  In this paper, we consider $d$-dimensional mixing random fields
$\bigl(X^{(n)}_{i}\bigr)_{i\in T_{n}}$ and study the convergence of the
empirical average $W_n:=\sigma_n^{-1} \sum_{i\in T_n}X^{(n)}_{i}. $ Under
$\alpha$-mixing and moment conditions, we obtain smooth Edgeworth expansions
for $W_n$ of any order $k\ge 1$ with better controlled remainder terms. We
exploit this to obtain rates for the convergence of $W_n$ to its normal limit
in the Wasserstein-$p$ distance for any $p\geq 1$. The bounds depend on the
moments of the random variables $\bigl(X^{(n)}_{i}\bigr)_{i\in T_{n}}$ and the
rate of polynomial decay of the mixing coefficients. Finally, we apply those
results to obtain tail inequalities and non-uniform Berry--Esseen bounds with
polynomial decay.

Probability