🚀 go-pugleaf

Let $n\geq1$, $N\geq2$, $(\alpha,\beta)\in{\bb R}^2$ and $C>0$.\\
Consider $X_1,\dots,X_N$ integer valued random variables
and $p_1,\dots,p_N$ strictly positive real numbers s.t.
$$
\sum_{k=1}^Np_k=1\,.
$$ 
Let the following three conditions
\begin{equation}\label{eq:cond1}
E[X_k]=Np_k\,,\quad\forall\,\,1\leq k\leq N\,.
\end{equation}
Also,
\begin{equation}\label{eq:cond2}
E[|\sum_{k=1}^NX_kf(\xi_k)-N\sum_{k=1}^Np_kf(\xi_k)|^\alpha]
\leq CN^\beta \|f\|^\alpha\,.
\end{equation}
where $f\in C_b({\bb R}^n)$ and $\{\xi_k\,,\,\,1\leq k\leq N\}$ are any independent 
${\bb R}^n$-valued random vectors.\\
And finally
\begin{equation}\label{eq:cond3}
0<\beta<\alpha\qquad and \quad \max(2,2\beta)\leq\alpha\,.
\end{equation}
$\mathbf{Question}$:\\

Under conditions (\ref{eq:cond1}), (\ref{eq:cond2}) and (\ref{eq:cond3}).\\
Give examples of discrete probabilities, i.e.\\
Define
$$
P(X_k=i_k\,,\,\,1\leq k\leq N)\,,\quad\forall(i_1,\dots,i_N)\in{\bb N}^N\,.
$$
$\mathbf{Solutions}$\\
I begin by the following remark:\\
If the r.v.'s $X_k$ are supposed independent and $\alpha=2$, then condition 
(\ref{eq:cond2}) is equivalent to 
\begin{equation}\label{eq:cond4}
E[\sum_{k=1}^N(X_k-Np_k)^2]\leq CN^\beta \,.
\end{equation}
I have already done the following examples, which are all associated to the case 
$(\alpha,\beta)=(2,1)$
\begin{quote}
1- Negative Binomial : independent and $C=2$.\\
2- Binomial : independent and $C=1$.\\
3- Poisson : independent and  $C=1$.\\
4- Bernoulli : independent and $C=1/4$.\\
5- Multinomial : Negatively correlated and $C=1$.
\end{quote}
I would like to have at least an "interessant " example for which 
$(\alpha,\beta)\neq(2,1)$, for example $(\alpha,\beta)\in\{(4,2), (2,1/2)\}$.\\