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 $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$ An analogous proof to that of the previous subsection allows us to show that, for any continuous and bounded function $f$ on $\mathbb{R}$, $$E_{n,f} \xrightarrow[n \rightarrow +\infty]{} \int_{-\infty}^{+\infty} f(u) \varphi_\infty(u) \mathrm{d}u$$ Let $K \in \mathbb{R}_+^*$, and let $f$ be a $K$-Lipschitz function and bounded on $\mathbb{R}$. Show that $$\left|E_{n,f} - \mathbb{E}\left(f\left(n^{1/4} M_n\right)\right)\right| \leqslant \frac{2K}{n^{1/4}\sqrt{2\pi}}$$ and deduce that $$\mathbb{E}\left(f\left(n^{1/4} M_n\right)\right) \xrightarrow[n \rightarrow +\infty]{} \int_{-\infty}^{+\infty} f(u) \varphi_\infty(u) \mathrm{d}u$$
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
$$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
An analogous proof to that of the previous subsection allows us to show that, for any continuous and bounded function $f$ on $\mathbb{R}$,
$$E_{n,f} \xrightarrow[n \rightarrow +\infty]{} \int_{-\infty}^{+\infty} f(u) \varphi_\infty(u) \mathrm{d}u$$
Let $K \in \mathbb{R}_+^*$, and let $f$ be a $K$-Lipschitz function and bounded on $\mathbb{R}$. Show that
$$\left|E_{n,f} - \mathbb{E}\left(f\left(n^{1/4} M_n\right)\right)\right| \leqslant \frac{2K}{n^{1/4}\sqrt{2\pi}}$$
and deduce that
$$\mathbb{E}\left(f\left(n^{1/4} M_n\right)\right) \xrightarrow[n \rightarrow +\infty]{} \int_{-\infty}^{+\infty} f(u) \varphi_\infty(u) \mathrm{d}u$$