## 2 Materials and Methods
### Deterministic Approximate EM Algorithm and Its Convergence for the Exponential Family
#### 2.1.3 Sketch of Proof  
The detailed proof of this result can be found in Appendix A.1, we propose here an abbreviated version where we highlight the key steps. Two intermediary propositions, introduced and proven in [PERSON] and Moulines [7], are instrumental in the proof of Theorem 1. These two propositions are called Propositions 9 and 11 by [PERSON] and Moulines, and used to prove their Theorem 3. Within our framework, with the new condition \(M2(a^{*})\) and the absence of Monte Carlo sum, the reasoning for verifying the conditions of applicability of the two proposition is quite different from [7] and will be highlighted below. Let us state these two propositions using the notations of this paper:  
**Proposition 1** (\"Proposition 9\"). : _Consider a parameter space \(\Theta\subseteq\mathbb{R}^{l}\), a point to point map \(T\) on \(\Theta\), a compact \(K\subset\Theta\) and a subset \(\mathcal{L}\subseteq\Theta\) such that \(\mathcal{L}\cap K\) is compact. Let \(g\) be a \(C^{0}\), Lyapunov function relative to \((T,\mathcal{L})\). Assume that there exist a \(K-\)valued sequence \(\left\{\theta_{n}\right\}_{n\in\mathbb{N}^{*}}\in K^{\mathbb{N}}\) such that:_  
\[\lim_{n\to\infty}\left|g(\theta_{n+1})-g\circ T(\theta_{n})\right|=0\,.\]  
_Then_  
* \(\left\{g(\theta_{n})\right\}_{n\in\mathbb{N}}\) _converges towards a connected component of_ \(g(\mathcal{L}\cap K)\)__
* _If_ \(g(\mathcal{L}\cap K)\) _has an empty interior, then_ \(\exists g^{*}\in\mathbb{R}^{*}_{+}\) _such that_ \(\left\{g(\theta_{n})\right\}_{n}\) _converges towards_ \(g^{*}\)_. Moreover,_ \(\left\{\theta_{n}\right\}_{n}\) _converges towards the set_ \(\mathcal{L}_{g^{*}}\cap K\)_. Where_ \(\mathcal{L}_{g^{*}}:=\left\{\theta\in\mathcal{L}|g(\theta)=g^{*}\right\}\)_._  
**Proposition 2** (\"Proposition 11\"). : _Consider a parameter space \(\Theta\subseteq\mathbb{R}^{l}\), a subset \(\mathcal{L}\subseteq\Theta\), a point to point map \(T\) on \(\Theta\) and a sequence of point to point maps \(\left\{F_{n}\right\}_{n\in\mathbb{N}^{*}}\) also on \(\Theta\). Let \(\left\{\theta_{n}\right\}_{n\in\mathbb{N}}\in\Theta^{\mathbb{N}}\) be a sequence defined from \(\left\{F_{n}\right\}_{n}\) by the Stable Approximate EM equation (3) for some increasing sequence \(\left\{K_{n}\right\}_{n\in\mathbb{N}}\) of compacts of \(\Theta\). Let \(\left\{j_{n}\right\}_{n\in\mathbb{N}}\) be the corresponding sequence of indices, also defined in (3), such that \(\forall n,\ \theta_{n}\in K_{j_{n}}\). We assume:_  
* _the_ \(C1\)_-_2 _conditions of Proposition_ 10 _of_ _[_7_]__._
* _the_ \(C2\)_-_3 _conditions of Proposition_ 10 _of_ _[_7_]__._
* _There exists_ \(g\)_, a_ \(C^{0}\) _Lyapunov function relative to_ \((T,\mathcal{L})\) _such that for all_ \(M>0\)_,_ \(\{\theta\in\Theta|g(\theta)>M\}\) _is compact, and:_ \[\Theta=\cup_{n\in\mathbb{N}}\left\{\theta\in\Theta|g(\theta)>n^{-1}\right\}.\]
* \(g(\mathcal{L})\) _is compact OR_ \((\textbf{C2})\,g(\mathcal{L}\cap K)\) _is finite for all compact_ \(K\subseteq\Theta\)_._
* \(\forall\theta\in K_{0},\quad\lim_{n\to\infty}|g\circ F_{n}-g\circ T|(\theta)=0\)__
* \(\forall\) _compact_ \(K\subseteq\Theta\)_:_ \[\lim_{n\to\infty}|g\circ F_{n}(\theta_{n})-g\circ T(\theta_{n})|\mathds{1}_{ \theta_{n}\in K}=0\,.\] _Then:_ _With probability_ \(1\)_,_ \(\limsup_{n\to\infty}j_{n}<\infty\) _and_ \(\{\theta_{n}\}_{n}\) _is a compact sequence._  
**Remark 2**. : _In [7], condition \(C1\) of Proposition 2 is mistakenly written as:_  
\[\Theta=\cup_{n\in\mathbb{N}}\left\{\theta\in\Theta|W(\theta)>n\right\}.\]  
_This is a typo that we have corrected here._  
The proof of Theorem 1 is structured as follows: verifying the conditions of Proposition 2, applying Proposition 2, verifying the conditions of Proposition 1 and finally applying Proposition 1.