Let $x > 0$. Using the study of a well-chosen function, show that $$\frac { x } { x ^ { 2 } + 1 } \varphi ( x ) \leqslant \int _ { x } ^ { + \infty } \varphi ( t ) \mathrm { d } t$$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
Show that if $\varphi \in \mathcal{C}_{c}(\mathbb{R})$ is a positive function, we have $\|T_{\mu}\varphi\|_{1} = \|\varphi\|_{1}$.

We define $Q_{>0} = (\mathbb{R}_+^*)^{I \times J}$ and $\mathscr{L} : Q_{>0} \times (\mathbb{R}^I \times \mathbb{R}^J) \rightarrow \mathbb{R}$ defined by $$\mathscr{L}(\boldsymbol{q}, (f, g)) = J_\epsilon(\boldsymbol{q}) + \sum_{i \in I} f_i \left(\alpha_i - \sum_{j \in J} q_{ij}\right) + \sum_{j \in J} g_j \left(\beta_j - \sum_{i \in I} q_{ij}\right).$$ (a) Verify that $Q_{>0}$ is an open convex set of $\mathbb{R}^{I \times J}$.
(b) Show that there exists $(f(\epsilon), g(\epsilon)) \in \mathbb{R}^I \times \mathbb{R}^J$ such that $\mathscr{L}(q(\epsilon), (f(\epsilon), g(\epsilon)))$ is a saddle point of $\mathscr{L}$. (Hint: One may identify $\mathbb{R}^{I \times J}$ with $\mathbb{R}^n$ and $\mathbb{R}^I \times \mathbb{R}^J$ with $\mathbb{R}^m$, for $n$ the cardinality of $I \times J$ and $m$ the sum of the cardinalities of $I$ and $J$, then use question 3 of part I.)

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A$ denote the event $\left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\}$.
Deduce that $$\mathbb { P } ( A ) \leqslant \mathbb { P } \left( \left\{ \left| R _ { n } \right| \geqslant x \right\} \right) + \max _ { 1 \leqslant p \leqslant n } \mathbb { P } \left( \left\{ \left| R _ { n } - R _ { p } \right| > 2 x \right\} \right) .$$

Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
An application $h$ from a non-trivial interval $J$ of $\mathbf{R}$ to $\mathbf{R}$ is said to be log-convex if, and only if, it takes values in $\mathbf{R}_+^*$ and $\ln \circ h$ is convex on $J$.
Verify that $f$ is an application from $I$ to $\mathbf{R}$ that is log-convex.

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$.
Conclude that $$\forall x > 0 , \quad \mathbb { P } \left( \left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\} \right) \leqslant 3 \max _ { 1 \leqslant p \leqslant n } \mathbb { P } \left( \left\{ \left| R _ { p } \right| \geqslant x \right\} \right) .$$

Show that $C^{0}(\mathbf{R}) \cap CL(\mathbf{R}) \subset L^{1}(\varphi)$.
We admit throughout the rest of the problem that $\int_{-\infty}^{+\infty} \varphi(t) \mathrm{d}t = 1$.

Show that $C^{0}(\mathbf{R}) \cap CL(\mathbf{R}) \subset L^{1}(\varphi)$, where $\varphi(x) = \frac{1}{\sqrt{2\pi}}\mathrm{e}^{-x^2/2}$ and $L^1(\varphi) = \{f \in C^0(\mathbf{R}),\, f\varphi \text{ integrable on } \mathbf{R}\}$.

Let $t \in \mathbf{R}_{+}$. Verify that the function $P_{t}(f)$ is well defined for $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ and verify that $P_{t}$ is linear on $C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$.
Recall that $\forall x \in \mathbf{R}, \quad P_{t}(f)(x) = \int_{-\infty}^{+\infty} f\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y.$

Let $t \in \mathbf{R}_+$. Verify that the function $P_t(f)$ is well defined for $f \in C^0(\mathbf{R}) \cap CL(\mathbf{R})$ and verify that $P_t$ is linear on $C^0(\mathbf{R}) \cap CL(\mathbf{R})$, where $$\forall x \in \mathbf{R}, \quad P_t(f)(x) = \int_{-\infty}^{+\infty} f\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y.$$

We denote by $\tilde{h}$ the restriction of the function $h : ]0,1[ \longrightarrow \mathbf{R},\; t \longmapsto \frac{1}{\sqrt{t(1-t)}}$ to the interval $\left]0, \frac{1}{2}\right]$. Verify that the function $\tilde{h}$ is decreasing on $]0, \frac{1}{2}[$, then show that the function $\tilde{h}$ belongs to $\mathscr{D}_{0, \frac{1}{2}}$.

Show that for all $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ and all $x \in \mathbf{R}$,
$$\lim_{t \rightarrow +\infty} P_{t}(f)(x) = \int_{-\infty}^{+\infty} f(y) \varphi(y) \mathrm{d}y$$

Show that for all $f \in C^0(\mathbf{R}) \cap CL(\mathbf{R})$ and all $x \in \mathbf{R}$, $$\lim_{t \rightarrow +\infty} P_t(f)(x) = \int_{-\infty}^{+\infty} f(y)\varphi(y)\,\mathrm{d}y,$$ where $P_t(f)(x) = \int_{-\infty}^{+\infty} f\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y$.

Let $t \in \mathbf{R}_{+}$. Show that if $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$, then $P_{t}(f) \in C^{0}(\mathbf{R})$. Also show that $P_{t}(f)$ is bounded in absolute value by a polynomial function in $|x|$ independent of $t$. Deduce that $P_{t}(f) \in L^{1}(\varphi)$.

Let $t \in \mathbf{R}_+$. Show that if $f \in C^0(\mathbf{R}) \cap CL(\mathbf{R})$, then $P_t(f) \in C^0(\mathbf{R})$. Also show that $P_t(f)$ is bounded in absolute value by a polynomial function in $|x|$ independent of $t$. Deduce that $P_t(f) \in L^1(\varphi)$.

Justify that the quantity $\operatorname{Ent}_{\varphi}(g)$ is well defined for all $g \in C^0(\mathbf{R}) \cap CL(\mathbf{R})$ with strictly positive values such that $\int_{-\infty}^{+\infty} g(x)\varphi(x)\,\mathrm{d}x = 1$, where $$\operatorname{Ent}_{\varphi}(g) = \int_{-\infty}^{+\infty} \ln(g(x))\,g(x)\,\varphi(x)\,\mathrm{d}x.$$
Hint: You may use question 11.

For $t \in \mathbf{R}_{+}$, we set $S(t) = \operatorname{Ent}_{\varphi}\left(P_{t}(f)\right)$. Justify that $S(t)$ is well defined.
Here $f$ is an element of $C^{2}(\mathbf{R})$ with strictly positive values such that the functions $f, f^{\prime}, f^{\prime\prime}$ and $\frac{f^{\prime 2}}{f}$ have slow growth, and $\int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x = 1$.

For $t \in \mathbf{R}_+$, we set $S(t) = \operatorname{Ent}_{\varphi}(P_t(f))$. Justify that $S(t)$ is well defined.

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

By admitting that the result of question 7 is valid for the functions $P_{t}(f)$ and $1 + \ln\left(P_{t}(f)\right)$, show that
$$\forall t \in \mathbf{R}_{+}^{*}, \quad -S^{\prime}(t) = \mathrm{e}^{-2t} \int_{-\infty}^{+\infty} \frac{P_{t}\left(f^{\prime}\right)(x)^{2}}{P_{t}(f)(x)} \varphi(x) \mathrm{d}x$$

By admitting that the result of question 7 is valid for the functions $P_t(f)$ and $1 + \ln(P_t(f))$, show that $$\forall t \in \mathbf{R}_+^*, \quad -S'(t) = \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} \frac{P_t(f')(x)^2}{P_t(f)(x)}\,\varphi(x)\,\mathrm{d}x.$$

By using the Cauchy-Schwarz inequality, show that
$$\forall t \in \mathbf{R}_{+}^{*}, \quad -S^{\prime}(t) \leq \mathrm{e}^{-2t} \int_{-\infty}^{+\infty} P_{t}\left(\frac{f^{\prime 2}}{f}\right)(x) \varphi(x) \mathrm{d}x$$

By using the Cauchy-Schwarz inequality, show that $$\forall t \in \mathbf{R}_+^*, \quad -S'(t) \leq \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} P_t\!\left(\frac{f'^2}{f}\right)(x)\,\varphi(x)\,\mathrm{d}x.$$

Deduce that we have:
$$\forall t \in \mathbf{R}_{+}^{*}, \quad -S^{\prime}(t) \leq \mathrm{e}^{-2t} \int_{-\infty}^{+\infty} \frac{f^{\prime 2}(x)}{f(x)} \varphi(x) \mathrm{d}x$$

Deduce that we have: $$\forall t \in \mathbf{R}_+^*, \quad -S'(t) \leq \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} \frac{f'^2(x)}{f(x)}\,\varphi(x)\,\mathrm{d}x.$$