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We present a short and elementary proof of the Ajtai-Koml\'os-Tusn\'ady (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. 

We explore upper bounds on Kantorovich transport distances between probability measures on the Euclidean spaces in terms of their Fourier-Stieltjes transforms, with focus on non-Euclidean metrics. The results are illustrated on empirical measures in the optimal matching problem on the real line. 

This work is devoted to the study of rates of convergence of the empirical measures μn over a sample (Xk) of independent identically distributed real-valued random variables towards the common distribution μ in Kantorovich transport distances Wp. The focus is on finite range bounds on the expected Kantorovich distances E(Wp(μn, μ)) in terms of moments and analytic conditions on the measure μ and its distribution function. The study describes a variety of rates, from the standard one to slower rates, and both lower and upper-bounds on E(Wp(μn,μ)) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log-concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation. 

Abstract Salient aspects of the commissioning, calibration, and performance of the CMS silicon strip tracker are discussed, drawing on experience during operation with proton-proton collisions delivered by the CERN LHC. The data were obtained with a variety of luminosities. The operating temperature of the strip tracker was changed several times during this period and results are shown as a function of temperature in several cases. Details of the system performance are presented, including occupancy, signal-to-noise ratio, Lorentz angle, and single-hit spatial resolution. Saturation effects in the APV25 readout chip preamplifier observed during early Run 2 are presented, showing the effect on various observables and the subsequent remedy. Studies of radiation effects on the strip tracker are presented both for the optical readout links and the silicon sensors. The observed effects are compared to simulation, where available, and they generally agree well with expectations.