A note on compatibility of conditional autoregressive models

Suppose that, to assess the joint distribution of a random vector $(X_1,ldots,X_n)$, one selects the kernels $Q_1,ldots,Q_n$ with $Q_i$ to be regarded as a possible conditional distribution for $X_i$ given $(X_j:j e i)$; $Q_1,ldots,Q_n$ are compatible if there exists a joint distribution for $(X_1,ldots,X_n)$ with conditionals $Q_1,ldots,Q_n$. Similarly, $Q_1,ldots,Q_n$ are improperly compatible if they can be obtained, according to the usual rule, with an improper distribution in place of a probability distribution. In this paper, compatibility and improper compatibility of $Q_1,ldots,Q_n$ are characterized under some assumptions on their functional form. The characterization applies, in particular, if each $Q_i$ belongs to a one parameter exponential family. Special attention is paid to Gaussian conditional autoregressive models.