Asympotic Properties of MLE of the Parameters for Point Processes

Suppase N（t）, t≥0 is a point process on probability space （Ω,F,P_θ） with intensity λ（θ,t）, where θ=（θ₁,…’,θ_m） is unknown parameter. Let θ^*=（θ₁^*,… θ_ｍ^*）' be the true value of θ. The observed data areN（t）, 0 ≤t≤T. Denote the MLE of θ as ^θ_T. In this paper we give that, under some conditions, the MLE of θ is consistent, and \beginaligned & \limsup _T \rightarrow \infty \frac\lambda_\min (T)\left\|\widehat\theta_T-\theta^*\right\|\sqrt2 h^2(T) \mathrmLLg\left(h^2(T)\right) \stackrel\text a.s. \leq 1 \\ & \limsup _T \rightarrow \infty \fracc \lambda_\max (T)\left\|\widehat\theta_T-\theta^*\right\|\sqrt2 h^2(T) \mathrmLLg\left(h^2(T)\right) \stackrel\text a.s. \geq 1\endaligned where c ＞ 0 and h（T） are obtained from λ（t,θ^*）, λ_max（T）（λ_min（T）） is the maximum （minimum） among theeigenvalues of the information matrix.