Asymptotic expansions for sums of weakly dependent random vectors

SummaryIt is shown that formal Edgeworth expansions are valid for sums of weakly dependent random vectors. The error of approximation has ordero(n−(s−2)/2) if(i)the moments of orders+1 are uniformly bounded(ii)a conditional Cramér-condition holds(iii)the random vectors can be approximated by other random vectors which satisfy a strong mixing condition and a Markov type condition. The strong mixing coefficients in (iii) are decreasing at an exponential rate. The above conditions can easily be checked and are often satisfied when the sequence of random vectors is a Gaussian, or a Markov, or an autoregressive process. Explicit formulas are given for the distribution of finite Fourier transforms of a strictly stationary time series.

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