Consider a sequence of random variables $(X_n)_{n\in \mathbb{N}}$ and a sequence of positive reals $(\sigma_n)_{n\in \mathbb{N}}$ such that: $$\frac{X_n}{\sigma_n}\stackrel{d}{\longrightarrow}Fréchet(\alpha,1)$$ where $\sigma_n\longrightarrow \infty$, especially for any subsequence $(k_n)_{n\in \mathbb{N}}$ of $(n)_{n\in \mathbb{N}}$, with $\frac{k_n}{n}\longrightarrow 1$, then $\frac{\sigma_{k_n}}{\sigma_n}\longrightarrow 1$ and $\alpha>0$. Now we assume that for some $\nu>0$ we have: $$\limsup_{n\longrightarrow\infty}\mathbb{E}\left(g_{\nu,\alpha}\left(\frac{\max\{X_n,1\}}{\sigma_n}\right)\right) < \infty$$
where $g_{\nu,\alpha}(x)=\left(x^{-\alpha}\chi_{\{x\leq e\}}+\log(x)\chi_{\{x>e\}}\right)^{2+\nu}$.
Now I am sure there has to be a way to conclude from the assumption that: $$\limsup_{n\longrightarrow\infty}\mathbb{E}\left(g_{\nu,\alpha}\left(\frac{\max\{X_n,c\}}{\sigma_n}\right)\right) < \infty$$ for all $c>0$.
To start of I tried to use the fact that: $$g_{\nu,\alpha}(x)\leq 2^{2+\nu}\left(g_1(x)^{2+\nu} + g_2(x)^{2+\nu}\right) $$, where $g_1(x)=x^{-\alpha}\chi_{\{x\leq e\}}$ and $g_2(x)=\log(x)\chi_{\{x>e\}}$.
Thus we can use monotonic properties of $g_1$ and $g_2$. But this doesn‘t seem to be useful, as I, no matter what I try, don't get the desired result.
What might come useful is that the weak convergence implies:
$$P(X_n < C) \longrightarrow 0, \ \forall C>0$$
Also rewriting $\frac{\max\{X_n,c\}}{\sigma_n} = \frac{\max\{X_n/c,1\}}{\sigma_n/c}$ looks effective, but I don‘t find a way to use this, to deduce the statement.
I think I am overlooking something plain in sight. I would be grateful just for a push in the right direction, no full answer - I would answer the question myself.
