Lennart Bondesson

Generalized Gamma convolutions and complete monotonicity

SummaryLetE be the class ofpdfsf on (0,∞) such that, for eachu>0,f(uv) f (u/v) is completely monotone as a function ofw=v+v−1. This class includes many familiarpdfs and is closed with respect to multiplication and division of independentrvs. Further,E ⊂T, whereT is the class of generalized Gamma convolutions (GGC) introduced by O. Thorin. Moreover,E coincides with the class ofpdfs of the form

A remarkable property of generalized gamma convolutions

C \cdot x^{\beta - 1} \cdot \mathop \Pi \limits_{i = 1}^N (1 + c_i x)^{y_i }

On a minimum correlation problem

(all parameters positive) or limits thereof. The Laplace transform ϕ of aGGC is characterized by complete monotonicity of ϕ(uv) ϕ(u/v) as a function ofw. This characterization has many consequences and applications. It follows that also the classT has simple multiplicative properties.

Traits related to stem volume yield capacity in sample trees of pinus sylvestris L. and pinus contorta dougl.

Abstract 1. Introduction and summary Part II of this paper is the result of the present authors efforts to show that a distribution on Z +={0, 1, 2, ... } with probability function of the form where the ai s and they γi s are strictly positive and , is a generalized negative binomial convolution (g.n.b.c.). The g.n.b.c.s were introduced in Part I of this paper. It will be shown that a distribution of the above type at least for N = 1 is a g.n.b.c. In particular the discrete Pareto distribution pj =constant·(j+a)-γ,j=0, 1, 2, ..., (a>0, γ>1) is a g.n.b.c. The proof will be based on a result for generalized gamma convolutions (g.g.c.) which is equivalent to Theorem 2 in Bondesson (1977).

A non-symmetric matrix with integer eigenvalues

SummaryIt is proved that f(x) is the density of a generalized Gamma convolution (GGC), then so is C(1+cx)-γf(x) provided that γ is a nonnegative integer. Some classes of GGCs are introduced.

New results on generalized gamma convolutions and the B-class

Abstract It is shown that the half-Cauchy distribution is infinitely divisible (ID). Some classes of ID and self-decomposable decreasing densities and probability functions on the nonnegative line are studied. These classes have many nice properties.

Factorization theory for probability distributions

A stationary sequence of Bernoulli(p) distributed variables is considered. For a general class of 1-dependent sequences the minimum and the maximum of the lag 1 autocorrelation is found. The correlation is never below -1/3 and attains this value only for certain values of p.

Shot-Noise Distributions

Species, provenances and even individual trees are known to differ in stem volume yield capacity, even when grown on the same site. Increased knowledge of the fundamental mechanisms for different productivity is essential for the success of breeding research and for the maximization of yield. Genetically homogeneous tree materials were selected on the basis of expected differences in stem volume yield capacity and trees were sampled and destructively analysed. The hypothesis to be tested was that yield capacity is correlated to certain traits of trees. High correlations with stem volume yield capacity were shown for harvest index, needle‐use efficiency and mineral‐nutrient‐use efficiency. Only weak correlations were shown for needle longevity and nitrogen concentration in needles. Also, indications were found that phenotypic plasticity was correlated positively to stem volume growth capacity. An attempt to correlate the ability to retranslocate nitrogen from dying needles with yield capacity failed.