On the maximum entropy principle for a class of stochastic processes
This paper extends results of \textit{E. Bolthausen} and the reviewer [ibid. 33, No. 1, 1-27 (1989; Zbl 0691.60023)] on the asymptotic behaviour of certain Laplace-type transformations of Markov chain path measures to a more general class of processes. Let \(P\) be a probability measure on \(\Omega = S^{\mathbb{N}_0}\), where \(S\) is a Polish space, and let \(L_n : \Omega \to {\mathcal M}_1 (\Omega)\) denote the \(n\)-cyclic empirical-process measure. For a suitable functional \(F : {\mathcal M}_1 (\Omega) \to \mathbb{R} \cup \{- \infty\}\) define the transformed path measures \(\{\widehat P_n\}_{n \in \mathbb{N}} \subset {\mathcal M}_1 (\Omega)\) by \(d \widehat P_n = Z_n^{-1} \exp (nF (L_n)) dP\). Assume that \(\{PL_n^{-1}\}_{n \in \mathbb{N}}\) satisfies a process-level large deviation principle with rate function \(I\) in the weak topology. If, in addition, there exist continuous, suitably bounded transition densities \(k_n : S^{n + 1} \to (0, \infty)\) of \(P\) satisfying an Orey-type fading condition, the authors prove that \(\{\widehat P_n\}_{n \in \mathbb{N}}\) is tight in \({\mathcal M}_1 (\Omega)\) and that every weak accumulation point is a mixture of those path measures in \({\mathcal M}_1 (\Omega)\) which maximize \(\overline F - I\), where \(\overline F\) is the upper-semicontinuous version of \(F\). This result includes certain conditional limit theorems.
scientific article; zbMATH DE number 757553