Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution function F s (x; t) at time t of the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution function F (⋅; t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for the nth derivative, ∂n F s (x; t) ∂ x n . The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

A simple technique for obtaining bounds in terms of means and variances for the expectations of certain functions of random variables in a given class is examined. The bounds given are sharp in the sense that they are attainable by at least one random variable in the class. This technique is applied to obtain bounds for moment generating functions, the coefficient of skewness and parameters associated with branching processes. In particular an improved lower bound for the Malthusian parameter in an age-dependent branching process is derived.

A churning transformation can be defined on probability measures by an infinite sequence of finite permutations of mass. Continuity and absolute continuity of measures are invariants for such transformations but it is shown that certain probability measures whose Fourier-Stieltjes transforms fail to vanish at infinity may be churned into measures whose transforms do vanish in this sense.

Let pα,θ be the Linnik density, that is, the probability density with the characteristic function . The following problem is studied: Let (α θ), (β, ϑ) be two point of PD. When is it possible to represent β,ϑ as a scale mixture of pαθ? A subset of the admissible pairs (α, θ), (β, ϑ) is described.

The key theme is converse forms of criteria for deciding determinateness in the classical moment problem. A method of proof due to Koosis is streamlined and generalized giving a convexity condition under which moments satisfying implies that c a positive constant. A contrapositive version is proved under a rapid variation condition on f (x), generalizing a result of Lin. These results are used to obtain converses of the Stieltjes versions of the Carleman and Krein criteria. Hamburger versions are obtained which relax the symmetry assumption of Koosis and Lin, respectively. A sufficient condition for Stieltjes determinateness of a discrete law is given in terms of its mass function. These criteria are illustrated through several examples.

The Mellin-Stieltjes convolution and related decomposition of distributions in M(α) (the class of distributions μ on (0, ∞) with slowly varying αth truncated moments ) are investigated. Maller shows that if X and Y are independent non-negative random variables with distributions μ and v, respectively, and both μ and v are in D2, the domain attraction of Gaussian distribution, then the distribution of the product XY (that is, the Mellin-Stieltjes convolution μ ^ v of μ and v) also belongs to it. He conjectures that, conversely, if μ ∘ v belongs to D2, then both μ and v are in it. It is shown that this conjecture is not true: there exist distributions μ ∈ D2 and v μ ∈ D2 such that μ ^ v belongs to D2. Some subclasses of D2 are given with the property that if μ ^ v belongs to it, then both μ and v are in D2.

Suppose Xi, i = 1,…,n are indepedent and identically distributed with E/X1/r < ∞, r = 1,2,…. If Cov (( − μ)r, S2) = 0 for r = 1, 2,…, where μ = EX1, S2 = , and , then we show X1 ~ N (μ, σ2), where σ2 = Var(X1). This covariance zero condition charaterizes the normal distribution. It is a moment analogue, by an elementary approach, of the classical characterization of the normal distribution by independence of and S2 using semi invariants. More generally, if Cov = 0 for r = 1,…, k, then E((X1 − μ)/σ)r+2 = EZr+2 for r = 1,… k, where Z ~ N(0, 1). Conversely Corr may be arbitrarily close to unity in absolute value, but for unimodal X1, Corr2( < 15/16, and this bound is the best possible.

A recent result of Rogozin on the relative stability of a distribution function is extended, by giving equivalences for relative stability in terms of truncated moments of the distribution and in terms of the real and imaginary parts of the characteristic function. As an application, the known results on centering distributions in the domain of attraction of a stable law are extended to the case of stochastically compact distributions.

Staring form a probability σ on the half-line moments of any order A. G. Pakes has defined probabilities σr, by length biasing order r and gr, by the stationary-excess operation of order r, r = 1, 2,…Examples are given to show that σ can bt determined in the Stieltjes sence while σ1 and g1 are indeterminate in the Stieltjes sence. This shows that a statement in a recent paper by Pakes does not hold.

The aim of this paper is to show that some of the known properties of distributions in the domain of attraction of a stable law have counterparts for distributions which are stochastically compact in the sense of Feller. This enables us to unify the ideas of Feller and Doeblin, who first studied the concept of stochastic compactness, and give new characterizations of stochastic compactness and the domain of attraction of the normal distribution.

Generalizing known results for special examples, we derive a Khintchine type decomposition of probability measures on symmetric hypergroups. This result is based on a triangular central limit theorem and a discussion of conditions ensuring that the set of all factors of a probability measure is weakly compact. By our main result, a probability measure satisfying certain restrictions can be written as a product of indecomposable factors and a factor in I0(K), the set of all measures having decomposable factors only. Some contributions to the classification of I0(K) are given for general symmetric hypergroups and applied to several families of examples like finite symmetric hypergroups and hypergroup joins. Furthermore, all results are discussed in detail for a class of discrete symmetric hypergroups which are generated by infinitely many joins, for a class of countable compact hypergroups, for Sturm-Liouville hypergroups on [0, ∞[ and, finally, for polynomial hypergroups.

We study negative definite functions on a Hilbert space and use their properties to give a proof of the Lévy-Khinchin formula for an infinitely divisible probability distribution on .

The class of subexponential distributions S is characterized by F(0) = 0, 1 − F(2)(x) ~ 2(1 − F(x)) as x → ∞. In this paper we consider a subclass of S for which the relation 1 − F(2)(x) − 2(1 − F(x)) + (1 − F(x))2 = o(a(x)) as x → ∞ holds, where α is a positive function satisfying α(X) = 0(1 − F(x)) (x → ∞).

Characterisations of the distribution of a non-negative random variable are sought for which the Liapunov moment inequality is extended to give inequalities between inverse powers of moment ratios, which are known as mean sizes in considerations of particle size distributions. A solution is found for continuous distributions, and the conditions applied to a number of well-known distributions. A further class of distributions is considered for which the new inequalities hold but the inequality direction is reversed for some orders of the moments. The study involves examination of the signs of the third central moments of a family of distributions, obtained by a log transformation, from the weighted, or moment, distributions induced by the non-negative random variable.

For a distribution function F on [0, ∞] we say F ∈ if {1 – F(2)(x)}/{1 – F(x)}→2 as x→∞, and F∈, if for some fixed γ > 0, and for each real , limx→∞ {1 – F(x + y)}/{1 – F(x)} ═ e– n. Sufficient conditions are given for the statement F ∈ F * G ∈ and when both F and G are in y it is proved that F*G∈pF + 1(1 – p) G ∈ for some (all) p ∈(0,1). The related classes ℒt are proved closed under convolutions, which implies the closure of the class of positive random variables with regularly varying tails under multiplication (of random variables). An example is given that shows to be a proper subclass of ℒ 0.

Distribution tails F(t) = F(t, ∞) are considered for which and as t → ∞. A real analytic proof is obtained of a theorem by Chover, Wainger and Ney, namely that .

In doing so, a technique is introduced which provides many other results with a minimum of analysis. One such result strengthens and generalizes the various known results on distribution tails of random sums.

Additionally, the closure and factorization properties for subexponential distributions are investigated further and extended to distributions with exponential tails.

Let X n (Λ) be the number of nonoverlapping occurrences of a simple pattern Λ in a sequence of independent and identically distributed (i.i.d.) multistate trials. For fixed k, the exact tail probability P{Xn (∧) < k} is difficult to compute and tends to 0 exponentially as n → ∞. In this paper we use the finite Markov chain imbedding technique and standard matrix theory results to obtain an approximation for this tail probability. The result is extended to compound patterns, Markov-dependent multistate trials, and overlapping occurrences of Λ. Numerical comparisons with Poisson and normal approximations are provided. Results indicate that the proposed approximations perform very well and do significantly better than the Poisson and normal approximations in many cases.

This paper is concerned with a nonstationary Markovian chain of cascading damage that constitutes an iterated version of a classical damage model. The main problem under study is to determine the exact distribution of the total outcome of this process when the cascade of damages finally stops. Two different applications are discussed, namely the final size for a wide class of SIR (susceptible → infective → removed) epidemic models and the total number of failures for a system of components in reliability. The starting point of our analysis is the recent work of Lefèvre (2007) on a first-crossing problem for the cumulated partial sums of independent parametric distributions, possibly nonstationary but stable by convolution. A key mathematical tool is provided by a nonstandard family of remarkable polynomials, called the generalised Abel–Gontcharoff polynomials. Somewhat surprisingly, the approach followed will allow us to relax some model assumptions usually made in epidemic theory and reliability. To close, approximation by a branching process is also investigated to a certain extent.

We study the convolution of compound negative binomial distributions with arbitrary parameters. The exact expression and also a random parameter representation are obtained. These results generalize some recent results in the literature. An application of these results to insurance mathematics is discussed. The sums of certain dependent compound Poisson variables are also studied. Using the connection between negative binomial and gamma distributions, we obtain a simple random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters. Applications to the reliability of m-out-of-n:G systems and to the shortest path problem in graph theory are also discussed.

Let X 1, X 2,… and Y 1, Y 2,… be two sequences of absolutely continuous, independent and identically distributed (i.i.d.) random variables with equal means E(X i)=E(Y i), i=1,2,… In this work we provide upper bounds for the total variation and Kolmogorov distances between the distributions of the partial sums ∑i=1 n X i and ∑i=1 n Y i. In the case where the distributions of the X is and the Y is are compared with respect to the convex order, the proposed upper bounds are further refined. Finally, in order to illustrate the applicability of the results presented, we consider specific examples concerning gamma and normal approximations.