Asymptotic normality of the maximum likelihood

We present conditions to obtain the asymptotic normality of the maximum likelihood estimator of a loss process presented in [2]. We shall use the notations of [2], write ‖ · ‖q for the standard L norm on an arbitrary space R, d ≥ 1, and let D φ denote the k−th order di erentiation with respect to φ. Let us introduce the following hypotheses: (A4) For all i ∈ {1, . . . , r}, λi(Φ0) > 0. (A5) For all e, if PΦ0(E = e) > 0, then for all j ∈ {1, . . . , r}, pj(e, Φ0) > 0. (A6) For all j ∈ {1, . . . , r}, all e 6= ∅ and all x, the map φ 7→ Pθ(j, e, φ)(m ∈ e⇒ Xm = xm) is twice continuously di erentiable in FΦ0 . (A7) For all i, j ∈ {1, . . . , r} and all e 6= ∅, there exists a neighborhood G of Φ0 in FΦ0 such that 1. ∫ sup φ∈G ∥∥Dφ lnPθ(i, e, φ)(m ∈ e⇒ Xm = xm)∥∥22 Pθ(j, e,Φ0)(m ∈ e⇒ Xm = xm) dx <∞. 2. ∫ sup φ∈G ∥∥D2 φ lnPθ(i, e, φ)(m ∈ e⇒ Xm = xm)∥∥1 Pθ(j, e,Φ0)(m ∈ e⇒ Xm = xm) dx <∞.