We give a functional central limit theorem for spatial birth and death processes based on the representation of such processes as solutions of stochastic equations. For any bounded and integrable function in Euclidean space, we define a family of processes which is obtained by integrals of this function with respect to the centered and scaled spatial birth and death process with constant death rate. We prove that this family converges weakly to a Gaussian process as the scale parameter goes to infinity. We do not need the birth rates to have a finite range of interaction. Instead, we require that the birth rates have a range of interaction that decays polynomially. In order to show the convergence of the finite-dimensional distributions of the above processes, we extend Penrose's multivariate spatial central limit theorem. An example of the asymptotic normalities of the time-invariance estimators for the birth rates of spatial point processes is given.

In this paper we consider a compound Poisson risk model where the insurer earns credit interest at a constant rate if the surplus is positive and pays out debit interest at another constant rate if the surplus is negative. Absolute ruin occurs at the moment when the surplus first drops below a critical value (a negative constant). We study the asymptotic properties of the absolute ruin probability of this model. First we investigate the asymptotic behavior of the absolute ruin probability when the claim size distribution is light tailed. Then we study the case where the common distribution of claim sizes are heavy tailed.

We study the convergence of at-the-money implied volatilities to the spot volatility in a general model with a Brownian component and a jump component of finite variation. This result is a consequence of the robustness of the Black-Scholes formula and of the central limit theorem for martingales.

The distributions of the run occurrences for a sequence of independent and identically distributed (i.i.d.) experiments are usually obtained by combinatorial methods (see Balakrishnan and Koutras (2002, Chapter 5)) and the resulting formulae are often very tedious, while the distributions for non i.i.d. experiments are generally intractable. It is therefore of practical interest to find a suitable approximate model with reasonable approximation accuracy. In this paper we demonstrate that the negative binomial distribution is the most suitable approximate model for the number of k-runs: it outperforms the Poisson approximation, the general compound Poisson approximation as observed in Eichelsbacher and Roos (1999), and the translated Poisson approximation in Rollin (2005). In particular, its accuracy of approximation in terms of the total variation distance improves when the number of experiments increases, in the same way as the normal approximation improves in the Berry-Esseen theorem.

The paper deals with the asymptotic behavior of the bridge of a Gaussian process conditioned to stay in n fixed points at n fixed past instants. In particular, functional large deviation results are stated for small time. Several examples are considered: integrated or not fractional Brownian motions and m-fold integrated Brownian motion. As an application, the asymptotic behavior of the exit probability is studied and used for the practical purpose of the numerical computation, via Monte Carlo methods, of the hitting probability up to a given time of the unpinned process.

We consider sequences of closed cycles of exponential single-server nodes with a single bottleneck. We study the cycle time and the successive sojourn times of a customer when the population sizes go to infinity. Starting from old results on the mean cycle times under heavy traffic conditions, we prove a central limit theorem for the cycle time distribution. This result is then utilised to prove a weak convergence characteristic of the vector of a customer's successive sojourn times during a cycle for a sequence of networks with population sizes going to infinity. The limiting picture is a composition of a central limit theorem for the bottleneck node and an exponential limit for the unscaled sequences of sojourn times for the nonbottleneck nodes.

The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.

For a linear random field (linear p-parameter stochastic process) generated by a dependent random field with zero mean and finite qth moments (q>2p), we give sufficient conditions that the linear random field converges weakly to a multiparameter standard Brownian motion if the corresponding dependent random field does so.

We describe the random meeting motion of a finite number of investors in markets with friction as a Markov pure-jump process with interactions. Using a sequence of these, we prove a functional law of large numbers relating the large motions with the finite market of the so-called continuum of agents.

Let S 0 := 0 and Sk := ξ 1 + ··· + ξk for k ∈ ℕ := {1, 2, …}, where {ξk : k ∈ ℕ} are independent copies of a random variable ξ with values in ℕ and distribution pk := P{ξ = k}, k ∈ ℕ. We interpret the random walk {Sk : k = 0, 1, 2, …} as a particle jumping to the right through integer positions. Fix n ∈ ℕ and modify the process by requiring that the particle is bumped back to its current state each time a jump would bring the particle to a state larger than or equal to n. This constraint defines an increasing Markov chain {Rk (n) : k = 0, 1, 2, …} which never reaches the state n. We call this process a random walk with barrier n. Let Mn denote the number of jumps of the random walk with barrier n. This paper focuses on the asymptotics of Mn as n tends to ∞. A key observation is that, under p 1 > 0, {Mn : n ∈ ℕ} satisfies the distributional recursion M 1 = 0 and for n = 2, 3, …, where In is independent of M 2, …, M n−1 with distribution P{In = k} = pk / (p 1 + ··· + p n−1), k ∈ {1, …, n − 1}. Depending on the tail behavior of the distribution of ξ, several scalings for Mn and corresponding limiting distributions come into play, including stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is to compare (couple) the number of jumps, Mn , with the first time, Nn , when the unrestricted random walk {Sk : k = 0, 1, …} reaches a state larger than or equal to n. The results are applied to derive the asymptotics of the number of collision events (that take place until there is just a single block) for β(a, b)-coalescent processes with parameters 0 < a < 2 and b = 1.

The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter α ∈ (0, 1]. We turn to the case of very heavy-tailed reproduction distribution α = 0 assuming that Zubkov's regularity condition holds with parameter β ∈ (0, ∞). Our main result gives a new asymptotic pattern for the reduced branching process conditioned on nonextinction during a long time interval.

Gabetta and Regazzini (2006b) have shown that finiteness of the initial energy (second moment) is necessary and sufficient for the solution of the Kac's model Boltzmann equation to converge weakly (C b -convergence) to a probability measure on R. Here, we complement this result by providing a detailed analysis of what does actually happen when the initial energy is infinite. In particular, we prove that such a solution converges vaguely (C 0-convergence) to the zero measure (which is identically 0 on the Borel sets of R). More precisely, we prove that the total mass of the limiting distribution splits into two equal masses (of value ½ each), and we provide quantitative estimates on the rate at which such a phenomenon takes place. The methods employed in the proofs also apply in the context of sums of weighted independent and identically distributed random variables x̃ 1, x̃ 2, …, where these random variables have an infinite second moment and zero mean. Then, with T n := ∑j=1 η n λj,n x̃ j , with max1 ≤ j ≤ η n λj,n → 0 (as n → +∞), and ∑j=1 η n λj,n 2 = 1, n = 1, 2, …, the classical central limit theorem suggests that T should in some sense converge to a ‘normal random variable of infinite variance’. Again, in this setting we prove quantitative estimates on the rate at which the mass splits into adherent masses to -∞ and +∞, or to ∞, that are analogous to those we have obtained for the Kac equation. Although the setting in this case is quite classical, we have not uncovered any previous results of a similar type.

In this paper we investigate the ‘local’ properties of a random mapping model, T n D̂, which maps the set {1, 2, …, n} into itself. The random mapping T n D̂ , which was introduced in a companion paper (Hansen and Jaworski (2008)), is constructed using a collection of exchangeable random variables D̂ 1, …, D̂ n which satisfy In the random digraph, G n D̂ , which represents the mapping T n D̂ , the in-degree sequence for the vertices is given by the variables D̂ 1, D̂ 2, …, D̂ n , and, in some sense, G n D̂ can be viewed as an analogue of the general independent degree models from random graph theory. By local properties we mean the distributions of random mapping characteristics related to a given vertex v of G n D̂ - for example, the numbers of predecessors and successors of v in G n D̂ . We show that the distribution of several variables associated with the local structure of G n D̂ can be expressed in terms of expectations of simple functions of D̂ 1, D̂ 2, …, D̂ n . We also consider two special examples of T n D̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine, for these examples, exact and asymptotic distributions for the local structure variables considered in this paper. These distributions are also of independent interest.

This article proves that the on-off renewal process with Weibull sojourn times satisfies the large deviation principle on a nonlinear scale. Unusually, its rate function is not convex. Apart from on a compact set, the rate function is infinite, which enables us to construct natural processes that satisfy the large deviation principle with nontrivial rate functions on more than one time scale.

We consider a sequential rule, where an item is chosen into the group, such as a university faculty member, only if his/her score is better than the average score of those already belonging to the group. We study four variables: the average score of the members of the group after k items have been selected, the time it takes (in terms of the number of observed items) to assemble a group of k items, the average score of the group after n items have been observed, and the number of items kept after the first n items have been observed. We develop the relationships between these variables, and obtain their asymptotic behavior as k (respectively, n) tends to ∞. The assumption throughout is that the items are independent and identically distributed with a continuous distribution. Though knowledge of this distribution is not needed to implement the selection rule, the asymptotic behavior does depend on the distribution. We study in some detail the exponential, Pareto, and beta distributions. Generalizations of the ‘better than average’ rule to the β better than average rules are also considered. These are rules where an item is admitted to the group only if its score is better than β times the present average of the group, where β > 0.

The copula of a multivariate distribution is the distribution transformed so that one-dimensional marginal distributions are uniform. We review a different transformation of a multivariate distribution which yields standard Pareto for the marginal distributions, and we call the resulting distribution the Pareto copula. Use of the Pareto copula has a certain claim to naturalness when considering asymptotic limit distributions for sums, maxima, and empirical processes. We discuss implications for aggregation of risk and offer some examples.

We make a correction to an important result by Cline [D. B. H. Cline, ‘Convolutions of distributions with exponential tails’, J. Austral. Math. Soc. (Series A)43 (1987), 347–365; D. B. H. Cline, ‘Convolutions of distributions with exponential tails: corrigendum’, J. Austral. Math. Soc. (Series A)48 (1990), 152–153] on the closure of the exponential class under convolution power mixtures (random summation).

Assume that there are k types of insurance contracts in an insurance company. The ith related claims are denoted by {X ij , j ≥ 1}, i = 1,…,k. In this paper we investigate large deviations for both partial sums S(k; n 1,…,n k ) = ∑i=1 k ∑j=1 n i X ij and random sums S(k; t) = ∑i=1 k ∑j=1 N i (t)X ij , where N i (t), i = 1,…,k, are counting processes for the claim number. The obtained results extend some related classical results.

Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite-dimensional diffusion processes on Δ∞ := {x ∈ [0, 1] N : ∑i≥1 x i = 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence of the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process, which does not satisfy the log-Sobolev inequality when S is infinite as observed by Stannat (2000).

We consider the tail behavior of the product of two independent nonnegative random variables X and Y. Breiman (1965) has considered this problem, assuming that X is regularly varying with index α and that E{Yα+ε} < ∞ for some ε > 0. We investigate when the condition on Y can be weakened and apply our findings to analyze a class of random difference equations.