Asymptotic Expansions of Central Limit Distances in Vaserstein Metrics

A. M. Davie ()
Additional contact information
A. M. Davie: University of Edinburgh, School of Mathematics and Maxwell Institute for Mathematical Sciences

A chapter in Stochastic Analysis and Applications 2025, 2026, pp 215-229 from Springer

Abstract: Abstract 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 .

Keywords: Primary 60F05; Secondary 49Q22 (search for similar items in EconPapers)
Date: 2026
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-03914-9_8

Ordering information: This item can be ordered from
http://www.springer.com/9783032039149

DOI: 10.1007/978-3-032-03914-9_8

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().