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 \(\Phi^{N}\).