Info about $h_w(s,p)$
Assume that $\left(s,p\right)$ is a random vector in $\mathbb{R}^{d_1 + d_2}$, which satisfies the $\beta$-modified logarithmic Sobolev inequality (Page $10$; Equation $15$) with constant $C_{\beta, D_{L S_\beta}}$
$w$ is a weight parameter. It is given $g(s,p)$ is defined as $E(y|(s,p))$. $y$ is a random variable, which is less than or equal to $1$ so it is evident that $\mathbb{E}(y|(s,p))$ $\leq 1$. So $g(s,p) \leq 1$. It is defined that $z_{i}=y_{i}-g\left(s_{i},p_{i}\right)$
Let $\mathcal{H}$ be a class of functions from $\mathbb{R}^{d_1} \times \mathbb{R}^{d_2} \rightarrow \mathbb{R}^q$ and let $\left(y_i,\left(s_i, p_i\right)\right)$ be iid input-output pair, And $y_{i} \in[-B,B]$. fix $\epsilon, \delta \in[0,1]$
All functions are neural net parameterized functions and hence following the Lipschitz property. $L_B$ and $L_T$ are Lipschitz constant.
$$ \mathcal{B}:=\left\{\mathrm{B}_{\boldsymbol{w}} \mid \mathrm{B}: \mathbb{R}^{d_{1}} \rightarrow \mathbb{R}^{q}, \operatorname{Lip}\left(\mathrm{B}_{\boldsymbol{w}}\right) \leq L_{B} \&\|\boldsymbol{w}\| \leq W_{B}\right\} $$ $$ \mathcal{T}:=\left\{\mathbf{T}_{\boldsymbol{w}} \mid \mathbf{T}: \mathbb{R}^{d_{2}} \rightarrow \mathbb{R}^{q}, \operatorname{Lip}\left(\mathbf{T}_{\boldsymbol{w}}\right) \leq L_{T} \&\|\boldsymbol{w}\| \leq W_{T}\right\} $$ Now consider the following function class of all possible inner-products between the two, $$ \mathcal{H}:=\left\{h_{\boldsymbol{w}_{b}, \boldsymbol{w}_{t}} = h_w\mid \mathbb{R}^{d_{1}} \times \mathbb{R}^{d_{2}} \ni(s, p) \mapsto h_w(s, p):=\left\langle\mathrm{B}_{\boldsymbol{w}_{b}}(s), \mathbf{T}_{\boldsymbol{w}_{t}}(p)\right\rangle \in \mathbb{R}, \mathrm{B}_{\boldsymbol{w}_{b}} \in \mathcal{B} \& \mathbf{T}_{\boldsymbol{w}_{t}} \in \mathcal{T}\right\} $$
I have this inequality,
$$\mathbb{P}\left(|(h_{w}\left(s_i,p_i\right)-\mathbb{E} h_{w}) z_i| \geq 2B \cdot C_{\beta, D_{L S_\beta}} m(\xi,h_{w} )\right) \leq e^{-\xi}\tag1$$
Assuming, $$X_i = (h_{w}\left(s_i,p_i\right)-\mathbb{E} h_{w}) z_i$$ & $ 2B \cdot C_{\beta, D_{L S_\beta}} m(\xi,h_{w} ) = t$
So, $$\mathbb{P}\left(|X_i| \geq t\right) \leq e^{-\xi}\tag2$$
How can I prove my var($X_i$) is bounded. I want to do this, so that I can apply Chebyshev’s inequality on $X_i$.