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

For each n >= 1, let X-n,X-1, . . . , X-n,X-Nn be real random variables and S-n = Sigma(Nn)(i=1) X-n,X-i. Let m(n) >= 1 be an integer. Suppose (X-n,X-1, . . . , X-n,X-Nn) is m(n)-dependent, E(X-ni) = 0, E(X-ni(2)) < infinity and sigma(2)(n) := E(S-n(2)) > 0 for all n and i. Then, d(W) (S-n/sigma(n), Z) <= 30 {c(1/3) + 12U(n)(c/2)(1/2)} for all n >= 1 and c > 0, where d(W) is Wasserstein distance, Z a standard normal random variable and U-n(c) = m(n)/sigma(2)(n) X Sigma(Nn)(i=1) E[X-n,i(2) 1{|X-n,X-i| > c sigma(n)/m(n)}]. Among other things, this estimate of d(W) (S-n/sigma(n), Z) yields a similar estimate of d(TV) (S-n/sigma(n), Z) where d(TV) is total variation distance.