I was reading this paper. I saw a fascinating Proposition.[Page-5 of that paper]
Now I have my $X$ as this which are not subgaussian. Can I get something like proposition $1.2$ for my non-subgaussian $X$?
$$\mathbb{P}\left(|(h_{w}\left(s_i,p_i\right)-\mathbb{E} h_{w}) z_i| \geq 2B \cdot C_{\beta, D_{L S_\beta}} m(\xi,h_{w} )\right) \leq e^{-\xi}\tag1$$
Let, $$X_i = (h_{w}\left(s_i,p_i\right)-\mathbb{E} h_{w}) z_i$$
$$\mathbb{P}\left(|X_i| \geq 2B \cdot C_{\beta, D_{L S_\beta}} m(\xi,h_{w})\right) \leq e^{-\xi}\tag2$$
$$ \mathbb{P}\left (\frac{1}{n}\left| \sum_{i=1}^n (h_{w}\left(s_i,p_i\right) - \mathbb{E}[h_w])z_i\right| \geq Constant \right ) \leq e^{-\kappa}\tag3$$ $\quad put \quad \xi = \kappa \quad and \quad thus \quad 2B \cdot C_{\beta, D_{L S_\beta}} m(\kappa,h_{w} ) = Constant $
How Can we write (2) to (3). Is it possible?