Let $Z$ be a discrete real random variable such that $\exp ( \lambda Z )$ has finite expectation for all $\lambda > 0$. Show that for all $\lambda > 0$ and $t \in \mathbb { R }$, $$P [ Z \geqslant t ] \leqslant \exp ( - \lambda t ) E [ \exp ( \lambda Z ) ] .$$

Let $n \geqslant 1$ be a natural integer, and let $\left( X _ { 1 } , \ldots , X _ { n } \right)$ be mutually independent discrete real random variables such that, for all $k \in \{ 1 , \ldots , n \}$, $$P \left[ X _ { k } = 1 \right] = P \left[ X _ { k } = - 1 \right] = \frac { 1 } { 2 }$$ We define $$S _ { n } = \frac { 1 } { n } \sum _ { k = 1 } ^ { n } X _ { k }$$ Show that $P \left[ S _ { n } \geqslant 0 \right] \geqslant \frac { 1 } { 2 }$.

Let $n \geqslant 1$ be a natural integer, and let $\left( X _ { 1 } , \ldots , X _ { n } \right)$ be mutually independent discrete real random variables such that, for all $k \in \{ 1 , \ldots , n \}$, $$P \left[ X _ { k } = 1 \right] = P \left[ X _ { k } = - 1 \right] = \frac { 1 } { 2 }$$ We define $$S _ { n } = \frac { 1 } { n } \sum _ { k = 1 } ^ { n } X _ { k }$$ as well as, for all $\lambda \in \mathbb { R }$, $$\psi ( \lambda ) = \log \left( \frac { 1 } { 2 } e ^ { \lambda } + \frac { 1 } { 2 } e ^ { - \lambda } \right)$$ Show that for all $t \in \mathbb { R }$, we have $$\frac { 1 } { n } \log P \left[ S _ { n } \geqslant t \right] \leqslant \inf _ { \lambda \geqslant 0 } ( \psi ( \lambda ) - \lambda t )$$

Let $n \geqslant 1$ be a natural integer, and let $\left( X _ { 1 } , \ldots , X _ { n } \right)$ be mutually independent discrete real random variables such that, for all $k \in \{ 1 , \ldots , n \}$, $$P \left[ X _ { k } = 1 \right] = P \left[ X _ { k } = - 1 \right] = \frac { 1 } { 2 }$$ We define $$S _ { n } = \frac { 1 } { n } \sum _ { k = 1 } ^ { n } X _ { k }$$ For each $\lambda \geqslant 0$, we set $$m ( \lambda ) = \frac { E \left[ X _ { 1 } \exp \left( \lambda X _ { 1 } \right) \right] } { E \left[ \exp \left( \lambda X _ { 1 } \right) \right] }$$ For all $n \geqslant 1 , \lambda \geqslant 0$ and $\varepsilon > 0$, we denote by $I _ { n } ( \lambda , \varepsilon )$ the random variable defined by $$I _ { n } ( \lambda , \varepsilon ) = \begin{cases} 1 & \text { if } \left| S _ { n } - m ( \lambda ) \right| \leqslant \varepsilon \\ 0 & \text { otherwise } \end{cases}$$ Show that $$P \left[ \left| S _ { n } - m ( \lambda ) \right| \leqslant \varepsilon \right] \geqslant E \left[ I _ { n } ( \lambda , \varepsilon ) \exp \left( \lambda n \left( S _ { n } - m ( \lambda ) - \varepsilon \right) \right] , \right.$$

Let $n \geqslant 1$ be a natural integer, and let $\left( X _ { 1 } , \ldots , X _ { n } \right)$ be mutually independent discrete real random variables such that, for all $k \in \{ 1 , \ldots , n \}$, $$P \left[ X _ { k } = 1 \right] = P \left[ X _ { k } = - 1 \right] = \frac { 1 } { 2 }$$ We define $$S _ { n } = \frac { 1 } { n } \sum _ { k = 1 } ^ { n } X _ { k }$$ as well as, for all $\lambda \in \mathbb { R }$, $$\psi ( \lambda ) = \log \left( \frac { 1 } { 2 } e ^ { \lambda } + \frac { 1 } { 2 } e ^ { - \lambda } \right)$$ For each $\lambda \geqslant 0$, we set $$m ( \lambda ) = \frac { E \left[ X _ { 1 } \exp \left( \lambda X _ { 1 } \right) \right] } { E \left[ \exp \left( \lambda X _ { 1 } \right) \right] }$$ as well as $$D _ { n } ( \lambda ) = \exp \left( \lambda n S _ { n } - n \psi ( \lambda ) \right)$$ For all $n \geqslant 1 , \lambda \geqslant 0$ and $\varepsilon > 0$, we denote by $I _ { n } ( \lambda , \varepsilon )$ the random variable defined by $$I _ { n } ( \lambda , \varepsilon ) = \begin{cases} 1 & \text { if } \left| S _ { n } - m ( \lambda ) \right| \leqslant \varepsilon \\ 0 & \text { otherwise } \end{cases}$$ Show that $$E \left[ I _ { n } ( \lambda , \varepsilon ) D _ { n } ( \lambda ) \right] \geqslant 1 - \frac { 4 } { n \varepsilon ^ { 2 } }$$

For every integer $n$ such that $n \geqslant 1$, we denote by $\widehat{M}_{n}(C) = \left(\widehat{X}_{ij}(C)\right)_{1 \leqslant i,j \leqslant n}$. For every $\omega \in \Omega$, we denote by $\widehat{\Lambda}_{1,n}(\omega) \geqslant \ldots \geqslant \widehat{\Lambda}_{n,n}(\omega)$ the eigenvalues of $\frac{1}{\sqrt{n}} \widehat{M}_{n}(C)(\omega)$ arranged in decreasing order. Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a $K$-Lipschitz function.
Show that $$\left|\mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} f\left(\Lambda_{i,n}\right)\right) - \mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} f\left(\widehat{\Lambda}_{i,n}\right)\right)\right| \leqslant \frac{K}{n} \mathbb{E}\left(\left\|M_{n} - \widehat{M}_{n}(C)\right\|_{F}\right).$$

We are given a centered real random variable $Y$ such that $Y ^ { 4 }$ has finite expectation.
Show successively that $Y ^ { 2 }$ and $| Y | ^ { 3 }$ have finite expectation, and that
$$\mathrm { E } \left( Y ^ { 2 } \right) \leq \left( \mathrm { E } \left( Y ^ { 4 } \right) \right) ^ { 1 / 2 } \quad \text { then } \quad \mathrm { E } \left( | Y | ^ { 3 } \right) \leq \left( \mathrm { E } \left( Y ^ { 4 } \right) \right) ^ { 3 / 4 }$$

With the notation and setup of the previous questions (mutually independent Rademacher variables, $S_N = \sum_{n=0}^N X_n a_n$, events $A_j$, $B_j$, $B_{j,m}$), explain how to deduce the formula $$\mathbb{P}(A_j) = \sum_{m=\phi(j)+1}^{\phi(j+1)} \mathbb{P}(A_j \cap B_{j,m}).$$

Let $X$ be a Bernoulli random variable with parameter $\lambda \in ]0,1[$. Show that $$d_{VT}\left(p_X, \pi_\lambda\right) = \lambda\left(1 - e^{-\lambda}\right).$$ Deduce that $$d_{VT}\left(p_X, \pi_\lambda\right) \leq \lambda^2.$$

Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Fix $\varepsilon \in ] 0 , 1 [$.
Show that there exists $x _ { 0 } \geqslant 1$ such that we have $$\forall x \geqslant x _ { 0 } , \quad \exists n _ { x } \in \mathbb { N } , \quad \forall n \geqslant n _ { x } , \quad x ^ { 2 } \mathbb { P } \left( \left\{ \left| S _ { n } \right| \geqslant x \sqrt { n } \right\} \right) \leqslant \varepsilon .$$

Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Fix $\varepsilon \in ] 0 , 1 [$.
For $x _ { 0 }$ and $x$ as in the previous question, we fix $N \geqslant \frac { n _ { x } } { \varepsilon }$ and we choose $n \geqslant N$. Show that then $$x ^ { 2 } \mathbb { P } \left( \left\{ \max _ { 1 \leqslant p \leqslant n } \left| S _ { p } \right| \geqslant 3 x \sqrt { n } \right\} \right) \leqslant 3 \varepsilon$$

We use the notations of the previous parts. We assume that there exists an eigenvalue $\lambda > 0$ and an associated eigenvector column $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ with $Mh = \lambda h$, and that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that $M_{i,j} \geqslant c\nu_j$ for all $i,j$. Let $\pi \in \mathscr{P}$ be such that $\pi M = \lambda \pi$, and let $C > 0$, $\gamma \in [0,1[$ be as in question 17.
(a) Show that for all $n \geqslant 0$ and $u \in \mathscr{M}_{d,1}(\mathbb{R})$ such that $\langle u, \pi \rangle = 0$, $$\left\| M^n u \right\|_1 \leqslant C(\lambda\gamma)^n \|u\|_1.$$
(b) Deduce that there exists $C_1 \geqslant 0$ such that for all $n \geqslant 0$ and $u \in \mathscr{M}_{d,1}(\mathbb{R})$ column vector such that $\langle u, \pi \rangle = 0$, $$\mathbb{E}\left(\langle X_n, u \rangle^2\right) \leqslant C_1 \|u\|_1^2 \left(\lambda^{2n} \left(\sum_{k=0}^{n-1} \lambda^{-k} \gamma^{2n-2k}\right) + (\lambda\gamma)^{2n}\right).$$

We suppose that $\lambda > 1$ and we use the random row vector $W_n = \lambda^{-n}\left(X_n - \|X_n\|_1 \pi\right)$.
Show that the event $\left\{\lim_{n \rightarrow +\infty} W_n = 0_{\mathbb{R}^d}\right\}$ is almost surely true. (One may begin by computing the probability of the event $$\left\{ \forall m \geqslant 0, \exists k \geqslant m \mid \|W_k\|_2 \geqslant \varepsilon \right\}$$ for all $\varepsilon > 0$.)

Let $X$ be a random variable defined on a probability space $(\Omega , \mathcal{A} , \mathbf{P})$ with values in $\mathbf{N}$ and admitting an expectation $\mathbf{E}(X)$ and a variance $\mathbf{V}(X)$. Show that if $\mathbf{E}(X) \neq 0$, then $\mathbf{P}(X = 0) \leq \frac{\mathbf{V}(X)}{(\mathbf{E}(X))^{2}}$. Hint: note that $(X = 0) \subset (|X - \mathbf{E}(X)| \geq \mathbf{E}(X))$.

Show that if $p _ { n } = o \left( \frac { 1 } { n ^ { 2 } } \right)$ in the neighborhood of $+ \infty$, then $\lim _ { n \rightarrow + \infty } \mathbf { P } \left( A _ { n } > 0 \right) = 0$.

Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. Let $X _ { n } ^ { 0 }$ be the discrete real random variable counting the number of copies of $G_0$ contained in $G \in \Omega_n$, and let $$\omega _ { 0 } = \min _ { \substack { H \subset G _ { 0 } \\ a _ { H } \geq 1 } } \frac { s _ { H } } { a _ { H } }$$ Deduce that if $p _ { n } = \mathrm { o } \left( n ^ { - \omega _ { 0 } } \right)$, then $\lim _ { n \rightarrow + \infty } \mathbf { P } \left( X _ { n } ^ { 0 } > 0 \right) = 0$. Hint: one may introduce $H _ { 0 } \subset G _ { 0 }$ achieving the minimum giving $\omega _ { 0 }$.

We now assume that the random variables $X_1, \ldots, X_N$ are independent, so that they are $k$-independent for all $k \in \{2, \ldots, N\}$. We now want to establish the following bound: there exist numerical constants $\alpha, \beta > 0$ (independent of $K \geq 1$ and $N$) such that for all $t \geq 0$, $$\mathbb{P}\left(|S_N| \geq t\sqrt{N}\right) \leq \beta \exp\left(-\alpha t^2/K^2\right)$$
I.6.a) Justify that it suffices to consider the case $K = 1$, which we will do in the next three questions.
I.6.b) Let $k$ be the largest even integer in $\{1, \ldots, N\}$ less than or equal to $\frac{2t^2}{e^2}$. Justify that (27) is satisfied if $$e \leq t \leq \frac{e}{\sqrt{2}}\sqrt{N}$$
I.6.c) Under hypothesis (32), prove that we have (31) with $$\beta = 2e, \quad \alpha = e^{-2}$$
I.6.d) Conclude that there exist numerical constants $\alpha, \beta > 0$ such that (31) is verified for all $t \geq 0$.

Let $\gamma$ be a positive numerical constant. We assume the following property is satisfied: for every convex set $A \subset Q^N$, $$\mathbb{P}(X \in A) \mathbb{E}\left[\exp\left(\gamma \frac{d(X,A)^2}{4K^2}\right)\right] \leq 1.$$ We are given a 1-Lipschitz and convex function $F : \mathbb{R}^N \rightarrow \mathbb{R}$. The purpose of this question is to prove that (37) is then verified: $$\mathbb{P}(F(X) \geq m) \geq \frac{1}{2} \Longrightarrow \mathbb{P}(F(X) \leq m - t) \leq \beta e^{-\alpha t^2/K^2}$$
II.1.a) Let $s, \sigma \in \mathbb{R}$ with $s < \sigma$. By considering the set $$A_s = \left\{x \in Q^N; F(x) \leq s\right\}$$ show that $$\mathbb{P}(F(X) \leq s)\mathbb{P}(F(X) \geq \sigma) \leq \exp\left(-\gamma \frac{(\sigma - s)^2}{4K^2}\right)$$
II.1.b) Prove that (37) is verified.

Let $x$ be an arbitrary point of $Q^N$. Let $P^N$ be the set of vertices of the hypercube $[0,1]^N$, that is the set of linear combinations of the $e_i$ for $i \in \{1, \ldots, N\}$ with coefficients 0 or 1. If $A$ is a non-empty subset of $Q^N$, we define the subsets $P_A(x)$ and $R_A(x)$ of $P^N$ as follows: let $H_i$ be the hyperplane orthogonal to $e_i$, generated by the $e_j$ for $j \neq i$. Then $z \in P_A(x)$ if there exists $a \in A$ such that $$\forall i \in \{1, \ldots, N\}, z \in H_i \Longrightarrow a - x \in H_i$$ while $z \in R_A(x)$ if there exists $a \in A$ such that $$\forall i \in \{1, \ldots, N\}, z \in H_i \Longleftrightarrow a - x \in H_i.$$
Given $A \subset Q^N$ non-empty and $x \in Q^N$, we also define the quantity $$q(x, A) := \inf\left\{|z|; z \in \Gamma(P_A(x))\right\}.$$ We moreover adopt the following convention: if $A$ is the empty set, we set $q(x, A) = 2N$.
II.2.a) If $z, z' \in P^N$, we denote by $z \leq z'$ when $\langle z, e_i \rangle \leq \langle z', e_i \rangle$ for all $i \in \{1, \ldots, N\}$. Prove that $$P_A(x) = \left\{z' \in P^N; \exists z \in R_A(x), z \leq z'\right\}$$
II.2.b) Let $x \in \mathbb{R}^N$. Justify the equivalences $$x \in A \Longleftrightarrow 0 \in P_A(x) \Longleftrightarrow P_A(x) = P^N$$
II.2.c) In dimension $N = 3$, give an example of a set $A$ for which $e_3 \notin P_A(0)$ and describe precisely the sets $R_A(0)$ and $P_A(0)$ corresponding.
II.2.d) Let $B$ be a non-empty subset of $\mathbb{R}^N$. We denote by $\Gamma_0(B)$ the set of convex combinations of at most $N+1$ elements of $B$: $$\Gamma_0(B) := \left\{\sum_{j=1}^{N+1} \theta_j z_j; \theta_j \in [0,1], z_j \in B, \sum_{j=1}^{N+1} \theta_j = 1\right\}.$$ Prove that $\Gamma(B) = \Gamma_0(B)$.
Hint: you may prove that any convex combination of $m+1$ elements of $B$ with $m > N$ can be rewritten as a convex combination of at most $m$ elements of $B$.
II.2.e) Let $B$ be a non-empty subset of $\mathbb{R}^N$. Prove that $\Gamma(B)$ is a convex set, and that it is compact if $B$ is.
II.2.f) Draw and name (as a geometric object) the convex hull $\Gamma(B)$ in dimension $N = 3$, in the three following cases: $$B = \{e_1, e_2, e_1 + e_2\}, \quad B = \{e_1, e_2, e_1 + e_2, e_2 + e_3\}, \quad B = P^3.$$ For each of these examples, say whether $B$ can correspond to a set $P_A(x)$.
II.2.g) Let $A \subset Q^N$ non-empty and $x \in Q^N$. Justify that the infimum in (48) is attained.
II.2.h) Let $A \subset Q^N$ non-empty and $x \in Q^N$. Justify that $q(x, A) \leq \sqrt{N}$. Under what condition do we have $q(x, A) = 0$?
II.2.i) Let $x \in Q^N$ and $A \subset Q^N$ non-empty. Justify that $$q(x, A) = \inf\left\{|z|; z \in \Gamma(R_A(x))\right\}$$
II.2.j) Let $x \in Q^N$ and $A \subset Q^N$ with $A$ convex. Prove that $$d(x, A) \leq 2K\, q(x, A).$$
II.2.k) Let $\gamma \geq 0$ be a numerical constant. Prove that the property: ``for every convex set $A \subset Q^N$, we have $\mathbb{P}(X \in A)\mathbb{E}\left[\exp\left(\gamma\, q(X,A)^2\right)\right] \leq 1$'' implies $$\mathbb{P}(X \in A)\mathbb{E}\left[\exp\left(\gamma \frac{d(X,A)^2}{4K^2}\right)\right] \leq 1.$$

Let $$\gamma_0 = \frac{1}{4}$$ The purpose of the end of this part II is the proof by induction on the dimension $N$ of the property: for every convex set $A \subset Q^N$, $$\mathbb{P}(X \in A)\mathbb{E}\left[\exp\left(\gamma_0\, q(X,A)^2\right)\right] \leq 1,$$ for $\gamma = \gamma_0$. For $N$ a positive integer, we introduce the following induction hypothesis $H_N$: ``Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. Let $N$ random variables $X_1, \ldots, X_N$ taking values in a finite set, independent and identically distributed, satisfying $\mathbb{P}(|X_n| \leq K) = 1$, and let $X = (X_i)_{1 \leq i \leq N}$ be the random vector with components $X_1, \ldots, X_N$. Then (53) is verified for $\gamma = \gamma_0 = \frac{1}{4}$''.
II.3.a) We consider the case $N = 1$. Prove that (53) is satisfied when $\gamma \leq \ln(2)$, and thus for $\gamma = \gamma_0$.
We now assume $N > 1$, and we fix $A \subset Q^N$ convex. We adopt the following notations: we decompose $$x = (\bar{x}, x_N) \quad \text{with} \quad \bar{x} = (x_i)_{1 \leq i \leq N-1} \in \mathbb{R}^{N-1}.$$ If $A \subset \mathbb{R}^N$ and $\theta \in \mathbb{R}$, we denote $$A_\theta := \left\{b \in \mathbb{R}^{N-1}; (b, \theta) \in A\right\}$$ the section of $A$ at level $\theta$. We also denote $$\bar{A} := \left\{\bar{a} \in \mathbb{R}^{N-1}; \exists \theta \in \mathbb{R}, (\bar{a}, \theta) \in A\right\}$$ the projection of $A$ onto $\mathbb{R}^{N-1}$.
II.3.b) Let $x \in Q^N$. Let $A \subset Q^N$ such that $A_{x_N}$ is non-empty. Let $\bar{z} \in P^{N-1}$. Prove that $$\bar{z} \in P_{A_{x_N}}(\bar{x}) \Longrightarrow (\bar{z}, 0) \in P_A(x)$$ and $$\bar{z} \in P_{\bar{A}}(\bar{x}) \Longrightarrow (\bar{z}, 1) \in P_A(x)$$
II.3.c) Prove that, for all $\lambda \in [0,1]$, we have $$q(x, A)^2 \leq (1-\lambda)^2 + \lambda\, q(\bar{x}, A_{x_N})^2 + (1-\lambda)\, q(\bar{x}, \bar{A})^2.$$
We fix $x_N \in \mathbb{R}$ such that $\mathbb{P}(X_N = x_N) > 0$ and we consider the probability $$\overline{\mathbb{P}}(B) := \mathbb{P}(B \mid X_N = x_N) = \frac{\mathbb{P}(B \cap \{X_N = x_N\})}{\mathbb{P}(X_N = x_N)} \quad \text{for } B \in \mathcal{A},$$ as well as the associated expectation $$\overline{\mathbb{E}}[Z] := \frac{1}{\mathbb{P}(X_N = x_N)} \mathbb{E}\left[Z\, \mathbf{1}_{\{X_N = x_N\}}\right]$$ for any random variable $Z$.
II.3.d) Assuming the induction hypothesis $H_{N-1}$, prove $$\overline{\mathbb{P}}(\bar{X} \in \bar{A})\, \overline{\mathbb{E}}\left[\exp\left(\gamma_0\, q(X,A)^2\right)\right] \leq e^{\gamma_0}$$ and justify that, for all $\lambda \in [0,1]$, we have $$\overline{\mathbb{P}}(\bar{X} \in A_{x_N})^\lambda\, \overline{\mathbb{P}}(\bar{X} \in \bar{A})^{1-\lambda}\, \overline{\mathbb{E}}\left[\exp\left(\gamma_0\, q(X,A)^2\right)\right] \leq e^{\gamma_0(1-\lambda)^2}.$$
Hint: you may assume H\"older's inequality: $$\overline{\mathbb{E}}\left[e^{\lambda Y} e^{(1-\lambda)Z}\right] \leq \left\{\overline{\mathbb{E}}\left[e^Y\right]\right\}^\lambda \left\{\overline{\mathbb{E}}\left[e^Z\right]\right\}^{(1-\lambda)}$$ for $\lambda \in [0,1]$ and $Y, Z$ random variables.
II.3.e) We assume $$\overline{\mathbb{P}}(\bar{X} \in \bar{A}) > 0,$$ and we define $$r = \frac{\overline{\mathbb{P}}(\bar{X} \in A_{x_N})}{\overline{\mathbb{P}}(\bar{X} \in \bar{A})}$$ Prove that $$r^\lambda e^{-\gamma_0(1-\lambda)^2}\, \overline{\mathbb{P}}(\bar{X} \in \bar{A})\, \overline{\mathbb{E}}\left[\exp\left(\gamma_0\, q(X,A)^2\right)\right] \leq 1.$$
II.3.f) We provisionally admit the following inequality: for all $\gamma \in [0, \gamma_0]$, for all $r \in ]0,1]$, $$\frac{1}{2-r} \leq \sup_{\lambda \in [0,1]} r^\lambda e^{-\gamma(1-\lambda)^2}$$ Justify that $$\overline{\mathbb{P}}(\bar{X} \in \bar{A})\, \overline{\mathbb{E}}\left[\exp\left(\gamma_0\, q(X,A)^2\right)\right] \leq (2 - r).$$ We shall distinguish the cases $\overline{\mathbb{P}}(\bar{X} \in A_{x_N}) > 0$ and $\overline{\mathbb{P}}(\bar{X} \in A_{x_N}) = 0$.
II.3.g) Prove that $$\mathbb{P}(X \in A)\, \mathbb{E}\left[\exp\left(\gamma_0\, q(X,A)^2\right) \mathbf{1}_{\{X_N = x_N\}}\right] \leq R(2-r)\, \mathbb{P}(X_N = x_N),$$ where $$R = \frac{\mathbb{P}(X \in A)}{\mathbb{P}(\bar{X} \in \bar{A})}$$
II.3.h) Prove that $$\mathbb{P}(X \in A)\, \mathbb{E}\left[\exp\left(\gamma_0\, q(X,A)^2\right)\right] \leq R(2-R),$$ where $R$ is defined in (72), then prove (53) and conclude the induction $H_{N-1} \Rightarrow H_N$. You should take care to account for the case where (66) is not verified.
II.3.i) Justify (69): for all $\gamma \in [0, \gamma_0]$, for all $r \in ]0,1]$, $$\frac{1}{2-r} \leq \sup_{\lambda \in [0,1]} r^\lambda e^{-\gamma(1-\lambda)^2}$$