Show that the series $\sum P \left( S _ { n } = 0 _ { d } \right)$ is divergent if and only if $P ( R \neq + \infty ) = 1$.

Let $\lambda > 0$ be fixed. We consider the set $\mathcal{G}$ of functions $g$ that can be written in the form $g = \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_\lambda)$ where $n$ is a strictly positive integer and $\left((x_i, \alpha_i)\right)_{1 \leq i \leq n}$ is a family of elements of $\mathbf{R}^2$: $$\mathcal{G} = \left\{ \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_\lambda) \mid n \in \mathbf{N}_*, \forall i \in \llbracket 1,n \rrbracket\ (x_i, \alpha_i) \in \mathbf{R} \times \mathbf{R} \right\}$$
Show that $\mathcal{G}$ is a vector subspace of $\mathcal{E}$ and that it is the smallest vector subspace of $\mathcal{E}$ that contains all functions $\tau_x(\gamma_\lambda)$ for arbitrary $x \in \mathbf{R}$.

For $i \in \mathbb{N}^{*}$, let $Y _ { i }$ be the Bernoulli random variable indicating the event $$Y _ { i } = \mathbf{1} \left( S _ { i } \notin \left\{ S _ { k } , 0 \leq k \leq i - 1 \right\} \right) .$$ Show that, for $i \in \mathbb{N}^{*}$: $$P \left( Y _ { i } = 1 \right) = P ( R > i )$$ Deduce that, for $n \in \mathbb{N}^{*}$: $$E \left( N _ { n } \right) = 1 + \sum _ { i = 1 } ^ { n } P ( R > i )$$

Let $\lambda > 0$ be fixed. We use the notation $\mathcal{G}$, $\mathcal{H} = C(\mathcal{G})$, $\gamma_\lambda$, $\tau_x$, and $(f \mid g) = \int_{-\infty}^{+\infty} f(y)g(y)\,\mathrm{d}y$ as defined previously.
(a) Show that there exists $c_\lambda > 0$ such that for all $(x, x') \in \mathbf{R} \times \mathbf{R}$ we have $$\left(\tau_x(\gamma_\lambda) \mid \tau_{x'}(\gamma_\lambda)\right) = c_\lambda \gamma_{2\lambda}(x - x')$$ Hint: One may note that $\frac{1}{\lambda}\left((y-x)^2 + (y-x')^2\right) = \frac{2}{\lambda}\left(y - (x+x')/2\right)^2 + \frac{1}{2\lambda}(x'-x)^2$.
(b) Deduce that for all $x \in \mathbf{R}$ $$C\left(\tau_x(\gamma_\lambda)\right) = c_\lambda \tau_x(\gamma_{2\lambda})$$ and that $$\mathcal{H} = \left\{ \sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_{2\lambda}) \mid n \in \mathbf{N}_*, \forall i \in \llbracket 1,n \rrbracket\ (x_i, \alpha_i) \in \mathbf{R} \times \mathbf{R} \right\}$$

We use the notation $\mathcal{G}$, $\mathcal{H}$, $\gamma_{2\lambda}$, $\tau_x$, $C$, $D$ as defined previously.
(a) Let $n \in \mathbf{N}_*$ and $(x_i)_{1 \leq i \leq n}$ a family of real numbers such that for all $i, j \in \llbracket 1,n \rrbracket$ we have $x_i \neq x_j$ when $i \neq j$. Show that the function $\sum_{i=1}^n \alpha_i \tau_{x_i}(\gamma_{2\lambda})$ is zero if and only if $\alpha_i = 0$ for all $1 \leq i \leq n$ (Hint: One may proceed by induction on $n$).
(b) Deduce that there exists a unique linear map $D$ from $\mathcal{H}$ to $\mathcal{G}$ such that $D \circ C(g) = g$ for all $g \in \mathcal{G}$ and $C \circ D(h) = h$ for all $h \in \mathcal{H}$.
(c) Show that for all $h \in \mathcal{H}$, we have for all $x \in \mathbf{R}$ that $h(x) = \left(\tau_x(\gamma_\lambda) \mid D(h)\right)$.

For all $(h_1, h_2) \in \mathcal{H} \times \mathcal{H}$, we denote $(h_1 \mid h_2)_{\mathcal{H}} = c_\lambda \left(D(h_1) \mid D(h_2)\right)$ where $c_\lambda$ is introduced in question (11a).
(a) Verify that $(\mid)_{\mathcal{H}}$ defines an inner product on $\mathcal{H}$.
(b) Show that for all $x \in \mathbf{R}$ and $h \in \mathcal{H}$ we have $h(x) = \left(\tau_x(\gamma_{2\lambda}) \mid h\right)_{\mathcal{H}}$.
(c) Show that for all $h \in \mathcal{H}$ we have $$\|h\|_\infty \leq \|h\|_{\mathcal{H}}$$ where we have set $\|h\|_\infty = \sup_{x \in \mathbf{R}} |h(x)|$ and $\|h\|_{\mathcal{H}} = (h \mid h)_{\mathcal{H}}^{1/2}$.

We assume that there exists $t _ { 0 } \in \mathbb { R } ^ { * }$ such that $\left| \phi _ { X } \left( t _ { 0 } \right) \right| = 1$. We assume further that $X ( \Omega )$ is countable and we use the notations of question 2 (i.e. $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$). Deduce that $\mathbb { P } \left( X \in a + \frac { 2 \pi } { t _ { 0 } } \mathbb { Z } \right) = 1$.

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$. We assume that $X ( \Omega )$ is finite and we use the notations of question 1: $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ with $a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$. Show that, for all $T \in \mathbb { R } _ { + } ^ { * }$, we have $V _ { m } ( T ) = \sum _ { n = 1 } ^ { r } \operatorname { sinc } \left( T \left( x _ { n } - m \right) \right) \mathbb { P } \left( X = x _ { n } \right)$.

For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t)\,\mathrm{d}t$$ Determine the eigenvalues of $T$ and show that the associated eigenspaces are one-dimensional.

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ Determine the eigenvalues of $T$ and show that the associated eigenspaces are one-dimensional.

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$. We assume that $X ( \Omega )$ is finite and we use the notations of question 1: $X ( \Omega ) = \left\{ x _ { 1 } , \ldots , x _ { r } \right\}$ with $a _ { k } = \mathbb { P } \left( X = x _ { k } \right)$. Using the result of Q15, deduce that $V _ { m } ( T ) \xrightarrow [ T \rightarrow + \infty ] { } \mathbb { P } ( X = m )$.

For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2}\sin(k\pi x)$. We denote by $G = \operatorname{Vect}\left((g_k)_{k \in \mathbb{N}^*}\right)$ and $H = G^\perp$. Justify that, for all $(f,g) \in E^2$, we have $$\langle T(f), g \rangle = \langle f, T(g) \rangle$$ One may use question 12.

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2} \sin(k\pi x)$. Justify that, for all $(f,g) \in E^2$, we have $$\langle T(f), g \rangle = \langle f, T(g) \rangle$$ One may use question 12.

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $T \in \mathbb { R } _ { + } ^ { * }$, we set $V _ { m } ( T ) = \frac { 1 } { 2 T } \int _ { - T } ^ { T } \phi _ { X } ( t ) \mathrm { e } ^ { - \mathrm { i } m t } \mathrm {~d} t$. We assume that $X ( \Omega )$ is countable and we use the notations of question 2: $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. For $n \in \mathbb { N }$ and $h \in \mathbb { R } _ { + } ^ { * }$, we set $g _ { n } ( h ) = \operatorname { sinc } \left( \frac { x _ { n } - m } { h } \right) \mathbb { P } \left( X = x _ { n } \right)$. Show that for all $T \in \mathbb { R } _ { + } ^ { * }$, we have $V _ { m } ( T ) = \sum _ { n = 0 } ^ { + \infty } g _ { n } \left( \frac { 1 } { T } \right)$.

For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2}\sin(k\pi x)$. We denote by $G = \operatorname{Vect}\left((g_k)_{k \in \mathbb{N}^*}\right)$ and $H = G^\perp$. We admit that, $$H \neq \{0\} \Longrightarrow \exists f \in H \text{ such that } \left\{ \begin{array}{l} \|f\| = 1 \\ \langle T(f), f \rangle = \sup_{h \in H, \|h\|=1} \langle T(h), h \rangle \end{array} \right.$$ Deduce that $H = \{0\}$.

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ For all $f \in E$, we set, $$\forall s \in [0,1], \quad T(f)(s) = \int_0^1 k_s(t) f(t) \, \mathrm{d}t$$ For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2} \sin(k\pi x)$. We denote by $G = \operatorname{Vect}\left((g_k)_{k \in \mathbb{N}^*}\right)$ and $H = G^\perp$. We admit that, $$H \neq \{0\} \Longrightarrow \exists f \in H \text{ such that } \left\{ \begin{array}{l} \|f\| = 1 \\ \langle T(f), f \rangle = \sup_{h \in H, \|h\|=1} \langle T(h), h \rangle \end{array} \right.$$ Deduce that $H = \{0\}$.

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $n \in \mathbb { N }$ and $h \in \mathbb { R } _ { + } ^ { * }$, we set $g _ { n } ( h ) = \operatorname { sinc } \left( \frac { x _ { n } - m } { h } \right) \mathbb { P } \left( X = x _ { n } \right)$. Show that the function $g _ { n }$ extends to a function $\tilde { g } _ { n }$ defined and continuous on $\mathbb { R } ^ { + }$.

For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2}\sin(k\pi x)$. Show that the family of vectors $(g_k)_{k \in \mathbb{N}^*}$ is orthonormal.

For all $k \in \mathbb{N}^*$, we set $g_k(x) = \sqrt{2} \sin(k\pi x)$. Show that the family of vectors $(g_k)_{k \in \mathbb{N}^*}$ is orthonormal.

Let $X$ be a real and discrete random variable and $m \in \mathbb { R }$. For $n \in \mathbb { N }$ and $h \in \mathbb { R } _ { + } ^ { * }$, we set $g _ { n } ( h ) = \operatorname { sinc } \left( \frac { x _ { n } - m } { h } \right) \mathbb { P } \left( X = x _ { n } \right)$, and $\tilde{g}_n$ denotes its continuous extension to $\mathbb{R}^+$. Show that the function $G = \sum _ { n = 0 } ^ { + \infty } \tilde { g } _ { n }$ is defined and continuous on $\mathbb { R } ^ { + }$.

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We assume that $\phi _ { X }$ is of class $C ^ { 2 }$ on $\mathbb { R }$. We denote $f$ the function which to all real $h > 0$ associates $f ( h ) = \frac { 2 \phi _ { X } ( 0 ) - \phi _ { X } ( 2 h ) - \phi _ { X } ( - 2 h ) } { 4 h ^ { 2 } }$. What is the limit of $f$ at 0 ?

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We assume that $\phi _ { X }$ is of class $C ^ { 2 }$ on $\mathbb { R }$. We denote $f ( h ) = \frac { 2 \phi _ { X } ( 0 ) - \phi _ { X } ( 2 h ) - \phi _ { X } ( - 2 h ) } { 4 h ^ { 2 } }$ for $h > 0$. Show that for all $h \in \mathbb { R } ^ { * } , f ( h ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \frac { \sin ^ { 2 } \left( h x _ { n } \right) } { h ^ { 2 } }$.

We fix a real random variable $X : \Omega \rightarrow \mathbb { R }$, whose image $X ( \Omega )$ is a countable set, with $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ and $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We assume that $\phi _ { X }$ is of class $C ^ { 2 }$ on $\mathbb { R }$. Using the results of Q31 and Q32, deduce that $X$ admits a moment of order 2.

We are given a real $a > 0$. Let $E_4$ be the space of functions continuous on $[0,a]$, taking values in $\mathbb{R}$, of class $\mathcal{C}^1$ piecewise and furthermore satisfying $f(a) = 0$. Let $\varphi:[0,a] \rightarrow \mathbb{R}$ be of class $\mathcal{C}^1$ satisfying $\varphi(a) = 0$ and, for all $x \in [0,a]$, $\varphi'(x) < 0$. Determine an inner product on $E_4$ such that the function $(x,y) \mapsto \min(\varphi(x), \varphi(y))$ is a reproducing kernel on the pre-Hilbert space $E_4$.

We are given a real $a > 0$. Let $E_4$ be the space of functions continuous on $[0,a]$, taking values in $\mathbb{R}$, of class $\mathcal{C}^1$ piecewise and furthermore satisfying $f(a) = 0$. Let $\varphi : [0,a] \rightarrow \mathbb{R}$ be of class $\mathcal{C}^1$ satisfying $\varphi(a) = 0$ and, for all $x \in [0,a]$, $\varphi'(x) < 0$. Determine an inner product on $E_4$ such that the function $(x,y) \mapsto \min(\varphi(x), \varphi(y))$ is a reproducing kernel on the pre-Hilbert space $E_4$.