Multivariate edgeworth expansions

Consider the Edgeworth expansion in powers of n − 1 2 for P ( Y n < x ) about Φ V ( x ), the multivariate normal distribution, where Y n ∈ R q is a standard estimate with asymptotic covariance V . Here, n typically denotes the sample size. For p ≤ q , one can often choose y ∈ R q such that P ( Y n < x + n − 1 2 y ) = Φ V ( x ) + O ( n − p + 1 2 ) .