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 $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$ We are given $x \in \mathbb{R}$ and $\varepsilon > 0$. Let $k$ be a non-zero natural integer such that $k \geqslant \frac{2}{\varepsilon Z_\infty}$. We define the function $$f_k : u \in \mathbb{R} \longmapsto \begin{cases} 1 & \text{if } u \leqslant x \\ 1 - k(u-x) & \text{if } x < u \leqslant x + \frac{1}{k} \\ 0 & \text{otherwise} \end{cases}$$ Deduce that there exists $n_0 \in \mathbb{N}$ such that, for all $n \geqslant n_0$, $$\mathbb{P}\left(n^{1/4} M_n \leqslant x\right) \leqslant \frac{\varepsilon}{2} + \int_{-\infty}^{x + \frac{1}{k}} \varphi_\infty(u) \mathrm{d}u$$
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
$$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
We are given $x \in \mathbb{R}$ and $\varepsilon > 0$. Let $k$ be a non-zero natural integer such that $k \geqslant \frac{2}{\varepsilon Z_\infty}$. We define the function
$$f_k : u \in \mathbb{R} \longmapsto \begin{cases} 1 & \text{if } u \leqslant x \\ 1 - k(u-x) & \text{if } x < u \leqslant x + \frac{1}{k} \\ 0 & \text{otherwise} \end{cases}$$
Deduce that there exists $n_0 \in \mathbb{N}$ such that, for all $n \geqslant n_0$,
$$\mathbb{P}\left(n^{1/4} M_n \leqslant x\right) \leqslant \frac{\varepsilon}{2} + \int_{-\infty}^{x + \frac{1}{k}} \varphi_\infty(u) \mathrm{d}u$$