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  • Author or Editor: M. Csörgő x
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Abstract  

Let G be a finite group. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω1(p). Zhang [5] proved that if P is a Sylow p-subgroup of G, Ω1(P) ≦ Z(P) and N G(Z(P)) has a normal p-complement, then G has a normal p-complement. The object of this paper is to generalize this result.

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Abstract  

Let X,X 1,X 2,… be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in D[0, 1] for the partial sum processes {S [nt], 0 ≦ t ≦ 1} where S n = Σj=1 n X j, under the assumption that X belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes {S [nt]=V n, 0 ≦ t ≦ 1}, where V n 2 = Σj=1 n X j 2. L p approximations of self-normalized partial sum processes are also discussed.

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Abstract  

This paper investigates weighted approximations for Studentized U-statistics type processes, both with symmetric and antisymmetric kernels, only under the assumption that the distribution of the projection variate is in the domain of attraction of the normal law. The classical second moment condition E|h(X 1, X 2)|2 < ∞ is also relaxed in both cases. The results can be used for testing the null assumption of having a random sample versus the alternative that there is a change in distribution in the sequence.

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Sample path properties of the Cauchy principal values of Brownian and random walk local times are studied. We establish LIL type results (without exact constants). Large and small increments are discussed. A strong approximation result between the above two processes is also proved.

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