Welcome to the resource topic for 2011/298
Title:
Local limit theorem for large deviations and statistical box-tests
Authors: Igor Semaev
Abstract:Let n particles be independently allocated into N boxes, where the l-th box appears with the probability a_l. Let \mu_r be the number of boxes with exactly r particles and \mu=[ \mu_{r_1},\ldots, \mu_{r_m}]. Asymptotical behavior of such random variables as N tends to infinity was studied by many authors. It was previously known that if Na_l are all upper bounded and n/N is upper and lower bounded by positive constants, then \mu tends in distribution to a multivariate normal low. A stronger statement, namely a large deviation local limit theorem for \mu under the same condition, is here proved. Also all cumulants of \mu are proved to be O(N). Then we study the hypothesis testing that the box distribution is uniform, denoted h, with a recently introduced box-test. Its statistic is a quadratic form in variables \mu-\mathbf{E}\mu(h). For a wide area of non-uniform a_l, an asymptotical relation for the power of the quadratic and linear box-tests, the statistics of the latter are linear functions of \mu, is proved. In particular, the quadratic test asymptotically is at least as powerful as any of the linear box-tests, including the well-known empty-box test if \mu_0 is in \mu.
ePrint: https://eprint.iacr.org/2011/298
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