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 $\mathrm{Var}[X]$.


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?

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

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