Existence of a Rate Function

Abstract

Let E be a Polish space and π(x,•) a transition probability function on E. Set Ω = E^η and let {P_x: x ∈ E} be the Markov family on Ω with transition function π (i.e. P_x (X(0) = x) = 1 and P_x is a Markov process with transition function π). For n ≥ 1 and ω ∈ Ω, define

$$ {L_n}(\Gamma, \omega ) = \frac{1}{n}\sum\limits_0^{{n - 1}} {{X_{\Gamma }}(X(k,\omega ))\quad, \quad \Gamma \in {B_E}} $$

, and

$$ L_n^l(\Gamma, \omega ) = L_n^l(\Gamma, {\theta_1}\omega ) = \frac{1}{n}\quad \sum\limits_1^n {{X_{\Gamma }}(X(k,\omega ))} $$

10. Finally, define Q_n,x and Q_n,x on m₁(m₁E)) to be the distribution of L_n and L ¹_n , respectively, under P_x.