Covariance matrices of self-affine measures

In this paper we derive a formula for the covariance matrix of any self-affine measure, i.e. a probability measure μμ satisfying μ=∑k=1lpkμ∘Sk−1, where {Sk(x)=Akx+bk}1≤k≤l is a family of affine contractive maps and {pk}1≤k≤l{pk}1≤k≤l is a set of probability weights. In particular if for every kk, Ak=AAk=A then the formula has the following form D2X=[I⊗I−A⊗A]−1D2B,D2X=[I⊗I−A⊗A]−1D2B, where D2XD2X denotes the covariance matrix of the measure μμ and D2BD2B denotes a covariance matrix of a discrete random variable BB with values bk, and corresponding probabilities pkpk.