Given reals $t > 0$ and $u$, we set
$$\zeta ( t , u ) = \exp \left( i \frac { u } { \sigma _ { t } } \left( m _ { t } - \frac { \pi ^ { 2 } } { 6 t ^ { 2 } } \right) \right) \quad \text { and } \quad j ( t , u ) = \zeta ( t , u ) h \left( t , \frac { u } { \sigma _ { t } } \right)$$
Conclude that
$$\int _ { - \pi \sigma _ { t } } ^ { \pi \sigma _ { t } } j ( t , u ) \mathrm { d } u \underset { t \rightarrow 0 ^ { + } } { \longrightarrow } \sqrt { 2 \pi }$$

We admit that $P \left( e ^ { - t } \right) \sim \sqrt { \frac { t } { 2 \pi } } \exp \left( \frac { \pi ^ { 2 } } { 6 t } \right)$ as $t$ tends to $0 ^ { + }$.
By applying formula (1) to $t = \frac { \pi } { \sqrt { 6 n } }$, prove that
$$p _ { n } \sim \frac { \exp \left( \pi \sqrt { \frac { 2 n } { 3 } } \right) } { 4 \sqrt { 3 } n } \quad \text { as } n \rightarrow + \infty$$

Let $p \in \left[ 1 , + \infty \right[$. Let $\left( X _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket}$ be a sequence of independent random variables all following a Rademacher distribution. Let $\left( c _ { 1 } , \ldots , c _ { n } \right) \in \mathbf { R } ^ { n }$. Suppose in this question that $\sum _ { i = 1 } ^ { n } c _ { i } ^ { 2 } = 1$. Show that the integral $\int _ { 0 } ^ { + \infty } t ^ { 3 } \mathrm { e } ^ { - t ^ { 2 } / 2 } \mathrm {~d} t$ converges, then that $$\mathbf { E } \left( \left( \sum _ { i = 1 } ^ { n } c _ { i } X _ { i } \right) ^ { 4 } \right) \leq 8 \int _ { 0 } ^ { + \infty } t ^ { 3 } \mathrm { e } ^ { - t ^ { 2 } / 2 } \mathrm {~d} t$$

In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We denote $Z_\infty = \int_{-\infty}^{+\infty} \exp\left(-\frac{x^4}{12}\right) \mathrm{d}x$ and we consider the function $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
Justify that, for any continuous and bounded function $f$ on $\mathbb{R}$, the quantity $$E_{n,f} = \int_{-\infty}^{+\infty} \mathbb{E}\left(f\left(\frac{t}{n^{1/4}} + n^{1/4} M_n\right)\right) \exp\left(-\frac{t^2}{2}\right) \frac{\mathrm{d}t}{\sqrt{2\pi}}$$ is well defined.

Consider a region $R$ defined by $0 < x < 1$ and $0 < y < 1$ in the $x y$-plane. We randomly select a point on $R$ and refer to the selected point as A. We assume that A is uniformly distributed on $R$. Let AB be a perpendicular line from A to the $y$-axis and AC be a perpendicular line from $A$ to the $x$-axis as shown in the figure. We call rectangle $OCAB$ as "the rectangle of A", where O denotes the origin. Let $S$ be a random variable representing the area of the rectangle of A. Answer the following questions.
(1) Calculate the expectation value of $S$.
(2) Calculate the probability that $S \leq r$ holds, where $0 < r < 1$.
(3) Calculate the probability density function of $S$.
Again consider the region $R$. Let $n$ be a positive integer. We select $n$ points on $R$ and refer to the selected points as $\mathrm{A}_1, \mathrm{~A}_2, \ldots, \mathrm{~A}_n$. We assume that each of the points is uniformly distributed on $R$, and $\mathrm{A}_i$ and $\mathrm{A}_j$ for $i \neq j$ are selected independently. Answer the following question.
(4) Let $S_i$ be a random variable representing the area of the rectangle of $\mathrm{A}_i$. Let $Z$ be a random variable which is the minimum of $S_1, S_2, \ldots, S_n$. Calculate the probability density function of $Z$.