## 2 Materials and Methods
### Deterministic Approximate EM Algorithm and Its Convergence for the Exponential Family
#### 2.1.2 Theorem  
In the following, we will state the convergence Theorem of Equation (3) and provide a brief description of the main steps of the proof. **Theorem 1** (Convergence of the Stable Approximate EM). : _Let \(\left\{\theta_{n}\right\}_{n\in\mathbb{N}}\) be a sequence of the Stable Approximate EM defined in Equation (3). Let us assume two sets of hypotheses:_  
* _Conditions M1-3 on the model._
* _M1._ \(\Theta\subseteq\mathbb{R}^{l}\)_,_ \(\mathcal{X}\subseteq\mathbb{R}^{d}\) _and_ \(\mu\) _is a_ \(\sigma\)_-finite positive Borel measure on_ \(\mathcal{X}\)_. Let_ \(\psi:\Theta\rightarrow\mathbb{R}\)_,_ \(\phi:\Theta\rightarrow\mathbb{R}^{d}\) _and_ \(S:\mathcal{X}\rightarrow\mathcal{S}\subseteq\mathbb{R}^{q}\)_. Define_ \(L:\mathcal{S}\times\Theta\rightarrow\mathbb{R}\)_,_ \(h:\mathcal{X}\times\Theta\rightarrow\mathbb{R}^{*}_{+}\) _and_ \(g:\Theta\rightarrow\mathbb{R}^{*}_{+}\) _as:_ \[L(s;\theta) :=\psi(\theta)+\left\langle s,\phi(\theta)\right\rangle,\] \[h(z;\theta) :=\exp(L(S(z);\theta))\,,\] \[g(\theta) :=\int_{z\in\mathcal{X}}h(z;\theta)\mu(dz)\,.\]
* _Assume that_
* _\(\psi\) _and_ \(\phi\) _are continuous on_ \(\Theta\)_;_
* _for all_ \(\theta\in\Theta\)_,_ \(\bar{S}(\theta):=\int_{z}S(z)p_{\theta}(z)\mu(dz)\) _is finite and continuous on_ \(\Theta\)_;_
* _there exists a continuous function_ \(\theta:\mathcal{S}\rightarrow\Theta\) _such that for all_ \(s\in\mathcal{S}\)_,_ \(L(s;\hat{\theta}(s))=\sup_{\theta\in\Theta}L(s;\theta)\)_;_
* _\(g\) _is positive, finite and continuous on_ \(\Theta\) _and the level set_ \(\left\{\theta\in\Theta,g(\theta)\geq M\right\}\) _is compact for any_ \(M>0\)_._
* _Define_ \(\mathcal{L}:=\left\{\theta\!\in\!\Theta|\hat{\theta}\circ\bar{S}(\theta)=\theta\right\}\) _and, for any_ \(g^{*}\in\mathbb{R}^{*}_{+}\)_,_ \(\mathcal{L}_{g^{*}}:=\left\{\theta\!\in\!\mathcal{L}|g(\theta)\!=\!g^{*}\right\}\)_. Assume either that:_
* _The set_ \(g(\mathcal{L})\) _is compact or_
* _for all compact sets_ \(K\subseteq\Theta\)_,_ \(g(K\cap\mathcal{L})\) _is finite._
* _The conditions on the approximation._ _Assume that_ \(\bar{p}_{\theta,n}(z)\) _is deterministic. Let_ \(S(z)=\left\{S_{u}(z)\right\}_{u=1}^{q}\)_. For all indices_ \(u\)_, for any compact set_ \(K\subseteq\Theta\)_, one of the two following configurations holds:_ \[ \begin{cases}\int_{z}S_{u}^{2}(z)dz<\infty\,,\\ \sup_{\theta\in K}\ \int_{z}(\bar{p}_{\theta,n}(z)-p_{\theta}(z))^{2}dz \underset{n\rightarrow\infty}{\longrightarrow}0\,.\end{cases} \] (4) _Or_ \[ \begin{cases}\sup_{\theta\in K}\ \int_{z}S_{u}^{2}(z)p_{\theta}(z)dz<\infty\,,\\ \sup_{\theta\in K}\ \int_{z}\left(\frac{\bar{p}_{\theta,n}(z)}{p_{\theta}(z)}-1 \right)^{2}p_{\theta}(z)dz\underset{n\rightarrow\infty}{\longrightarrow}0\,. \end{cases} \] (5) _Then,_
* \(\lim_{n\rightarrow\infty}j_{n}<\infty\) _and_ \(\sup_{n\in\mathbb{N}}\|\theta_{n}\|<\infty\)_, with probability_ \(1\)_;_
* \(g(\theta_{n})\) _converges towards a connected component of_ \(g(\mathcal{L})\)_._
* _Let_ \(Cl:\mathcal{P}(\Theta)\rightarrow\Theta\) _be the set closure function and_ \(d:\Theta\times\mathcal{P}(\Theta)\rightarrow\mathbb{R}_{+}\) _be any point-to-set distance function within_ \(\Theta\)_. If_ \(g\big{(}\mathcal{L}\cap Cl\big{(}\left\{\theta_{n}\right\}_{n\in\mathbb{N}} \big{)}\big{)}\) _has an empty interior, then_ \(\exists g^{*}\in\mathbb{R}^{*}_{+}\) _such that:_ \[g(\theta_{n})\underset{n\rightarrow\infty}{\longrightarrow}g^{*}\,,\] \[d(\theta_{n},\mathcal{L}_{g^{*}})\underset{n\rightarrow\infty}{ \longrightarrow}0\,.\]  
**Remark 1**. :
* _When_ \(g\) _is differentiable on_ \(\Theta\)_, its stationary points are the stable points of the EM, i.e.,_ \(\mathcal{L}=\{\theta\in\Theta|\
abla g(\theta)=0\}\)_._
* _The M_1_-_3 _conditions in Theorem_ 1 _are almost identical to the similarly named M_1_-_3 _conditions in_ _[_7_]__.