# Given that $X_{i+1} = \rho X_{i}$, determine the dispersion matrix $Var[\textbf{X}]$

If $$X_{1},X_{2},\ldots,X_{n}$$ are random variables and $$X_{i+1} = \rho X_{i}$$ $$(i = 1,2,\ldots,n)$$, where $$\rho$$ is constant, and $$\mathrm{Var}[X_1] = \sigma^2$$, find $$\mathrm{Var}[X]$$.

MY ATTEMPT

According to the problem setting, we conclude that $$X_{k} = \rho^{k-1}X_{1}$$. Consequently, we have \begin{align*} \mathrm{Cov}(X_{i},X_{j}) = \mathrm{Cov}(\rho^{i-1}X_{1},\rho^{j-1}X_{1}) = \rho^{i+j-2}\mathrm{Var}(X_{1}) = \sigma^{2}\rho^{i+j-2} \end{align*}

From whence the answer becomes clear. Could someone double-check my reasoning?

• What is $X$? Otherwise you're just using the bilinearity of the covariance which is valid. – Tony S.F. Apr 14 at 8:55