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One-dimensional empirical measures, order statistics, and Kantorovich transport distances
This work is devoted to the study of rates of convergence of the empirical measures \\mu_{n} = \\frac {1}{n} \\sum_{k=1}^n \\delta_{X_k}, n \\geq 1, over a sample (X_{k})_{k \\geq 1} of independent identically distributed real-valued random variables towards the common distribution \\mu in Kantorovich transport distances W_p. The focus is on finite range bounds on the expected Kantorovich distances \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) or \\big [ \\mathbb{E}(W_{p}^p(\\mu_{n},\\mu )) \\big ]^1/p in terms of moments and analytic conditions on the measure \\mu and its distribution function. The study describes a variety of rates, from the standard one \\frac {1}{\\sqrt n} to slower rates, and both lower and upper-bounds on \\mathbb{E}(W_{p}(\\mu_{n},\\mu )) 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.
eBook
Additive Differentials for ARX Mappings with ProbabilityExceeding 1/4
We consider the additive differential probabilities of functions and , where and . The probabilities are used for the differential cryptanalysis of ARX ciphers that operate only with addition modulo , bitwise XOR ( ), and bit rotations ( ). A complete characterization of differentials whose probability exceeds is obtained. All possible values of their probabilities are for . We describe differentials with each of these probabilities and calculate the number of these values. We also calculate the number of all considered differentials. It is for and for , where . We compare differentials of both mappings under the given constraint.
Journal Article
The Pseudo-Marginal Approach for Efficient Monte Carlo Computations
We introduce a powerful and flexible MCMC algorithm for stochastic simulation. The method builds on a pseudo-marginal method originally introduced in [Genetics 164 (2003) 1139-1160], showing how algorithms which are approximations to an idealized marginal algorithm, can share the same marginal stationary distribution as the idealized method. Theoretical results are given describing the convergence properties of the proposed method, and simple numerical examples are given to illustrate the promising empirical characteristics of the technique. Interesting comparisons with a more obvious, but inexact, Monte Carlo approximation to the marginal algorithm, are also given.
Journal Article
Chance in the world : a Humean guide to objective chance
Whether something happens randomly, by chance; or from a series of events.
Book
Nilspace Factors for General Uniformity Seminorms, Cubic Exchangeability and Limits
We study a class of measure-theoretic objects that we call
eBook