Anthony G. Pakes

Asymptotics of random contractions

In this paper we discuss the asymptotic behaviour of random contractions X=RS, where R, with distribution function F, is a positive random variable independent of S[set membership, variant](0,1). Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of X assuming that F is in the max-domain of attraction of an extreme value distribution and the distribution function of S satisfies a regular variation property. We apply our result to derive the asymptotics of the probability of ruin for a particular discrete-time risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.

Limit laws for the number of near maxima via the Poisson approximation

Given a sequence of i.i.d. random variables, new proofs are given for limit theorems for the number of observations near the maximum up to time n, as n --> [infinity]. The proofs rely on a Poisson approximation to conditioned binomial laws, and they reveal the origin in the limit laws of mixing with respect to extreme value laws. For the case of attraction to the Frechet law, the effects of relaxing a technical condition are examined. The results are set in the broader context of counting observations near upper order statistics. This involves little extra effort.

CRITERIA FOR THE UNIQUE DETERMINATION OF PROBABILITY DISTRIBUTIONS BY MOMENTS

Summary A positive probability law has a density function of the general form Q(x) exp(−x 1/λ L(x)), where Q is subject to growth restrictions, and L is slowly varying at infinity. This law is determined by its moment sequence when λ 2. It is still determined when λ = 2 and L(x) does not tend to zero too quickly. This paper explores the consequences for the induced power and doubled laws, and for mixtures. The proofs couple the Carleman and Krein criteria with elementary comparison arguments.

Limit theorems for the population size of a birth and death process allowing catastrophes

The linear birth and death process with catastrophes is formulated as a right continuous random walk on the non-negative integers which evolves in continuous time with an instantaneous jump rate proportional to the current value of the process. It is shown that distributions of the population size can be represented in terms of those of a certain Markov branching process. The ergodic theory of Markov branching process transition probabilities is then used to develop a fairly complete understanding of the behaviour of the population size of the birth-death-catastrophe process.

Characterization of discrete laws via mixed sums and Markov branching processes

Abstract Let (Zt) be a subordinator independent of 0 ≤ U ≤ 1 and let u and v be positive constants. Solutions to the “in law” equation Z u = d UZ u+v exist under certain conditions and they have a distribution function which is continuous on the positive reals. A discrete version of this equation is here formulated in which ordinary multiplication is replaced by a lattice-preserving operation whose definition involves a subcritical Markov branching process. It is shown that the existence, uniqueness and representation theory for the continuous problem transfers to the discrete problem. Specific examples are exhibited, and extension to two-sided discrete laws is explored.

Tail asymptotics under beta random scaling

Abstract Let X , Y , B be three independent random variables such that X has the same distribution function as YB. Assume that B is a beta random variable with positive parameters α , β and Y has distribution function H with H ( 0 ) = 0 . In this paper we derive a recursive formula for calculation of H, if the distribution function H α , β of X is known. Furthermore, we investigate the relation between the tail asymptotic behaviour of X and Y, which is closely related to asymptotics of Weyl fractional-order integral operators. We present three applications of our asymptotic results concerning the extremes of two random samples with underlying distribution functions H and H α , β , respectively, and the conditional limiting distribution of bivariate elliptical distributions.

Quasi-stationary laws for Markov processes: Examples of an always proximate absorbing state

Under consideration is a continuous-time Markov process with non-negative integer state space and a single absorbing state 0. Let T be the hitting time of zero and suppose Pi(T t)=1 for all t>0. Most known cases satisfy (*). The Markov process has a quasi-stationary distribution iff Ei(eeT) 0. The published proof of this fact makes crucial use of (*). By means of examples it is shown that (*) can be violated in quite drastic ways without destroying the existence of a quasi-stationary distribution.

Remarks on converse Carleman and Krein criteria for the classical moment problem

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.

Mixture representations for symmetric generalized Linnik laws

This paper offers a simple proof based on random variable representations for a mixture representation of symmetric Linnik laws previously derived by purely analytic means. The new approach can be set in a much more general context which embraces the symmetric and the positive generalized Linnik laws.

Characterization by invariance under length-biasing and random scaling

Abstract A positive random variable X with law L(X) and finite moment of order r > 0 has an induced length-biased law of order r, denoted by L(Xr). Let V ⩾ 0 be independent of Xr. A characterization problem seeks solution pairs (L(X), L(V)) for the “in-law” equation X ≅ VXr, where ≅ denotes equality in law. A renewal process interpretation asks when is the random rescaling of the stationary total lifetime VX1 equal in law to a typical lifetime X? Solutions are known in special cases. A comprehensive existence/uniqueness theory is presented, and many consequences are explored. Unique solutions occur when − log X and − log V have spectrally positive infinitely divisible laws. Particular cases are explored. Connections with the stationary lifetime law of renewal theory also are investigated.