated )?!

I'd really appreciate a small hint to know exactly where to start

lolypop

 

[lippyka]

's attempt at a solutionLet F(s,t) be the joint characteristic function of Sn and Sm. Then we can express F(s,t) in terms of the individual characteristic functions of Xi as follows:F(s,t) = E[exp(isSn + itSm)] = E[exp(is(X1 + X2 + ... + Xm) + it(X1 + X2 + ... + Xm))] = E[exp(isX1 + itX1 + isX2 + itX2 + ... + isXm + itXm)] = E[exp(isX1 + itX1)]E[exp(isX2 + itX2)]...E[exp(isXm + itXm)] = F1(s,t)F2(s,t)...Fm(s,t)where F1(s,t), F2(s,t), ..., Fm(s,t) are the individual characteristic functions of Xi. From the given information, we know that the covariance between Xi and Xj is equal to ρσ^|i-j|, so we can use this to determine the individual characteristic functions of Xi. Specifically, we have F1(s,t) = E[exp(isX1 + itX1)] = E[exp(isX1)exp(itX1)] = E[exp(isX1)]E[exp(itX1)] = exp(ρσs^2/2)exp(ρσt^2/2) Similarly, for i > 1, we have Fi(s,t) = E[exp(isXi + itXi)] = E[exp(isXi)exp(itXi)] = E[exp(isXi)]E[exp(itXi)] = exp(ρσs^i)exp(ρσt^i) Therefore, the joint characteristic function of Sn and Sm can be expressed as