We denote
$$p_{+} = \mathbb{P}(X' \in C_{+1}) \quad \text{and} \quad p_{-} = \mathbb{P}(X' \in C_{-1})$$
We will assume, without loss of generality, that $p_{+} \geqslant p_{-}$. Using the induction hypothesis, justify that
$$\mathbb{E}\left(\left.\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right\rvert \, \varepsilon_{n} = 1\right) \leqslant \frac{1}{p_{+}}$$

We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ We denote by $u$ and $v$ the real and imaginary parts of $f$, so that, for any $(x,y) \in D(0,R)$, $$u(x,y) \in \mathbb{R}, \quad v(x,y) \in \mathbb{R}, \quad f(x,y) = u(x,y) + \mathrm{i} v(x,y).$$ Show that $u$ and $v$ are harmonic functions on $D(0,R)$.

We denote
$$p_{+} = \mathbb{P}(X' \in C_{+1}) \quad \text{and} \quad p_{-} = \mathbb{P}(X' \in C_{-1})$$
We will assume, without loss of generality, that $p_{+} \geqslant p_{-}$.
Deduce from the above that for all $\lambda$ in $[0, 1]$
$$\mathbb{E}\left(\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right) \leqslant \frac{1}{2}\left(\frac{1}{p_{+}} + \exp\left(\frac{\lambda^{2}}{2}\right) \frac{1}{(p_{-})^{1-\lambda}} \cdot \frac{1}{(p_{+})^{\lambda}}\right)$$

We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ We admit the following result: a function $h$ from $D(0,R)$ to $\mathbb{C}$ expands in a power series on $D(0,R)$ if and only if $h$ is of class $\mathcal{C}^1$ on $D(0,R)$ and for all $(x,y) \in D(0,R)$, $\frac{\partial h}{\partial y}(x,y) = \mathrm{i} \frac{\partial h}{\partial x}(x,y)$.
Show that if $f$ does not vanish on $D(0,R)$ then $1/f$ expands in a power series on $D(0,R)$.

We denote
$$p_{+} = \mathbb{P}(X' \in C_{+1}) \quad \text{and} \quad p_{-} = \mathbb{P}(X' \in C_{-1})$$
We will assume, without loss of generality, that $p_{+} \geqslant p_{-}$. We set $\lambda = 1 - \frac{p_{-}}{p_{+}}$. Show that
$$\mathbb{E}\left(\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right) \leqslant \frac{1}{2p_{+}}\left(1 + \exp\left(\frac{\lambda^{2}}{2}\right) (1 - \lambda)^{\lambda - 1}\right)$$

We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ We denote by $u$ and $v$ the real and imaginary parts of $f$. Show that the function $uv$ is harmonic on $D(0,R)$.

Show that for all $x \in [0, 1[$
$$\frac{x^{2}}{2} + (x - 1) \ln(1 - x) \leqslant \ln(2 + x) - \ln(2 - x)$$
One may perform a function study.

Let $g$ be a function from $D(0,R) \subset \mathbb{R}^2$ to $\mathbb{R}$. We assume that $g$ is harmonic. Show that the function $h$ defined on $D(0,R)$ by $$h : (x,y) \longmapsto \frac{\partial g}{\partial x}(x,y) - \mathrm{i} \frac{\partial g}{\partial y}(x,y)$$ expands in a power series on $D(0,R)$.

Deduce that for all $x \in [0, 1[$
$$1 + \exp\left(\frac{x^{2}}{2}\right)(1 - x)^{x - 1} \leqslant \frac{4}{2 - x}$$

Let $g$ be a function from $D(0,R) \subset \mathbb{R}^2$ to $\mathbb{R}$. We assume that $g$ is harmonic. Show that if $g$ belongs to $\mathcal{H}(D(0,R))$ then there exists a function $H$ that expands in a power series on $D(0,R)$ such that $g$ is the real part of $H$.
One may consider a power series that is a primitive of the power series associated with the function $h$ from the previous question.

Complete the proof of inequality
$$\mathbb{P}(X \in C) \cdot \mathbb{E}\left(\exp\left(\frac{1}{8} d(X, C)^{2}\right)\right) \leqslant 1 \tag{II.1}$$

Throughout this part, $f$ denotes a function that expands in a power series on $D(0,R)$, i.e., $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ Show that for all $r \in [0, R[$, we have $f(0) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos(t), r\sin(t)) \, \mathrm{d}t$.

Deduce Talagrand's inequality: For every non-empty closed convex set $C$ of $E$ and for every strictly positive real number $t$
$$\mathbb{P}(X \in C) \cdot \mathbb{P}(d(X, C) \geqslant t) \leqslant \exp\left(-\frac{t^{2}}{8}\right)$$

Show an analogous result to Q32 for harmonic functions: for a harmonic function $g$ on $D(0,R)$, show that for all $r \in [0, R[$, $g(0) = \frac{1}{2\pi} \int_0^{2\pi} g(r\cos(t), r\sin(t)) \, \mathrm{d}t$.

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We denote by $\|\cdot\|_{F}$ the associated Euclidean norm. We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define
$$g : \left\lvert \, \begin{aligned} & \mathcal{M}_{k,d}(\mathbb{R}) \rightarrow \mathbb{R} \\ & M \mapsto \|M \cdot u\| \end{aligned} \right.$$
Show that $C = \left\{M \in \mathcal{M}_{k,d}(\mathbb{R}) \mid g(M) \leqslant r\right\}$ is a convex and closed subset of $\mathcal{M}_{k,d}(\mathbb{R})$.

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We denote by $\|\cdot\|_{F}$ the associated Euclidean norm. We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define
$$g : \left\lvert \, \begin{aligned} & \mathcal{M}_{k,d}(\mathbb{R}) \rightarrow \mathbb{R} \\ & M \mapsto \|M \cdot u\| \end{aligned} \right.$$
Show that for every matrix $M$ in $\mathcal{M}_{k,d}(\mathbb{R})$
$$\|M \cdot u\| \leqslant \|M\|_{F}$$

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We denote by $\|\cdot\|_{F}$ the associated Euclidean norm. We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define
$$g : \left\lvert \, \begin{aligned} & \mathcal{M}_{k,d}(\mathbb{R}) \rightarrow \mathbb{R} \\ & M \mapsto \|M \cdot u\| \end{aligned} \right.$$
Let $C = \left\{M \in \mathcal{M}_{k,d}(\mathbb{R}) \mid g(M) \leqslant r\right\}$. Let $r$ and $t$ be two real numbers, with $t > 0$. Show that for every matrix $M$ in $\mathcal{M}_{k,d}(\mathbb{R})$
$$d(M, C) < t \quad \Longrightarrow \quad g(M) < r + t$$

Show that for all natural integer $p$, the integral $$I _ { p } = \int _ { - \infty } ^ { + \infty } e ^ { - ( t - p \pi ) ^ { 2 } } \sin t \mathrm {~d} t$$ is absolutely convergent and that it equals zero.

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We denote by $\|\cdot\|_{F}$ the associated Euclidean norm. We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define
$$g : \left\lvert \, \begin{aligned} & \mathcal{M}_{k,d}(\mathbb{R}) \rightarrow \mathbb{R} \\ & M \mapsto \|M \cdot u\| \end{aligned} \right.$$
Let $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ be a random variable taking values in $\mathcal{M}_{k,d}(\mathbb{R})$, whose coefficients $\varepsilon_{ij}$ are independent Rademacher random variables. Let $C = \left\{M \in \mathcal{M}_{k,d}(\mathbb{R}) \mid g(M) \leqslant r\right\}$. Deduce that
$$\mathbb{P}(g(X) \leqslant r) \cdot \mathbb{P}(g(X) \geqslant r + t) \leqslant \exp\left(-\frac{1}{8} t^{2}\right)$$

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define $g(M) = \|M \cdot u\|$. Let $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ be a random variable taking values in $\mathcal{M}_{k,d}(\mathbb{R})$, whose coefficients $\varepsilon_{ij}$ are independent Rademacher random variables. We say that a real number $m$ is a median of $g(X)$ when
$$\mathbb{P}(g(X) \geqslant m) \geqslant \frac{1}{2} \quad \text{and} \quad \mathbb{P}(g(X) \leqslant m) \geqslant \frac{1}{2}$$
Justify that $g(X)$ admits at least one median. One may consider the function $G$ from $\mathbb{R}$ to $\mathbb{R}$ such that, for every real number $t$, $G(t) = \mathbb{P}(g(X) \leqslant t)$, and examine the set $G^{-1}([1/2, 1])$.

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define $g(M) = \|M \cdot u\|$. Let $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ be a random variable taking values in $\mathcal{M}_{k,d}(\mathbb{R})$, whose coefficients $\varepsilon_{ij}$ are independent Rademacher random variables. Deduce from the above that, for every strictly positive real number $t$
$$\mathbb{P}(|g(X) - m| \geqslant t) \leqslant 4 \exp\left(-\frac{1}{8} t^{2}\right)$$
where $m$ is a median of $g(X)$.

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define $g(M) = \|M \cdot u\|$. Let $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ be a random variable taking values in $\mathcal{M}_{k,d}(\mathbb{R})$, whose coefficients $\varepsilon_{ij}$ are independent Rademacher random variables. Let $m$ be a median of $g(X)$. Deduce that $\mathbb{E}\left((g(X) - m)^{2}\right) \leqslant 32$.

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define $g(M) = \|M \cdot u\|$. Let $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ be a random variable taking values in $\mathcal{M}_{k,d}(\mathbb{R})$, whose coefficients $\varepsilon_{ij}$ are independent Rademacher random variables.
Show that $\mathbb{E}\left(g(X)^{2}\right) = k$, and deduce that $\mathbb{E}(g(X)) \leqslant \sqrt{k}$.

Show that $\mathcal{S}(\Delta_{k+1}) \subset \Delta_k$ and $\mathcal{S}^*(\Delta_k) \subset \Delta_{k+1}$.

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define $g(M) = \|M \cdot u\|$. Let $X = (\varepsilon_{ij})_{1 \leqslant i \leqslant k, 1 \leqslant j \leqslant d}$ be a random variable taking values in $\mathcal{M}_{k,d}(\mathbb{R})$, whose coefficients $\varepsilon_{ij}$ are independent Rademacher random variables. Let $m$ be a median of $g(X)$.
Deduce that $(\sqrt{k} - m)^{2} \leqslant \mathbb{E}\left((g(X) - m)^{2}\right)$.