On the speed of convergence in the central limit theorem of log-likelihood ratio processes

Abstract

Let$\bigcirc H$, the parameter space, be an open subset ofR ^k,k⩾1. For each$\theta \in \bigcirc H$, let the r.v.'sX _n ,n=1, 2,... be defined on the probability space (X, ℱP _θ) and take values in (S,S,L) whereS is a Borel subset of a Euclidean space andL is the σ-field of Borel subsets ofS. Forh∈R _k and a sequence of p.d. normalizing matrices ∂ _n = ∂ ^{k × k} _n (θ₀ set θ ^* _n = θ^* = θ₀ + ∂ _n h, where θ₀ is the true value of θ, such that θ^*,$\theta _0 \in \bigcirc H$. Let Δ _n (θ^*, θ_*)( be the log-likelihood ratio of the probability measure$P_{n\theta ^* }$ with respect to the probability measure$P_{n\theta _0 }$, whereP _nθ is the restriction ofP _θ over ℱ _n = σ(X ₁,X ₂,...,X _n . In this paper we, under a very general dependence setup obtain a rate of convergence of the normalized log-likelihood ratio statistic to Standard Normal Variable. Two examples are taken into account.