## Appendix A Proofs of the Two Main Theorems
### Proof of the General Theorem
#### a.1.1 Verifying the Conditions of Proposition 2  
\(g\) is the likelihood function of a model of the curved exponential family. Let \(T\) be the point to point map describing the transition between \(\theta_{n}\) and \(\theta_{n+1}\) in the exact EM algorithm. \(\mathcal{L}\) the set of stationary points by \(T\): \(\mathcal{L}:=\{\theta\in\Theta|T(\theta)=\theta\}\). (Note that if \(g\) is differentiable, the general properties of the EM tell us that its critical points of \(g\) are the stationary points: \(\mathcal{L}=\{\theta\in\Theta|\
abla g(\theta)=0\}\)). Additionally, \(g\) is a \(C^{0}\) Lyapunov function associated to \((T,\mathcal{L})\). Let \(\{\theta_{n}\}_{n}\) be the sequence defined by the stable approximate EM with \(\{F_{n}\}_{n\in\mathbb{N}}\) our sequence of point to point maps. We verify that under this framework--and with the assumptions of Theorem 1--we check the conditions of Proposition 2. As in [7], \(M1\)-3 directly implies \(C1\)-2. Let us show that we have the last two conditions for Proposition 2:  
\[\forall\theta\in K_{0},\quad\lim_{n\to\infty}|g\circ F_{n}-g\circ T|(\theta)= 0\,, \tag{101}\]  
and  
\[\forall\text{ compact }K\subseteq\Theta,\quad\lim_{n\to\infty}|g\circ F_{n}( \theta_{n})-g\circ T(\theta_{n})|\mathds{1}_{\theta_{n}\in K}=0\,. \tag{102}\]  
We focus on (102), since (101) is easier to verify and will come from the same reasoning. The first steps are similar to [7]. We underline the most consequent deviations from the proof of [7] when they occur. Equivalent Formulation of the Convergence  
We write Equation (102) under an equivalent form. Let \(\tilde{S}_{n}(\theta):=\{\int_{z}S_{u}(z)\hat{p}_{\theta,n}(z)dz\}_{u=1}^{q}\) and \(\tilde{S}(\theta):=\big{\{}\int_{z}S_{u}(z)p_{\theta}(z)dz\big{\}}_{u=1}^{q}\). Then \(F_{n}(\theta_{n})=\hat{\theta}(\tilde{S}_{n}(\theta_{n}))\) and \(T(\theta_{n})=\hat{\theta}(\tilde{S}(\theta_{n}))\). Hence \(|g\circ F_{n}(u_{n})-g\circ T(u_{n})|=|g\circ\hat{\theta}(\tilde{S}_{n}(\theta _{n}))-g\circ\hat{\theta}(\tilde{S}(\theta_{n}))|\). To show Equation (102):  
\[|g\circ\hat{\theta}(\tilde{S}_{n}(\theta_{n}))-g\circ\hat{\theta}(\tilde{S}( \theta_{n}))|\mathds{1}_{\theta_{n}\in K}\underset{n\to\infty}{\to}0\,,\]  
it is sufficient and necessary to have:  
\[\forall\epsilon>0,\,\exists N\in\mathbb{N},\,\forall n\geq N,\,|g\circ\hat{ \theta}(\tilde{S}_{n}(\theta_{n}))-g\circ\hat{\theta}(\tilde{S}(\theta_{n})) |\mathds{1}_{\theta_{n}\in K}\leq\epsilon\,.\]  
An other equivalent formulation is that there are a finite number of integers \(n\) such that \(|g\circ\hat{\theta}(\tilde{S}_{n}(\theta_{n}))-g\circ\hat{\theta}(\tilde{S}( \theta_{n}))|\mathds{1}_{\theta_{n}\in K}>\epsilon\), in other words:  
\[\forall\epsilon>0,\,\sum_{n=1}^{\infty}\mathds{1}_{|g\circ\hat{\theta}(\tilde {S}_{n}(\theta_{n}))-g\circ\hat{\theta}(\tilde{S}(\theta_{n}))|\mathds{1}_{ \theta_{n}\in K}>\epsilon}<\infty\,.\]  
We aim to relate the proximity between the images \(g\circ\hat{\theta}\) of to the proximity between the antecedents of \(g\circ\hat{\theta}\). The function \(g\circ\hat{\theta}:\mathbb{R}^{q}\to\mathbb{R}\) is continuous, but not necessarily uniformly continuous on \(\mathbb{R}^{q}\). As a consequence, we will need to restrict ourselves to a compact to get uniform continuity properties. We already have a provident compact \(K\). \(\tilde{S}:\Theta\to\mathbb{R}^{l}\) is continuous, hence \(S(K)\) is a compact as well. Let \(\delta\) be a strictly positive real number. Let \(\tilde{S}(K,\delta):=\left\{s\in\mathbb{R}^{q}\middle|\underset{t\in K}{inf} \middle|\left|\tilde{S}(t)-s\right|\right|\leq\delta\right\}\). Where we use any norm \(\left|\left|\cdot\right|\right|\) on \(\mathbb{R}^{q}\) since they are all equivalent. \(\tilde{S}(K,\delta)\) is a compact set as well.