In this paper, we derive higher order Edgeworth expansions for the finite sample distributions of the subsampling-based $t$-statistic and the Wald statistic in the Gaussian location model under the so-called fixed-smoothing paradigm. In particular, we show that the error of asymptotic approximation is at the order of the reciprocal of the sample size and obtain explicit forms for the leading error terms in the expansions. The results are used to justify the second-order correctness of a new bootstrap method, the Gaussian dependent bootstrap, in the context of Gaussian location model.
"Fixed-smoothing asymptotics for time series." Ann. Statist. 41 (3) 1329 - 1349, June 2013. https://doi.org/10.1214/13-AOS1113