*Bounty: 50*

*Bounty: 50*

Burg’s Theorem characterizes the form of an entropy-maximizing time series, subject to constraints on the autocorrelation. More precisely, the theorem states that the autoregressive Gaussian process $X_i=sum_{j=1}^{K} a_j X_{i-j}+epsilon_i$ attains the maximal entropy rate among all stochastic processes satisfying the conditions $E X_iX_{i+k}=alpha_k$, $1leq kleq K$. Here $K$ is fixed, and $epsilon_i$ are iid mean zero Gaussian.

I am interested in a generalization in which the cross-correlation with a second,fixed time series ${Y_i}_i$ is also constrained. More precisely, the problem is to characterize the time series $X_i$ with maximal entropy rate satisfying the constraints $EX_iX_{i+k}=alpha_k, 1leq kleq K_1$ and $EX_iY_{i+k}=beta_k, 1leq kleq K_2$. (By "characterize", I mean an explicit functional form akin to the above theorem).

Can the time series ${X_i}$ be explicitly charaterized in this case, and is there a reference to such a theorem in the literature?