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A class of Stieltjes functions of finite type is introduced. These satisfy Widder's conditions on the successive derivatives up to some finite order and are not necessarily smooth. We show that such functions have a unique integral representation along some generic kernel which is a truncated Laurent series approximating the standard Stieltjes kernel. We then obtain a two-to-one correspondence, via the logarithmic derivative, between these functions and a subclass of hyperbolically monotone functions of finite type. This correspondence generalizes a representation of HCM functions in terms of two Stieltjes transforms earlier obtained by the first author.

Let k > 0 be an integer and Y a standard Gamma(k) distributed random variable. Let X be an independent positive random variable with a density that is hyperbolically monotone (HIM) of order k. Then Y . X and Y/X both have distributions that are generalized gamma convolutions (GGCs). This result extends a result of Roynette et al. from 2009 who treated the case k = 1 but without use of the HM-concept. Applications in excursion theory of diffusions and in the theory of exponential functionals of Levy processes are mentioned.

Thorin’s class of generalized gamma convolutions (GGCs) is closed with respect to change in scale, weak limits, and addition of independent random variables. Here, it is shown that the GGC class also has the remarkable property of being closed with respect to multiplication of independent random variables. This novel result, which has a simple extension to symmetric distributions on R, has many consequences and applications. In particular, it follows that X∼ GGC implies that exp(X)∼ GGC. The latter result is used to find a large class of explicit probability functions on {0,1,2,…} which are generalized negative binomial convolutions (GNBCs). The paper ends with several open problems.

Fixed size without replacement sampling designs with probability functions that are linear or quadratic functions of the sampling indicators are defined and studied. Generality, simplicity, remarkable properties, and also somewhat restricted flexibility characterize these designs. It is shown that the families of linear and quadratic designs are closed with respect to sample complements and with respect to conditioning on sampling outcomes for specific units. Relations between inclusion probabilities and parameters of the probability functions are derived and sampling procedures are given. 

In this paper we consider a family of sampling designs for which increasing first-order inclusion probabilities imply, in a specific sense, increasing conditional inclusion probabilities. It is proved that the complementary Midzuno, the conditional Poisson, and the Sampford designs belong to this family. It is shown that designs of the family are more efficient than a comparable with-replacement design. Furthermore, the efficiency gain is explicitly given for these designs.

Methods to perform fixed size sampling with prescribed second-order inclusion probabilities are presented. The focus is on a conditional Poisson design of order 2, a CP(2) design. It is an exponential design of quadratic type and it is carefully studied. In particular, methods to find the suitable values of the parameters and methods to sample are described. Small examples illustrate.

Sampford's (1967) unequal probability sampling method is extended to the case that the inclusion probabilities do not sum to an integer. In this case the sampling outcome is left open for exactly one randomly chosen unit and that unit gets a new inclusion probability. Three applications are presented. Two of them challenge traditional sampling routines. The simple Pareto sampling design, which was introduced by Rosén (1997a,b), is also extended. The extended Pareto design is shown to be close to the extended Sampford design.

Two new unequal probability sampling methods are introduced: conditional and restricted Pareto sampling. The advantage of conditional Pareto sampling compared with standard Pareto sampling, introduced by Rosén (J. Statist. Plann. Inference, 62, 1997, 135, 159), is that the factual inclusion probabilities better agree with the desired ones. Restricted Pareto sampling, preferably conditioned or adjusted, is able to handle cases where there are several restrictions on the sample and is an alternative to the recent cube method for balanced sampling introduced by Deville and Tillé (Biometrika, 91, 2004, 893). The new sampling designs have high entropy and the involved random numbers can be seen as permanent random numbers.

Simple recursion formulas are presented for the calculation of inclusion probabilities of allorders for conditional Poisson, Sampford, Pareto, and other sampling designs. For CPsampling, the formulas date partially back to work by Chen et al. in 1994.