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

For each $n\ge 1$, let $X_{n,1},\ldots,X_{n,N_n}$ be real random variables and $S_n=\sum_{i=1}^{N_n}X_{n,i}$. Let $m_n\ge 1$ be an integer. Suppose $(X_{n,1},\ldots,X_{n,N_n})$ is $m_n$-dependent, $E(X_{ni})=0$, $E(X_{ni}^2)<\infty$ and $σ_n^2:=E(S_n^2)>0$ 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 $$U_n(c)=\frac{m_n}{σ_n^2}\,\sum_{i=1}^{N_n}E\Bigl[X_{n,i}^2\,1\bigl\{\abs{X_{n,i}}>c\,σ_n/m_n\bigr\}\Bigr].$$ Among other things, this estimate of $d_W\bigl(S_n/σ_n,\,Z\bigr)$ yields a similar estimate of $d_{TV}\bigl(S_n/σ_n,\,Z\bigr)$ where $d_{TV}$ is total variation distance.

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