Assuming that $\int_{-\infty}^{+\infty} \exp\left(-x^{2}\right) \mathrm{d}x = \sqrt{\pi}$, give the value of $\int_{-\infty}^{+\infty} g_{\sigma}(x) \mathrm{d}x$.
Study the variations of $g_{\sigma}$. Show that the second derivative of $g_{\sigma}$ vanishes and changes sign at exactly two points. Give the shape of the graph of $g_{\sigma}$ and mark the two points mentioned.
Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, continuous and integrable on $\mathbb{R}$. Show that, for any real $\xi$, the function $\left\lvert\, \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & x \mapsto f(x) \exp(-\mathrm{i} 2\pi \xi x) \end{aligned}\right.$ is integrable on $\mathbb{R}$.
Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, continuous and integrable on $\mathbb{R}$. The Fourier transform is defined as $\mathcal{F}(f) : \left\lvert\, \begin{aligned} & \mathbb{R} \rightarrow \mathbb{C} \\ & \xi \mapsto \int_{-\infty}^{+\infty} f(x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x \end{aligned}\right.$. Show that $\mathcal{F}(f)$ is continuous on $\mathbb{R}$.
Let $f$ be a function from $\mathbb{R}$ to $\mathbb{C}$, of class $\mathcal{C}^{1}$. We assume that $f$ and its derivative $f^{\prime}$ are integrable on $\mathbb{R}$. Show that $f$ tends to zero at $+\infty$ and at $-\infty$.
We define $\hat{f}$ on $\mathbb{R}_{+}^{*} \times \mathbb{R}$ by: $\forall(t, \xi) \in \mathbb{R}_{+}^{*} \times \mathbb{R},\ \hat{f}(t, \xi) = \int_{-\infty}^{+\infty} f(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x$. Show that, for any real number $\xi$, $\lim_{t \rightarrow 0^{+}} \hat{f}(t, \xi) = \widehat{g_{\sigma}}(\xi)$. One may use any sequence $\left(t_{n}\right)_{n \in \mathbb{N}}$ of strictly positive reals converging to zero.
Show that, for any real $\xi$ and any real $t > 0$, $\frac{\partial \hat{f}}{\partial t}(t, \xi) = \int_{-\infty}^{+\infty} \frac{\partial f}{\partial t}(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x$.
By noting that $\int_{-\infty}^{+\infty} \frac{\partial f}{\partial t}(t, x) \exp(-\mathrm{i} 2\pi \xi x) \mathrm{d}x = \mathcal{F}\left(\frac{\partial f}{\partial t}(t, \cdot)\right)(\xi)$ and using question 7, show that, for any real $\xi$ and any real $t > 0$, $\frac{\partial \hat{f}}{\partial t}(t, \xi) = -4\pi^{2} \xi^{2} \hat{f}(t, \xi)$.
Show that, for all $\xi \in \mathbb{R}$, there exists a real $K(\xi)$ such that for all $t \in \mathbb{R}_{+}^{*}$, $\hat{f}(t, \xi) = K(\xi) \exp\left(-4\pi^{2} \xi^{2} t\right)$.
Let $t$ be a strictly positive real number. Using questions 20 and 12, and the result that if $u$ and $v$ are functions from $\mathbb{R}$ to $\mathbb{R}$, continuous and integrable on $\mathbb{R}$ and satisfying $\mathcal{F}(u) = \mathcal{F}(v)$, then $u = v$, deduce the existence of a real $\lambda_{t,\sigma}$ such that $$f(t, \cdot) = \lambda_{t,\sigma} g_{\sqrt{\sigma^{2}+2t}}$$
Show that the function $I : \left\lvert\, \begin{aligned} & \mathbb{R}_{+}^{*} \rightarrow \mathbb{R} \\ & t \mapsto \int_{-\infty}^{+\infty} f(t, x) \mathrm{d}x \end{aligned}\right.$ is constant. One may use the result of question 17.
Let $\tau$ be a strictly positive real and $q$ a natural integer greater than or equal to 2. We set $\delta = \frac{1}{q+1}$ and $r = \frac{\tau}{\delta^{2}}$. The numerical scheme imposes, for any natural integer $n$ and any $k \in \llbracket 1, q \rrbracket$: $$\frac{f_{n+1}(k) - f_{n}(k)}{\tau} = \frac{f_{n}(k+1) - 2f_{n}(k) + f_{n}(k-1)}{\delta^{2}}$$ as well as $f_{n}(0) = f_{n}(q+1) = 0$. We set $F_{n} = \left(\begin{array}{c} f_{n}(1) \\ \vdots \\ f_{n}(q) \end{array}\right)$, $I_{q}$ is the identity matrix of order $q$, $B$ is the square matrix of order $q$ with coefficient $(i,j)$ equal to 1 if $|i-j|=1$ and 0 otherwise, and $A = (1-2r)I_{q} + rB$. Show that, for all $n \in \mathbb{N}$, $F_{n+1} = A F_{n}$.
With the same setup as Q27 (matrices $A$, $B$, $I_q$, $r$, $F_n$), justify that the matrices $A$ and $B$ are diagonalizable over $\mathbb{R}$ and that, for all $n \in \mathbb{N}$, $F_{n} = A^{n} F_{0}$.
With the same setup as Q27 (matrices $A$, $B$, $I_q$, $r$, $F_n$), show that the sequence $\left(F_{n}\right)_{n \in \mathbb{N}}$ is bounded regardless of the choice of $F_{0}$ if and only if the eigenvalues of $A$ belong to $[-1, 1]$.
Let $\lambda$ be an eigenvalue of $B$ (the square matrix of order $q$ with coefficient $(i,j)$ equal to 1 if $|i-j|=1$ and 0 otherwise) and let $Y = \left(\begin{array}{c} y_{1} \\ \vdots \\ y_{q} \end{array}\right)$ be an associated eigenvector. By considering a coefficient of $Y$ whose absolute value is maximal, show that $\lambda \in [-2, 2]$ and justify the existence of an element $\theta$ of $[0, \pi]$, such that $\lambda = 2\cos\theta$.
Let $\lambda$ be an eigenvalue of $B$ and $Y = \left(\begin{array}{c} y_{1} \\ \vdots \\ y_{q} \end{array}\right)$ an associated eigenvector. Show that, if we impose $y_{0} = y_{q+1} = 0$, then, for all $k \in \llbracket 1, q \rrbracket$, $y_{k-1} - \lambda y_{k} + y_{k+1} = 0$.
Determine the spectrum of $B$ (the square matrix of order $q$ with coefficient $(i,j)$ equal to 1 if $|i-j|=1$ and 0 otherwise) and a basis of eigenvectors of $B$.
With $A = (1-2r)I_{q} + rB$, $r = \frac{\tau}{\delta^2}$, $\delta = \frac{1}{q+1}$, give a necessary and sufficient condition on $r$ for the sequence $\left(F_{n}\right)_{n \in \mathbb{N}}$ to be bounded regardless of the choices of $q$ and $F_{0}$.
Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables (taking values in $\{1,-1\}$ each with probability $1/2$). For every $n \in \mathbb{N}^{*}$, we set $Y_{n} = \frac{1}{2}\left(X_{n}+1\right)$ and $Z_{n} = \sum_{j=1}^{n} Y_{j}$. Determine the distribution of the random variable $Y_{n}$ and that of the random variable $Z_{n}$.
Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables. We denote, for every integer $n \geqslant 1$, $S_{n} = \sum_{j=1}^{n} X_{j}$. Let $k$ be an integer such that $-n \leqslant k \leqslant n$. Show that, if $n$ and $k$ do not have the same parity, then $\mathbb{P}\left(S_{n} = k\right) = 0$.
Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables. We denote $S_{n} = \sum_{j=1}^{n} X_{j}$. Let $k$ be an integer such that $-n \leqslant k \leqslant n$. Show that, if $n$ and $k$ have the same parity, $\mathbb{P}\left(S_{n} = k\right) = \binom{n}{(k+n)/2} \frac{1}{2^{n}}$.
Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. For $x$ real, $\lfloor x \rfloor$ denotes the integer part of $x$. For all real numbers $\delta > 0$ and $\tau > 0$, calculate $\mathbb{V}\left(\delta S_{\lfloor 1/\tau \rfloor}\right)$, the variance of the random variable $\delta S_{\lfloor 1/\tau \rfloor}$.
Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. Show that, for every real number $\delta$, $\mathbb{V}\left(\delta S_{\lfloor 1/\tau \rfloor}\right)$ is equivalent to $\frac{\delta^{2}}{\tau}$, as $\tau$ tends to 0 from above.
Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. For every $n \in \mathbb{N}^{*}$ and every $k \in \mathbb{Z}$, by setting $p_{n}(k) = \mathbb{P}\left(S_{n} = k\right)$, show that $$\frac{p_{n+1}(k) - p_{n}(k)}{\tau} = \frac{\delta^{2}}{2\tau} \frac{p_{n}(k+1) - 2p_{n}(k) + p_{n}(k-1)}{\delta^{2}}$$
Using the result of Q40, deduce a probabilistic interpretation of the stability condition studied in Part III (i.e., the condition on $r = \frac{\tau}{\delta^2}$ found in Q34).