Let {Xn} be a sequence of independent identically distributed random variables with a common continuous distribution function and let Mj;n denote the jth upper order statistic among X1,X2, . . . ,Xn, n ≥ j. For a large class of distributions, we obtain the law of the iterated logarithm for {M1,n,M2,n}, properly normalized. As a consequence, we establish a law of the iterated logarithm for the spacings {M1,n −M2,n}.
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We study a system of pseudodifferential equations which is elliptic in the Petrovskiî sense on a closed smooth manifold. We prove that the operator generated by the system is a Fredholm operator in a refined two-sided scale of Hilbert function spaces. Elements of this scale are special isotropic spaces of Hörmander-Volevich-Paneah.
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