A SIMPLE FOURIER ANALYTIC PROOF OF THE AKT OPTIMAL MATCHING THEOREM

A SIMPLE FOURIER ANALYTIC PROOF OF THE AKT OPTIMAL MATCHING THEOREM

We present a short and elementary proof of the Ajtai–Komlós–Tusnády (AKT) optimal matching theorem in dimension 2 via Fourier analysis and a smoothing argument. The upper bound applies to more general families of samples, including dependent variables, of interest in the study of rates of convergenc...

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Veröffentlicht in:The Annals of applied probability 2021-12, Vol.31 (6), p.2567-2584
Hauptverfasser: Bobkov, Sergey G., Ledoux, Michel
Format: Artikel
Sprache:eng
Schlagworte:
Convergence
Dependent variables
Empirical analysis
Fourier analysis
Fourier transforms
Lower bounds
Matching
Mathematical functions
Theorems
Upper bounds
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Beschreibung
Zusammenfassung:We present a short and elementary proof of the Ajtai–Komlós–Tusnády (AKT) optimal matching theorem in dimension 2 via Fourier analysis and a smoothing argument. The upper bound applies to more general families of samples, including dependent variables, of interest in the study of rates of convergence for empirical measures. Following the recent pde approach by L. Ambrosio, F. Stra and D. Trevisan, we also adapt a simple proof of the lower bound.
ISSN:1050-5164
2168-8737
DOI:10.1214/20-AAP1656