## 2 Materials and Methods
### Deterministic Approximate EM Algorithm and Its Convergence for the Exponential Family
#### 2.1.3 Sketch of Proof
Verifying the conditions of Proposition 2.Let \(g\) be the observed likelihood function defined in hypothesis \(M1\) of Theorem 1, \(T\) the exact EM point to point map defined in (1), \(\mathcal{L}:=\{\theta\in\Theta|T(\theta)=\theta\}\) the set of its stable points, \(\{F_{n}\}_{n}\) a sequence of approximated point to point map as defined in (2). With some increasing sequence \(\{K_{n}\}_{n\in\mathbb{N}}\) of compacts of \(\Theta\), let \(\{\theta_{n},j_{n}\}_{n\in\mathbb{N}}\) be defined from the Stable Approximate EM equation (3). By design of the regular EM, the following two properties are true. First, \(g\) is a \(C^{0}\), Lyapunov function relative to \((T,\mathcal{L})\). Second, the map \(T\) can be written \(T:=\hat{\theta}\circ\tilde{S}\), with the \(\hat{\theta}\) and \(\tilde{S}\) defined in condition \(M2\) of Theorem 1. These properties, in conjunction with hypotheses \(M1-3\) of Theorem 1, directly imply that condition \((a)\) of Proposition 2 is verified. The steps to verify conditions \((b)\) and \((c)\) differ from those in the proof of [7]. Let \(\tilde{S}_{n}(\theta_{n})=\int_{2}S(z)\,\bar{\rho}_{\theta,n}(z)\mu(dz)\) be the approximated E step in the Stable Approximate EM, such that \(F_{n}=\hat{\theta}\circ\tilde{S}_{n}\). By using uniform continuity properties on compacts, we prove that the following condition is sufficient to verify both \((a)\) and \((b)\):  
\[\forall\text{ compact }K,\ \sup_{\theta\in K}\,\left\|\tilde{S}_{n}(\theta)- \tilde{S}(\theta)\right\|\,\underset{n\to\infty}{\longrightarrow}\,0\,. \tag{6}\]  
Then, upon replacing \(\tilde{S}_{n}\) and \(\tilde{S}\) by their integral forms, it becomes clear that each of the hypotheses (4) or (5) of Theorem 1 is sufficient to verify (6). In the end, all conditions are verified to apply Proposition 2. Applying Proposition 2.The application of Proposition 2 to the Stable Approximate EM tells us that with probability \(1\):  
\(\limsup_{n\to\infty}j_{n}<\infty\) and \(\{\theta_{n}\}_{n\in\mathbb{N}}\) is a compact sequence. Which is specifically the result \((i)(a)\) of Theorem 1. Verifying the conditions of Proposition 1.We already have that the likelihood \(g\) is a \(C^{0}\), Lyapunov function relative to \((T,\mathcal{L})\). Thanks to Proposition 2, we have that \(K:=Cl(\{\theta_{n}\}_{n})\) is a compact. Then, \(\mathcal{L}\cap K\) is also compact thanks to hypothesis \(M3\). Moreover, by definition,the EM sequence verifies: \(\left\{\theta_{n}\right\}_{n}\in K^{\mathbb{N}}\). The last condition that remains to be shown to apply Proposition 1 is that:  
\[\lim_{n\to\infty}\lvert g(\theta_{n+1})-g\circ T(\theta_{n})\rvert=0\,.\]  
If we apply \((c)\) of Proposition 2 with \(K=Cl(\left\{\theta_{n}\right\}_{n})\), we get an almost identical result, but with \(\theta_{n+1}\) replaced by \(F_{n}(\theta_{n})\). However, \(F_{n}(\theta_{n})\
eq\theta_{n+1}\) only when \(j_{n+1}=j_{n}+1\):  
\[\lvert g(\theta_{n+1})-g\circ T(\theta_{n})\rvert=\lvert g(\theta_{0})-g\circ T (\theta_{n})\rvert\mathds{1}_{j_{n+1}=j_{n}+1}+\lvert g\circ F_{n}(\theta_{n}) -g\circ T(\theta_{n})\rvert\mathds{1}_{j_{n+1}=j_{n}}\,.\]  
We have proven with Proposition 2 that there is only a finite number of such increments. Hence, when \(n\) is large enough, \(F_{n}(\theta_{n})=\theta_{n+1}\) always, and we have the desired result. **Applying Proposition 1**. : Since we verify all we need to apply the conclusions of Proposition 1:  
* \(\left\{g(\theta_{n})\right\}_{n\in\mathbb{N}}\) converges towards a connected component of \(g(\mathcal{L}\cap Cl(\left\{\theta_{n}\right\}_{n}))\subset g(\mathcal{L})\). * If \(g(\mathcal{L}\cap Cl(\left\{\theta_{n}\right\}_{n}))\) has an empty interior, then the sequence \(\left\{g(\theta_{n})\right\}_{n\in\mathbb{N}}\) converges towards a \(g^{*}\in\mathbb{R}\) and \(\left\{\theta_{n}\right\}_{n}\) converges towards the set \(\mathcal{L}_{g^{*}}\cap Cl(\left\{\theta_{n}\right\}_{n})\subseteq\mathcal{L} _{g^{*}}\). Which are both respectively exactly \((i)(b)\) and \((ii)\) of Theorem 1 and conclude the proof of the Theorem.