Asymptotic Expansions of Central Limit Distances in Vaserstein Metrics

For a pair of i.i.d. sequences of random variables $$X_1,X_2,\ldots $$ X 1 , X 2 , … and $$\tilde{X}_1,\tilde{X}_2,\ldots $$ X ~ 1 , X ~ 2 , … , under suitable conditions we show the existence of an asymptotic expansion in powers of $$m^{-1}$$ m - 1 of the Vaserstein distance $$\mathbb {W}_r(Y_m,\tilde{Y}_m)$$ W r ( Y m , Y ~ m ) where $$r\ge 1$$ r ≥ 1 , $$Y_m=m^{-1/2}(X_1+\cdots +X_m)$$ Y m = m - 1 / 2 ( X 1 + ⋯ + X m ) and $$\tilde{Y}_m$$ Y ~ m is defined similarly. This is obtained by applying methods from optimal transport theory to a Cornish–Fisher expansion. The terms in the expansion are determined by the moments of the random variables, and we give some sample calculations of such terms. This is an extension of a central limit problem, to which it reduces when the means are 0 and $$\tilde{Y}_1$$ Y ~ 1 is normal with the same variance as $$X_1$$ X 1 . We also describe an application to the question of eventual monotonicity in m of the sequence $$\mathbb {W}_r(Y_m,\tilde{Y}_m)$$ W r ( Y m , Y ~ m ) , related to a question of Villani for the case $$r=2$$ r = 2 .