Quantitative bounds in the central limit theorem for $m$-dependent random variables

For each $n\ge 1$, let $X_{n,1},\dots ,X_{n,N_n}$ be real random variables and $S_n={\sum }_{i=1}^{N_n}{X}_{n,i}$. Let $m_n\ge 1$ be an integer. Suppose $(X_{n,1},\dots ,X_{n,N_n})$ is $m_n$-dependent, $E(X_{ni})=0$, $E(X_{ni}^2)<\mathrm{\infty }$ and ${\sigma }_n^2:=E(S_n^2)>0$ for all $n$ and $i$. Then, \begin{gather*} d_W\Bigl(\frac{S_n}{σ_n},,Z\Bigr)\le 30,\bigl{c^{1/3}+12,U_n(c/2)^{1/2}\bigr}\quad\quad$\text{for all }$n\ge 1$\text{ and }$c>0, \end{gather*} where $d_W$ is Wasserstein distance, $Z$ a standard normal random variable and $$U_n(c)=\frac{m_n}{{\sigma }_n^2}\sum _{i=1}^{N_n}E\left[X_{n,i}^{2}1\right\{\text{abs}{X}_{n,i}>c{\sigma }_n\mathrm{/}{m}_n\left\}\right]\mathrm{.}$$ Among other things, this estimate of $d_W(S_n\mathrm{/}{\sigma }_n,Z)$ yields a similar estimate of $d_{TV}(S_n\mathrm{/}{\sigma }_n,Z)$ where $d_{TV}$ is total variation distance.