$\mathcal{F}$. The conditional expectation of $X$ relative to the $\sigma$-algebra $\mathcal{G}$ is a random variable denoted by $\mathbb{E}(X/\mathcal{G})$ satisfying

1.  $ \mathbb{E}(X/\mathcal{G})$ is $\mathcal{G}$-measurable.

2.  $\int_G\mathbb{E}(X/\mathcal{G})d\mathbb{P}=\int_GXd\mathbb{P}, \hspace{0.5cm}\forall \hspace{0.2cm} G\in \mathcal{G}$.

In the litterature, $\mathbb{E}(X/\mathcal{G})$ is called the projection of $X$ upon $\mathcal{G}$.

The proof of the following theorem can be seen in [@Phi].

1.  $ \mathbb{E}(\mathbb{E}(X/\mathcal{G}))=\mathbb{E}(X)$.

2.  If $X$ is $\mathcal{G}$-measurable, then $\mathbb{E}(X/\mathcal{G})=X$.

3.  $\mathbb{E}((X+Y)/\mathcal{G})=\mathbb{E}(X/\mathcal{G})+\mathbb{E}(Y/\mathcal{G})$.

4.  If $\mathcal{G}\subset \mathcal{G}'$ then $\mathbb{E}(X/\mathcal{G}')=\mathbb{E}.(\mathbb{E}(X/\mathcal{G})/\mathcal{G}')$.

5.  If $\sigma(X)$ and $\mathcal{G}$ are independent, then $\mathbb{E}(X/\mathcal{G})=\mathbb{E}(X)$.

6.  If $X\leq Y$ a.s, then $\mathbb{E}(X/\mathcal{G})\leq \mathbb{E}(Y/\mathcal{G})$.

7.  If $X$ is $\mathcal{G}$ measurable, then $\mathbb{E}(XY/\mathcal{G})=X\mathbb{E}(Y/\mathcal{G})$.

### Convergence of random variables

.

Let $p\in[1,\infty)$, we denote by $\mathbb{L}^p(\Omega, \mathbb{R}^n)$ the equivalence class of measurable functions $X :\Omega : \longrightarrow \mathbb{R}^n $, $\mathcal{F}_t$-measurable such that $$\begin{aligned}
||X||^p_{\mathbb{L}^p(\Omega,\mathbb{R}^n)} := \mathbb{E}(||X||^p) =\int_{\Omega}||X(\omega)||^pd\mathbb{P}(\omega)<+\infty.\end{aligned}$$ Let $(X_n)\subset \mathbb{L}^p(\Omega, \mathbb{R}^n)$ be a sequence of random variables and $X\in \mathbb{L}^p(\omega, \mathbb{R}^n)$ a random variable. Let $$\begin{aligned}
N:=\{\omega : \lim_{n\longrightarrow \infty}X_n(\omega)=X(\omega)\}\end{aligned}$$

1.  $(X_n)$ converges to $X$ almost surely if $N^c$ is negligeable.

2.  $(X_n)$ converges in probability to $X$ if $$\begin{aligned}
    \forall\hspace{0.2cm}\epsilon >0\hspace{0.3cm}\lim_{n\longrightarrow\infty}\mathbb{P}(||X_n-X||>\epsilon)=0.\end{aligned}$$

3.  $(X_n)$ converges in $\mathbb{L}^p$ to $X$ if $$\begin{aligned}
     \lim_{n\longrightarrow +\infty}\mathbb{E}(||X_n-X||^p)=0.
     \end{aligned}$$

: **\[Frobenius norm\]**

The Frobenius norm of a $m\times n$ matrix $A=(a_{ij})_{1\leq i\leq n; 