The LIL of the MLE of the Nonhomogenous Poisson Processes

On the interval of time 0, T, a realization of a random process X（T） = X（t）, 0＜t＜T, characteristic of which, （for instance, mean） depends on a parameterθ∈Θ unknown to the observer, is observed. It is required to construct sufficiently good estimates θ_T = θ_T （X_T） of the parameter θ from a realization of X（T） and describo its properties. A commonly used estimator for θ is t e maximun likelihood estimator （MLE）. The properties of MLE of non-homogenous processes have been discussed in Kutoyants （1）, where he established the weak consistency, limit distributions and the convergence of the moments of the MLE. But he demanded that the parameter space was a bounded interval （α, β）. In this paper, we will consider the non-homogenous Poisson processes. \hat\theta_T is the MLE on Θ_T= （α, β_T）, where lim_T→∞β_T. We will give some conditions, under which, the LIL of the MLB \hat\theta_T holds.