Large deviation for empirical field of a symmetric measure (Q1192387)

Let \(F\) be a finite set and let \(\Omega=F^ N\) be the compact product space. Denote by \(P(\Omega)\) the compact space of probability measures on \(\Omega\) and by \(P_ s(\Omega)\) its subspace in which all the elements are invariant under shifts, \(\theta_ i: \Omega\to\Omega\), \(i\in\mathbb{N}\). Let \(\mu\) denote a symmetric measure on \(\Omega\). It has the representation \(\mu=\int_{P(F)}\rho^ N m(d\rho)\), where \(m\) is a probability measure on the space \(P(F)\) of all probability measures on \(F\). Define the empirical field \(R_ n\) on \(\Omega\) by \(R_ n(\omega)={1\over n}\sum^{n-1}_{i=0}\delta_{\theta_ i\omega}\), where \(\delta_ \omega\) is the Dirac measure at \(\omega\), \(\omega\in\Omega\). Let \(h(\nu,\rho^ N)\) denote Donsker-Varadhan's specific entropy of \(\nu\) with respect to \(\rho^ N\) and let \(I(\nu)=\inf_{\rho\in\text{supp}(m)}h(\nu,\rho^ N)\). The author obtains the lower bound (upper bound) for \(\lim_{n\to\infty}{1\over n}\log\mu(R_ n,A)\) when \(A\) is an open subset (a closed subset) of \(P(\Omega)\). The bounds are given in terms of \(\inf_{\nu\in A\cap P_ s(\Omega)}I(\nu)\). While obtaining the lower bound, the author extends the Shannon-McMillan theorem.