Polynomial birth–death processes and the second conjecture of Valent

The conjecture of Valent about the type of Jacobi matrices with polynomially growing weights is proved.     

On the cycle maximum of birth–death processes and networks of queues

This paper considers the cycle maximum in birth–death processes as a stepping stone to characterisation of the asymptotic behaviour of the maximum number of customers in single queues and open Kelly–Whittle networks of queues. For positive recurrent birth–death processes we show that the sequence of sample maxima is stochastically compact. For transient birth–death processes we show that the sequence of sample maxima conditioned on the maximum being finite is stochastically compact.We show that the Markov chain recording the total number of customers in a Kelly–Whittle network is a birth–death process with birth and death rates determined by the normalising constants in a suitably defined sequence of closed networks. Explicit or asymptotic expressions for these normalising constants allow asymptotic evaluation of the birth and death rates, which, in turn, allows characterisation of the cycle maximum in a single busy cycle, and convergence of the sequence of sample maxima for Kelly–Whittle networks of queues.     

Well-posedness of Hamilton–Jacobi equations in population dynamics and applications to large deviations

We prove Freidlin–Wentzell type large deviation principles for various rescaled models in populations dynamics that have immigration and possibly harvesting: birth–death processes, Galton–Watson trees, epidemic SI models, and prey–predator models.The proofs are carried out using a general analytic approach based on the well-posedness of a class of associated Hamilton–Jacobi equations. The notable feature for these Hamilton–Jacobi equations is that the Hamiltonian can be discontinuous at the boundary. We prove a well-posedness result for a large class of Hamilton–Jacobi equations corresponding to one-dimensional models, and give partial results for the multi-dimensional setting.     

Berry–Esseen bounds and Cramér-type large deviations for the volume distribution of Poisson cylinder processes

A stationary Poisson cylinder process Π _cyl ^(d,k) is composed of a stationary Poisson process of k-flats in ℝ ^d that are dilated by i.i.d. random compact cylinder bases taken from the corresponding orthogonal complement. We study the accuracy of normal approximation of the d-volume V _ϱ ^(d,k) of the union set of Π _cyl ^(d,k) that covers ϱW as the scaling factor ϱ becomes large. Here W is some fixed compact star-shaped set containing the origin as an inner point. We give lower and upper bounds of the variance of V _ϱ ^(d,k) that exhibit long-range dependence within the union set of cylinders. Our main results are sharp estimates of the higher-order cumulants of V _ϱ ^(d,k) under the assumption that the (d − k)-volume of the typical cylinder base possesses a finite exponential moment. These estimates enable us to apply the celebrated “Lemma on large deviations” of Statulevičius.     

Sharp asymptotics of large deviations in ℝ d

Given a sequence {X₁}i=1,2,3,... of i.i.d. random variables taking values in ? ^d ,d≥2, letS _n =Σ _i=1 ⁿ X _t=1. For Λ a Borel set in ? ^d having smooth boundary, witha=inf_x∈ΛI(x) the minimal value of the large deviation rate functionI(x) over Λ, we find, under suitable hypotheses, asymptotic results asn→∞, of the form $$P(S_n \in n\Gamma ) = n^\gamma e^{ - na} (d_0 + o(1))$$ where the constant γ depends sensitively on the geometry of Λ and the dimensiond, and takes values ?∞<γ≤(d?2/2). For fixeda=inf_x∈ΛI(x), we construct examples having any specific γ in this range.     

On approximation of functions in terms of Φ-variation

Получен аналог теоре мы Джексона о приближ ении полиномами в метрике Ф-вариации; для некото рых классов функций (в ыпуклых, конечной вариации, с заданным модулем неп рерывности) установл ено соотношение типа сла бой эквивалентности, которое связывает ме жду собой рациональн ые аппроксимации, в этой и равномерной метриках Даны оценкиε-энтропи и в метрике Ф-вариации классов функций с заданным мо дулем Ф-абсолютной непреры вности и с заданным мо дулем непрерывности.     

Conditions for the existence of quasi-stationary distributions for birth–death processes with killing

We consider birth–death processes on the nonnegative integers, where {1,2,…}$\{1,2,\dots \}$ is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 (killing) may occur from any state. Assuming that absorption at 0 is certain we are interested in additional conditions on the transition rates for the existence of a quasi-stationary distribution. Inspired by results of Kolb and Steinsaltz [M. Kolb, D. Steinsaltz, Quasilimiting behavior for one-dimensional diffusions with killing, Ann. Probab. 40 (2012) 162–212] we show that a quasi-stationary distribution exists if the decay rate of the process is positive and exceeds at most finitely many killing rates. If the decay rate is positive and smaller than at most finitely many killing rates then a quasi-stationary distribution exists if and only if the process one obtains by setting all killing rates equal to zero is recurrent.     

Poisson convergence,in large deviations,for the superposition of independent point processes

For a small buffer queueing system fed by many flows of a large class of traffic processes we show the single server queue and associated sample paths behave as if fed by marked Poisson traffic in a large deviations limit. The timescale of events of interest tends to zero, so we study the log moment generating function as time tends to zero. The associated rate function depends only on the mean arrival rate and the moment generating function of the arrivals. These results are useful in estimating drop probabilities while studying the effect of small buffers on communication protocols. Research supported by EPSRC Grant GR/S86266/01.     

Existence and approximation of fixed points of multivalued generalized α-nonexpansive mappings in Banach spaces

Numerical Algorithms - We consider the theoretical and numerical aspects of the quadrature rules associated with a sequence of polynomials generated by a special RII recurrence relation. We also...     

Irreducibility of dynamics and representation of KMS states in terms of Lévy processes

Quantum systems described by the Schr?dinger operators with Φ being continuous functions such that the pseudo-differential operators Φ(p_j) generate Lévy processes, are considered. It is proven that the linear span of the operators is dense in the algebra of all observables in the σ-strong and hence in the σ-weak and strong topologies. Here are time automorphisms and the F’s are taken from families of multiplication operators obeying conditions described in the paper. This result implies that a linear functional continuous in either of these topologies is fully determined by its values on such products. In the case of KMS states this yields a representation of such states in terms of path integrals. Received: 22 December 2004     

First-passage times and related moments for continuous-time birth–death chains

First-passage time problems for continuous-time birth–death chains are considered. Recursive formulas for the moments of the first-exit time and of the first-passage time in terms of the potential coefficients are explicitly obtained. Making use of the probability current, some functional relations between transition probabilities for unrestricted and restricted continuous-time birth–death chains are determined. Finally, two continuous-time birth–death chains with constant rates are taken in account; for them, closed form results on the first-exit time and on the first-passage time are explicitly obtained.

     

Comparison of α-Condorcet points with median and center locations

The usual concept of solution in single voting location is the Condorcet point. A Condorcet solution is the location such that no other location is preferred by a strict majority of voters; i.e. a half of them. It is assumed that each user always prefers closer locations. Because a Condorcet point does not necessarily exist, the α-Condorcet point is defined in the same way but assuming that two locations are indifferent for a user if the distances to both differ at most in α. We give bounds for the value of the objective function in an α-Condorcet point in the median and center problems. These results, for a general graph and for a tree, extend previous bounds for the objective function in a Condorcet point. We also provide a set of instances where these bounds are asymptotically reached. This research has been partially supported by DGICYT through project PB95-1237-C03-02 and by Gobierno de Canarias through the projects CO-1/97 and PI1999/116.     

A bivariate non-homogeneous birth and death model for predator–prey interactions

We propose a bivariate non-homogeneous birth and death process as a model for predator–prey interactions. Its expectation is periodic, as it is a solution to the classical Lotka–Volterra system. Moreover, the mean age at extinction, as defined in Kendall (1948), is infinite.     

Periodic homogenization of strongly nonlinear reaction–diffusion equations with large reaction terms

We study in this article the periodic homogenization problem related to a strongly nonlinear reaction–diffusion equation. Owing to the large reaction term, the homogenized equation has a rather quite different form which puts together both the reaction and convection effects. We show in a special case that the homogenized equation is exactly of a convection-diffusion type. This study relies on a suitable version of the well-known two-scale convergence method.     

On the distribution of points of maximal deviation in complex Čebyšev approximation

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On the distribution of points of maximal deviation in complex ebyev approximation