Local convergence for exponential mixture family

From Redner, Walker 1984

Theorem 5.2.  Suppose that the Fisher information matrix $I(\Phi)$ is positive definite at the true parameter $\Phi^*$ and that $\Phi^* = (\alpha_1^*, \dotsc, \alpha_m^*, \phi_1^*, \dotsc, \phi_m^*)$ is such that $\alpha_i^* > 0 \text{ for } i = 1,\dotsc,m$.  For $\Phi^{(0)} \in \Omega$, denote by $\{\Phi^{(j)}\}_{j=0,1,2,\dotsc}$ the sequence in $\Omega$ generated by the EM iteration.  Then with probability 1, whenever N is sufficiently large, the unique strongly consistent solution $\Phi^N = (\alpha_1^N, \dotsc, \alpha_m^N, \phi_1^N, \dotsc, \phi_m^N)$ of the likelihood equations is well defined and there is a certain norm on $\Omega$ in which  $\{\Phi^{(j)}\}_{j=0,1,2,\dotsc}$ converges linearly to $\Phi^N$ whenever $\Phi^{(0)}$ is sufficiently near $\Phi^N$, i.e. there is a constant $0 \leq \lambda < 1$, for which
$$\lVert \Phi^{(j+1)} - \Phi^N \rVert \leq \lambda \lVert \Phi^{(j)} - \Phi^N \rVert, \quad j = 0,1,2,\dotsc$$ whenever $\Phi^{(0)}$ is sufficiently near $\Phi^{N}$.