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We consider the approximation of a convolution of possibly different probability measures by (compound) Poisson distributions and also by related signed measures of higher order. We present new total variation bounds having a better structure than those from the literature. A numerical example illustrates the usefulness of the bounds, and an application in the Poisson process approximation is given. The proofs use arguments from Kerstan (1964) and Roos (1999b) in combination with new smoothness inequalities, which could be of independent interest. [ABSTRACT FROM AUTHOR]

In an experiment of treatment selections, random samples are drawn from k populations with ordered means. The probability that a sample statistic from the population with the highest mean turns out to be ranked the highest is referred to as the probability of correct selection (PCS). An inequality was proved previously that shows the monotonicity of PCS with respect to change in variance of the samples. In this article, we first present a more general form of the probability inequality to be used to investigate PCS. An extension of the monotonicity of PCS to order statistics is considered. We show that the PCS of the smallest order statistic preserves the monotonicity. Additionally, a normal approximation method is used to further generalize the theory. The general order statistics will not enjoy the same properties, as we reveal the obstacles, and a numerical counter example. [ABSTRACT FROM AUTHOR]

Abstract: Using Schauder''s fixed point theorem, with the help of an integral representation in ‘Sharp conditions for weighted 1-dimensional Poincaré inequalities’, Indiana Univ. Math. J., 49 (2000) 143–175, by Chua and Wheeden, we obtain existence and uniqueness theorems and ‘continuous dependence of average condition’ for average value problem: under the usual conditions for initial value problem. We also extend our existence and uniqueness theorems in the case where v is just a signed measure with and [Copyright &y& Elsevier]