bluuebunny/crossref_metadata_2025 huggingface dataset:
Large Deviations for the Graph Distance in Supercritical Continuum Percolation
Authors: Yao Chang-Long , Chen Ge , Guo Tian-De
Published: July 2016

Denote the Palm measure of a homogeneous Poisson processHλwith two points 0 andxby P0,x. We prove that there exists a constant μ ≥ 1 such that P0,x(D(0,x) / μ||x||2∉ (1 − ε, 1 + ε) | 0,x∈C∞) exponentially decreases when ||x||2tends to ∞, whereD(0,x) is the graph distance between 0 andxin the infinite componentC∞of the random geometric graphG(Hλ; 1). We derive a large deviation inequality for an asymptotic shape result. Our results have applications in many fields and especially in wireless sensor networks.
Pricing the Zero-Coupon Bond and its Fair Premium Under a Structural Credit Risk Model with Jumps
Authors: Dong Yinghui , Wang Guojing , Wu Rong
Published: July 2016

In this paper we consider a structural form credit risk model with jumps. We investigate the credit spread, the price, and the fair premium of the zero-coupon bond for the proposed model. The price and the fair premium of the bond are associated with the Laplace transform of default time and the firm's expected present market value at default. We give sufficient conditions under which the Laplace transform and the expected present market value of a firm at default are twice continuously differentiable. We derive closed-form expressions for them when the jumps have a hyperexponential distribution. Using the closed-form expressions, we obtain numerical solutions for the default probability, the credit spread, and the fair premium of the bond.
On the Maximal Offspring in a Critical Branching Process with Infinite Variance
Authors: Bertoin Jean
Published: July 2016

We investigate the maximal number M

k
of offspring amongst all individuals in a critical Galton-Watson process started with k ancestors. We show that when the reproduction law has a regularly varying tail with index -α for 1 < α < 2, then k
-1
M

k
converges in distribution to a Frechet law with shape parameter 1 and scale parameter depending only on α.
Heavy-Traffic Limits for Nearly Deterministic Queues
Authors: Sigman Karl , Whitt Ward
Published: July 2016

We establish heavy-traffic limits for nearly deterministic queues, such as the G/D/n many-server queue. Since waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated G
n
/D/1 model, where the G
n
interarrival times are the sum of n consecutive interarrival times in the original model, we focus on the G
n
/D/1 model and the generalization to G
n
/G
n
/1, where ‘cyclic thinning’ is applied to both the arrival and service processes. We establish different limits in two cases: (i) when (1 − ρ
n
)√n → β as n → ∞ and (ii) when (1 − ρ
n
)n → β as n → ∞, where ρ
n
is the traffic intensity in model n. The nearly deterministic feature leads to interesting nonstandard scaling.
Some Results for Vacation Systems with Sojourn Time Limits
Authors: Katayama Tsuyoshi
Published: July 2016

In this paper we deal with an M/G/1 vacation system with the sojourn time (wait plus service) limit and two typical vacation rules, i.e. multiple and single vacation rules. Using the level crossing approach, explicit expressions for the steady-state distributions of the virtual waiting times are obtained in vacation systems with exponential and constant service times, a general vacation time, and two vacation rules.
Efficient Estimation of One-Dimensional Diffusion First Passage Time Densities via Monte Carlo Simulation
Authors: Ichiba Tomoyuki , Kardaras Constantinos
Published: July 2016

We propose a method for estimating first passage time densities of one-dimensional diffusions via Monte Carlo simulation. Our approach involves a representation of the first passage time density as the expectation of a functional of the three-dimensional Brownian bridge. As the latter process can be simulated exactly, our method leads to almost unbiased estimators. Furthermore, since the density is estimated directly, a convergence of order 1 / √N, where N is the sample size, is achieved, which is in sharp contrast to the slower nonparametric rates achieved by kernel smoothing of cumulative distribution functions.
Ruin Probability with Parisian Delay for a Spectrally Negative Lévy Risk Process
Authors: Czarna Irmina , Palmowski Zbigniew
Published: July 2016

In this paper we analyze the so-called Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time ζ > 0. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we derive an expression for the ruin probability in terms of quantities that can be calculated explicitly in many models. We find its Cramér-type and convolution-equivalent asymptotics when reserves tend to ∞. Finally, we analyze some explicit examples.
On the Monitoring Error of the Supremum of a Normal Jump Diffusion Process
Authors: Chen Ao , Feng Liming , Song Renming
Published: July 2016

We derive an expansion for the (expected) difference between the continuously monitored supremum and evenly monitored discrete maximum over a finite time horizon of a jump diffusion process with independent and identically distributed normal jump sizes. The monitoring error is of the forma0/N1/2+a1/N3/2+ · · · +b1/N+b2/N2+b4/N4+ · · ·, whereNis the number of monitoring intervals. We obtain explicit expressions for the coefficients {a0,a1, …,b1,b2, …}. In particular,a0is proportional to the value of the Riemann zeta function at ½, a well-known fact that has been observed for Brownian motion in applied probability and mathematical finance.
On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications
Authors: Cai Jun , Tang Qihe
Published: July 2016

In this paper, we discuss max-sum equivalence and convolution closure of heavy-tailed distributions. We generalize the well-known max-sum equivalence and convolution closure in the class of regular variation to two larger classes of heavy-tailed distributions. As applications of these results, we study asymptotic behaviour of the tails of compound geometric convolutions, the ruin probability in the compound Poisson risk process perturbed by an α-stable Lévy motion, and the equilibrium waiting-time distribution of the M/G/k queue.
Large deviations for super-Brownian motion with immigration
Authors: Zhang Mei
Published: July 2016

We derive a large-deviation principle for super-Brownian motion with immigration, where the immigration is governed by the Lebesgue measure. We show that the speed function ist1/2ford= 1,t/logtford= 2 andtford≥ 3, which is different from that of the occupation-time process counterpart (without immigration) and the model of random immigration.
Generalized integrated telegraph processes and the distribution of related stopping times
Authors: Zacks S.
Published: July 2016

Let {X(t),V(t),t≥ 0} be a telegraph process, withV(0+) = 1. The distribution ofX(t) is derived for the general case of an alternating renewal process, describing the length of time a particle is moving to the right or to the left. The distributions of the first-crossing times of given levelsaand −aare studied for M/G and for G/M processes.
Convergence Properties of Perturbed Markov Chains
Authors: Roberts Gareth O. , Rosenthal Jeffrey S. , Schwartz Peter O.
Published: July 2016

In this paper, we consider the question of which convergence properties of Markov chains are preserved under small perturbations. Properties considered include geometric ergodicity and rates of convergence. Perturbations considered include roundoff error from computer simulation. We are motivated primarily by interest in Markov chain Monte Carlo algorithms.
Asymptotic Expansions for Array Branching Processes with Applications to Bootstrapping
Authors: Sriram T. N.
Published: July 2016

Asymptotic expansions are obtained for the distribution function of a studentized estimator of the offspring mean sequence in an array branching process with immigration. The expansion result is shown to hold in a test function topology. As an application of this result, it is shown that the bootstrapping distribution of the estimator of the offspring mean in a sub-critical branching process with immigration also admits the same expansion (in probability). From these considerations, it is concluded that the bootstrapping distribution provides a better approximation asymptotically than the normal distribution.
Multivariate Convex Orderings, Dependence, and Stochastic Equality
Authors: Scarsini Marco
Published: July 2016

We consider the convex ordering for random vectors and some weaker versions of it, like the convex ordering for linear combinations of random variables. First we establish conditions of stochastic equality for random vectors that are ordered by one of the convex orderings. Then we establish necessary and sufficient conditions for the convex ordering to hold in the case of multivariate normal distributions and sufficient conditions for the positive linear convex ordering (without the restriction to multi-normality).
Optimal Stopping of a Risk Reserve Process with Interest and Cost Rates
Authors: Schöttl A.
Published: July 2016

The risk reserve process of an insurance company within a deteriorating Markov-modulated environment is considered. The company invests its capital with interest rate α; the premiums and claims are increasing with rates β and γ. The problem of stopping the process at a random time which maximizes the expected net gain in order to calculate new premiums is investigated. A semimartingale representation of the risk reserve process yields, under certain conditions, an explicit solution of the problem.
Subexponential asymptotics of a Markov-modulated random walk with queueing applications
Authors: Jelenković Predrag R. , Lazar Aurel A.
Published: July 2016

Let {(X

n
,J

n
)} be a stationary Markov-modulated random walk on ℝ x E (E is finite), defined by its probability transition matrix measure F = {F

ij
}, F

ij
(B) = ℙ[X
1 ∈ B, J
1 = j | J
0 = i], B ∈
B
(ℝ), i, j ∈ E. If F

ij
([x,∞))/(1-H(x)) → W

ij
∈ [0,∞), as x → ∞, for some long-tailed distribution function H, then the ascending ladder heights matrix distribution G
+(x) (right Wiener-Hopf factor) has long-tailed asymptotics. If 𝔼X

n
< 0, at least one W

ij
> 0, and H(x) is a subexponential distribution function, then the asymptotic behavior of the supremum of this random walk is the same as in the i.i.d. case, and it is given by ℙ[sup
n≥0
S

n
> x] → (−𝔼X

n
)−1 ∫
x

∞ ℙ[X

n
> u]du as x → ∞, where S

n
= ∑1

n

X

k
, S
0 = 0. Two general queueing applications of this result are given.

First, if the same asymptotic conditions are imposed on a Markov-modulated G/G/1 queue, then the waiting time distribution has the same asymptotics as the waiting time distribution of a GI/GI/1 queue, i.e., it is given by the integrated tail of the service time distribution function divided by the negative drift of the queue increment process. Second, the autocorrelation function of a class of processes constructed by embedding a Markov chain into a subexponential renewal process, has a subexponential tail. When a fluid flow queue is fed by these processes, the queue-length distribution is asymptotically proportional to its autocorrelation function.
On closed support T-Invariants and the traffic equations
Authors: Boucherie Richard J. , Sereno Matteo
Published: July 2016

The traffic equations are the basis for the exact analysis of product form queueing networks, and the approximate analysis of non-product form queueing networks. Conditions characterising the structure of the network that guarantees the existence of a solution for the traffic equations are therefore of great importance. This note shows that the new condition stating that each transition is covered by a minimal closed support T-invariant, is necessary and sufficient for the existence of a solution for the traffic equations for batch routing queueing networks and stochastic Petri nets.
Stochastic analysis of partitioning algorithms for matching problems
Authors: Rüschendorf Ludger , Sachs Gernot
Published: July 2016

Partitioning algorithms for the Euclidean matching and for the semi-matching problem in the plane are introduced and analysed. The algorithms are analogues of Karp's well-known dissection algorithm for the travelling salesman problem. The algorithms are proved to run in time nlogn and to approximate the optimal matching in the probabilistic sense. The analysis is based on the techniques developed in Karp (1977) and on the limit theorem of Redmond and Yukich (1993) for quasiadditive functionals.
Evaluating policies for generalized bandits via a notion of duality
Authors: Crosbie J. H. , Glazebrook K. D.
Published: July 2016

Nash's generalization of Gittins’ classic index result to so-called generalized bandit problems (GBPs) in which returns are dependent on the states of all arms (not only the one which is pulled) has proved important for applications. The index theory for special cases of this model in which all indices are positive is straightforward. However, this is not a natural restriction in practice. An earlier proposal for the general case did not yield satisfactory index-based suboptimality bounds for policies — a central feature of classical Gittins index theory. We develop such bounds via a notion of duality for GBPs which is of independent interest. The index which emerges naturally from this analysis is the reciprocal of the one proposed by Nash.
Hypothesis testing and Skorokhod stochastic integration
Authors: Privault Nicolas
Published: July 2016

We define a class of anticipative flows on Poisson space and compute its Radon-Nikodym derivative. This result is applied to statistical testing in an anticipative queueing problem.
Ornstein-Uhlenbeck type processes and branching processes with immigration
Authors: Li Zeng-Hu
Published: July 2016

It is shown that an Ornstein-Uhlenbeck type process associated with a spectrally
positive Lévy process can be obtained as the fluctuation limits of both discrete state and continuous state branching processes with immigration.
Explicit bounds for geometric convergence of Markov chains
Authors: Kolassa John E.
Published: July 2016

This paper presents bounds on convergence rates of Markov chains in terms of quantities calculable directly from chain transition operators. Bounds are constructed by creating a probability distribution that minorizes the transition kernel over some region, and by examining bounds on an expectation conditional on lying within and without this region. These are shown to be sharper in most cases than previous similar results. These bounds are applied to a Markov chain useful in frequentist conditional inference in canonical generalized linear models.
A best-choice problem with multiple selectors
Authors: Glickman Hagit
Published: July 2016

Consider a situation where a known number, n, of objects appear sequentially
in a random order. At each stage, the present object is presented to d ≥ 2 different selectors, who must jointly decide whether to select or reject it, irrevocably. Exactly one object must be chosen. The observation at stage j is a d-dimensional vector
R
(j) = (R
1(j),…, R

d
(j)), where R

i
(j) is the relative rank of the jth object, by the criterion of the ith selector. The decision whether to stop or not at time j is based on the d-dimensional random vectors
R
(1),…,
R
(j). The criteria according to which each selector ranks the objects can either be dependent or independent. Although the goal of each selector is to maximize the probability of choosing the best object from his/her point of view, all d selectors must cooperate and chose the same object. The objective studied here is the maximization of the minimum over the d individual probabilities of choosing the best object. We exhibit the structure of the optimal rule. For independent criteria we give a full description of the rule and show that the optimal value tends to d
-d/(d-1), as n → ∞. Furthermore, we show that as n → ∞, the liminf of the values under negatively associated criteria is bounded below by d
-d/(d-1).
Families of birth-death processes with similar time-dependent behaviour
Authors: Lenin R. B. , Parthasarathy P. R. , Scheinhardt W. R. W. , van Doorn E. A.
Published: July 2016

We consider birth-death processes taking values in but allow the death rate in state 0 to be positive, so that escape from is possible. Two such processes with transition functions are said to be similar if, for all there are constants c

ij
such that for all t ≥ 0. We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family. These issues are also resolved in the more general setting in which the two processes are called similar if there are constants c

ij
and ν such that for all t ≥ 0.
The xlogx condition for general branching processes
Authors: Olofsson Peter
Published: July 2016

The xlogx condition is a fundamental criterion for the rate of growth of a general branching process, being equivalent to non-degeneracy of the limiting random variable. In this paper we adopt the ideas from Lyons, Pemantle and Peres (1995) to present a new proof of this well-known theorem. The idea is to compare the ordinary branching measure on the space of population trees with another measure, the size-biased measure.
Un principe d'invariance pour une classe de marches p-corrélées sur ℤd
Authors: Bienvenüe Alexis
Published: July 2016

Let ζ be a Markov chain on a finite state spaceD,fa function fromDto ℝd, andSn= ∑k=1nf(ζk). We prove an invariance theorem forSand derive an explicit expression of the limit covariance matrix. We give its exact value forp-reinforced random walks on ℤ2withp= 1, 2, 3.
Wald's equations, first passage times and moments of ladder variables in Markov random walks
Authors: Der Fuh Cheng , Leung Lai Tze
Published: July 2016

Previous work in extending Wald's equations to Markov random walks involves finiteness of moment generating functions and uniform recurrence assumptions. By using a new approach, we can remove these assumptions. The results are applied to establish finiteness of moments of ladder variables and to derive asymptotic expansions for expected first passage times of Markov random walks. Wiener–Hopf factorizations for Markov random walks are also applied to analyse ladder variables and related first passage problems.
Reward processes for semi-Markov processes: asymptotic behaviour
Authors: Soltani A. Reza , Khorshidian K.
Published: July 2016

The asymptotic behaviour of the cumulative mean of a reward process 𝒵ρ, where the reward function ρ belongs to a rather large class of functions, is obtained. It is proved that E𝒵
ρ
(t) = C
0 + C
1
t + o(1), t → ∞, where C
0 and C
1 are fully specified. A section is devoted to the dual process of a semi-Markov process, and a formula is given for the mean of the first passage time from a state i to a state j of the dual process, in terms of the means of passage times of the original process.
On the stability of polling models with multiple servers
Authors: Down D.
Published: July 2016

The stability of polling models is examined using associated fluid limit models. Examples are presented which generalize existing results in the literature or provide new stability conditions while in both cases providing simple and intuitive proofs of stability. The analysis is performed for both general single server models and specific multiple server models.
Idle and busy periods in stable M/M/k queues
Authors: Daley D. J. , Servi L. D.
Published: July 2016

This paper finds the first and second moments of the number of arrivals in a stable M/M/k queue during an idle period, i.e. in a period when at least one server is idle. These and similar results are used along with renewal theory asymptotics to find the first and second moments of the proportion of arrivals occurring while there are at least ξ customers waiting for service. The asymptotics of the second moment for large ξ are established.
Modelling random linear nucleation and growth by a Markov chain
Authors: Quine M. P. , Law J. S.
Published: July 2016

In an attempt to investigate the adequacy of the normal approximation for the number of nuclei in certain growth/coverage models, we consider a Markov chain which has properties in common with related continuous-time Markov processes (as well as being of interest in its own right). We establish that the rate of convergence to normality for the number of ‘drops’ during times 1,2,…n is of the optimal ‘Berry–Esséen’ form, as n → ∞. We also establish a law of the iterated logarithm and a functional central limit theorem.
Stationarity of a stochastic population flow model
Authors: Stadje Wolfgang
Published: July 2016

We consider a classical population flow model in which individuals pass through n strata with certain state-dependent probabilities and at every time t = 0,1,2,…, there is a stochastic inflow of new individuals to every stratum. For a stationary inflow process we prove the convergence of the joint distribution of group sizes and derive the limiting Laplace transform.
The Brownian excursion multi-dimensional local time density
Authors: Gittenberger Bernhard , Louchard Guy
Published: July 2016

Expressions for the multi-dimensional densities of Brownian excursion local time are derived by two different methods: a direct method based on Kac's formula for Brownian functionals and an indirect one based on a limit theorem for Galton–Watson trees.
Prediction of shot noise
Authors: Lund Robert B. , Butler Ronald W. , Paige Robert L.
Published: July 2016

Prediction of future values of a shot noise process observed on a discrete lattice of points is considered. The shot magnitudes are assumed to be independent and identically distributed and to arrive via a Poisson process; the effect of each shot dissipates and/or accumulates according to a known shot function. Conditional mean and linear point predictors of future process values are developed. Distributional prediction, obtained through saddlepoint approximation of the conditional distributions of the process, is also explored.
Loss networks and Markov random fields
Authors: Zachary Stan , Ziedins Ilze
Published: July 2016

This paper examines the connection between loss networks without controls and Markov random field theory. The approach taken yields insight into the structure and computation of network equilibrium distributions, and into the nature of spatial dependence in networks. In addition, it provides further insight into some commonly used approximations, enables the development of more refined approximations, and permits the derivation of some asymptotically exact results.
Mass transportation problems with capacity constraints
Authors: Rachev S. T. , Olkin I.
Published: July 2016

We exhibit solutions of Monge–Kantorovich mass transportation problems with constraints on the support of the feasible transportation plans and additional capacity restrictions. The Hoeffding–Fréchet inequalities are extended for bivariate distribution functions having fixed marginal distributions and satisfying additional constraints. Sharp bounds for different probabilistic functionals (e.g.Lp-distances, covariances, etc.) are given when the family of joint distribution functions has prescribed marginal distributions, satisfies restrictions on the support, and is bounded from above, or below, by other distributions.
Optimal control of a truncated general immigration process through total catastrophes
Authors: Kyriakidis E. G.
Published: July 2016

A Markov decision model is considered for the control of a truncated general immigration process, which represents a pest population, by the introduction of total catastrophes. The optimality criterion is that of minimizing the expected long-run average cost per unit time. Firstly, a necessary and sufficient condition is found under which the policy of never controlling is optimal. If this condition fails, a parametric analysis, in which a fictitious parameter is varied over the entire real line, is used to establish the optimality of a control-limit policy. Furthermore, an efficient Markov decision algorithm operating on the class of control-limit policies is developed for the computation of the optimal policy.
Recurrent ring dynamics in two-dimensional excitable cellular automata
Authors: Gravner Janko
Published: July 2016

The Greenberg–Hastings model (GHM) is a simple cellular automaton which emulates two properties of excitable media: excitation by contact and a refractory period. We study two ways in which external stimulation can make ring dynamics in the GHM recurrent. The first scheme involves the initial placement of excitation centres which gradually lose strength, i.e. each time they become inactive (and then stay inactive forever) with probability 1 − p
f. In this case, the density of excited sites must go to 0; however, their long–term connectivity structure undergoes a phase transition as p
f increases from 0 to 1. The second proposed rule utilizes continuous nucleation in that new rings are started at every rested site with probability p

s
. We show that, for small p

s
, these dynamics make a site excited about every p

s

−1/3 time units. This result yields some information about the asymptotic shape of a closely related random growth model.
On some equalities of laws for Brownian motion with drift
Authors: Imhof J. P.
Published: July 2016

A trivariate equality of laws given by Doney and Yor is extended to a quadrivariate version. Some related explicit density calculations are carried out. It is shown that a bivariate case which has been used to establish the Dassios–Port–Wendel equality of laws is in fact equivalent to it.
Excursions of birth and death processes, orthogonal polynomials, and continued fractions
Authors: Guillemin Fabrice , Pinchon Didier
Published: July 2016

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.
Epidemic processes on digraphs of random mappings
Authors: Jaworski Jerzy
Published: July 2016

A random mapping (T

n
;q) of a finite set V, V = {1,2,…,n}, into itself assigns independently to each i ∊ V its unique image j ∊ V with probability q if i = j and with probability P = (1-q)/(n−1) if i ≠ j. Three versions of epidemic processes on a random digraph G

T
representing (T

n
;q) are studied. The exact probability distributions of the total number of infected elements as well as the threshold functions for these epidemic processes are determined.
Outcomes of epidemic models with general infection and removal rate functions at certain stopping times
Authors: Clancy Damian
Published: July 2016

This paper aims to show how certain known martingales for epidemic models may be derived using general techniques from the theory of stochastic integration, and hence to extend the allowable infection and removal rate functions of the model as far as possible. Denoting by x, y the numbers of susceptible and infective individuals in the population, then we assume that new infections occur at rate β
xy

xy and infectives are removed at rate γ
xy

y, where the ratio β
xy
/ γ
xy
can be written in the form q(x+y) / xp(x) for appropriate functions p,q. Under this condition, we find equations giving the distribution of the number of susceptibles remaining in the population at appropriately defined stopping times. Using results on Abel–Gontcharoff pseudopolynomials we also derive an expression for the expectation of any function of the number of susceptibles at these times, as well as considering certain integrals over the course of the epidemic. Finally, some simple examples are given to illustrate our results.
Search schemes for random optimization algorithms that preserve the asymptotic distribution
Authors: Dorea Chang C. Y. , Gonçalves Cátia R.
Published: July 2016

Markovian algorithms for estimating the global maximum or minimum of real valued functions defined on some domain Ω ⊂ ℝ
d
are presented. Conditions on the search schemes that preserve the asymptotic distribution are derived. Global and local search schemes satisfying these conditions are analysed and shown to yield sharper confidence intervals when compared to the i.i.d. case.
The random triangle model
Authors: Jonasson Johan
Published: July 2016

The random triangle model is a Markov random graph model which, for parameters p ∊ (0,1) and q ≥ 1 and a graph G = (V,E), assigns to a subset, η, of E, a probability which is proportional to p
|η|(1-p)|E|-|η|
q

t(η), where t(η) is the number of triangles in η. It is shown that this model has maximum entropy in the class of distributions with given edge and triangle probabilities.

Using an analogue of the correspondence between the Fortuin-Kesteleyn random cluster model and the Potts model, the asymptotic behavior of the random triangle model on the complete graph is examined for p of order n
−α, α > 0, and different values of q, where q is written in the form q = 1 + h(n) / n. It is shown that the model exhibits an explosive behavior in the sense that if h(n) ≤ c log n for c < 3α, then the edge probability and the triangle probability are asymptotically the same as for the ordinary G(n,p) model, whereas if h(n) ≥ c' log n for c' > 3α, then these quantities both tend to 1. For critical values, h(n) = 3α log n + o(log n), the probability mass divides between these two extremes.

Moreover, if h(n) is of higher order than log n, then the probability that η = E tends to 1, whereas if h(n) = o(log n) and α > 2/3, then, with a probability tending to 1, the resulting graph can be coupled with a graph resulting from the G(n,p) model. In particular these facts mean that for values of p in the range critical for the appearance of the giant component and the connectivity of the graph, the way in which triangles are rewarded can only have a degenerate influence.
Hitting time in an M/G/1 queue
Authors: Ross Sheldon M. , Seshadri Sridhar
Published: July 2016

We study the expected time for the work in an M/G/1 system to exceed the level x, given that it started out initially empty, and show that it can be expressed solely in terms of the Poisson arrival rate, the service time distribution and the stationary delay distribution of the M/G/1 system. We use this result to construct an efficient simulation procedure.
Asymptotics of pooling design performance
Authors: Percus J. K. , Percus O. E. , Bruno W. J. , Torney D. C.
Published: July 2016

We analyse the expected performance of various group testing, or pooling, designs. The context is that of identifying characterized clones in a large collection of clones. Here we choose as performance criterion the expected number of unresolved ‘negative’ clones, and we aim to minimize this quantity. Technically, long inclusion–exclusion summations are encountered which, aside from being computationally demanding, give little inkling of the qualitative effect of parametric control on the pooling strategy. We show that readily-interpreted re-summation can be performed, leading to asymptotic forms and systematic corrections. We apply our results to randomized designs, illustrating how they might be implemented for approximating combinatorial formulae.
Phase transition in the random triangle model
Authors: Häggström Olle , Jonasson Johan
Published: July 2016

The random triangle model was recently introduced as a random graph model that captures the property of transitivity that is often found in social networks, i.e. the property that given that two vertices are second neighbors, they are more likely to be neighbors. For parameters p ∊ [0,1] and q ≥ 1, and a finite graph G = (V, E), it assigns to elements η of {0,1}
E
probabilities which are proportional to

where t(η) is the number of triangles in the open subgraph. In this paper the behavior of the random triangle model on the two-dimensional triangular lattice is studied. By mapping the system onto an Ising model with external field on the hexagonal lattice, it is shown that phase transition occurs if and only if p = (q−1)−2/3 and q > q

c
for a critical value q

c
which turns out to equal It is furthermore demonstrated that phase transition cannot occur unless p = p

c
(q), the critical value for percolation of open edges for given q. This implies that for q ≥ q

c
, p

c
(q) = (q−1)−2/3.
Heavy traffic analysis of a queueing system with bounded capacity for two types of customers
Authors: Perry David , Stadje Wolfgang
Published: July 2016

We study a service system with a fixed upper bound for its workload and two independent inflows of customers: frequent ‘small’ ones and occasional ‘large’ ones. The workload process generated by the small customers is modelled by a Brownian motion with drift, while the arrival times of the large customers form a Poisson process and their service times are exponentially distributed. The workload process is reflected at zero and at its upper capacity bound. We derive the stationary distribution of the workload and several related quantities and compute various important characteristics of the system.
On the rates of convergence of Erlang's model
Authors: Fricker Christine , Robert Philippe , Tibi Danielle
Published: July 2016

The convergence to equilibrium of the renormalized M/M/N/N queue is analysed. Upper bounds on the distance to equilibrium are obtained and the cut-off property for two regimes of this queue is proved. Simple probabilistic methods, such as coupling techniques and martingales, are used to obtain these results.
Limiting conditional and conditional invariant distributions for the Poisson process with negative drift
Authors: Fierro Raúl , Martínez Servet , San Martín Jaime
Published: July 2016

In this paper we study the conditional limiting behaviour for the virtual waiting time process for the queueM/D/1. We describe the family of conditional invariant distributions which are continuous and parametrized by the eigenvalues λ ∊ (0, λc], as it happens for diffusions. In this case, there is a periodic dependence of the limiting conditional distributions on the initial point and the minimal conditional invariant distribution is a mixture, according to an exponential law, of the limiting conditional distributions.
A central limit theorem for super-Brownian motion with super-Brownian immigration
Authors: Hong Wen-Ming , Li Zeng-Hu
Published: July 2016

We prove a central limit theorem for the super-Brownian motion with immigration governed by another super-Brownian. The limit theorem leads to Gaussian random fields in dimensionsd≥ 3. Ford= 3 the field is spatially uniform; ford≥ 5 its covariance is given by the potential operator of the underlying Brownian motion; and ford= 4 it involves a mixture of the two kinds of fluctuations.
Extensions of the bifurcating autoregressive model for cell lineage studies
Authors: Huggins R. M. , Basawa I. V.
Published: July 2016

The bifurcating autoregressive model has been used previously to model cell lineage data. A feature of this model is that each line of descendants from an initial cell follows an AR(1) model, and that the environmental effects on sisters are correlated. However, this model concentrates on modelling the correlations between mother and daughter cells and between sister cells, and does not explain the large correlations between more distant relatives observed by some authors. Here the model is extended, firstly by allowing lines of descent to follow an ARMA(p,q) model rather than an AR(1) model, and secondly by allowing correlations between the environmental effects of relatives more distant than sisters. The models are applied to several data sets consisting of independent cell lineage trees.
Martingales versus PDEs in finance: an equivalence result with examples
Authors: Heath David , Schweizer Martin
Published: July 2016

We provide a set of verifiable sufficient conditions for proving in a number of practical examples the equivalence of the martingale and the PDE approaches to the valuation of derivatives. The key idea is to use a combination of analytic and probabilistic assumptions that covers typical models in finance falling outside the range of standard results from the literature. Applications include Heston's stochastic volatility model and the Black-Karasinski term structure model.
Preservation results for life distributions based on comparisons with asymptotic remaining life under replacements
Authors: Bhattacharjee M. C. , Abouammoh A. M. , Ahmed A. N. , Barry A. M.
Published: July 2016

We investigate some preservation properties of two nonparametric classes of survival distributions and their duals, under appropriate reliability operations. The aging properties defining these nonparametric classes are based on comparing the mean life of a new unit to the mean residual life function of the asymptotic remaining survival time of the unit under repeated perfect repairs. They are motivated from a point of view that realistic notions of degradation, applicable to repairable systems, should be based on contrasting some aspect of the remaining life of a repairable unit (under a given repair strategy, such as renewals) to the life of a new unit.
Some new results on stochastic comparisons of parallel systems
Authors: Khaledi Baha-Eldin , Kochar Subhash
Published: July 2016

Let X
1,…,X

n
be independent exponential random variables with X

i
having hazard rate . Let Y
1,…,Y

n
be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏
i=1

n
λ
i
)1/n
, the geometric mean of the λis. Let X

n:n
= max{X
1,…,X

n
}. It is shown that X

n:n
is greater than Y

n:n
according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of X

n:n
and an upper bound on the hazard rate function of X

n:n
in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference
65, 203–211), which are in terms of the arithmetic mean of the λ
i
s. Furthermore, let X
1
*,…,X

n

∗ be another set of independent exponential random variables with X

i

∗ having hazard rate λ
i

∗, i = 1,…,n. It is proved that if (logλ1,…,logλ
n
) weakly majorizes (logλ1
∗,…,logλ
n

∗, then X

n:n
is stochastically greater than X

n:n

∗.
One-dimensional branching random walks in a Markovian random environment
Authors: Machado F. P. , Popov S. Yu.
Published: July 2016

We study a one-dimensional supercritical branching random walk in a non-i.i.d. random environment, which considers both the branching mechanism and the step transition. This random environment is constructed using a recurrent Markov chain on a finite or countable state space. Criteria of (strong) recurrence and transience are presented for this model.
An asymptotic expansion for the optimal stopping boundary in problems with non-linear costs of observation
Authors: Irle A. , Kubillus O. , Paulsen V.
Published: July 2016

The assumption of linear costs of observation usually leads to optimal stopping boundaries which are straight lines. For non-linear costs of observation, the question arises of how the shape of cost functions influences the shape of optimal stopping boundaries. In Irle (1987), (1988) it was shown that, under suitable assumptions on c, for the problem of optimal stopping (W

t
+ x)+ - c(s + t), t ∊ [0,∞), the optimal stopping boundary h(t) can be enscribed between k
1/c'(t) and k
2/c'(t) for some constants k
1, k
2. In this paper we find the exact asymptotic expansion h(t) = 1/(4c'(t))(1 + o(1)).
Boundary crossing probability for Brownian motion
Authors: Pötzelberger Klaus , Wang Liqun
Published: July 2016

Wang and Pötzelberger (1997) derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary crossing probability for general nonlinear boundaries. The present paper gives a sharper asymptotic upper bound of the approximation error for the formula, and generalizes the results to two-sided boundaries. Numerical computations are easily carried out using the Monte Carlo simulation method. A rule is proposed for choosing optimal nodes for the approximating piecewise linear boundaries, so that the corresponding approximation errors of boundary crossing probabilities converge to zero at a rate of O(1/n
2).
Regenerative processes in the infinite mean cycle case
Authors: Mitov K. V. , Yanev N. M.
Published: July 2016

A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the waiting time and the lifetime belong to the domain of attraction of stable laws with parameters in the interval (½,1]. An integral representation for the distribution functions of the regenerative process is obtained, using the spent time distributions of the corresponding alternating renewal process. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new results for the corresponding renewal process and two limit theorems for the convergence in distribution.
Strong ergodicity for single-birth processes
Authors: Zhang Yu-Hui
Published: July 2016

An explicit and computable criterion for strong ergodicity of single-birth processes is presented. As an application, some sufficient conditions are obtained for strong ergodicity of an extended class of continuous-time branching processes and multi-dimensional Q-processes by comparison methods respectively. Consequently strong ergodicity of the Q-process corresponding to the finite-dimensional Schlögl model is proven.
Optimization of inspection intervals based on cost
Authors: Badía F. G. , Berrade M. D. , Campos Clemente A.
Published: July 2016

We present a new inspection policy useful when testing is needed to detect failures of a single-unit system. It is supposed that tests may fail and give an erroneous result. The inspection policy minimizing cost per unit of time for an infinite time span is also discussed. In addition, we study the behaviour of the optimum policy for some time to failure distributions often assumed in reliability: exponential and Pareto.
Energy of a string driven by a two-parameter Gaussian noise white in time
Authors: Belinskiy Boris P. , Caithamer Peter
Published: July 2016

In this paper we consider the stochastic wave equation in one spatial dimension driven by a two-parameter Gaussian noise which is white in time and has general spatial covariance. We give conditions on the spatial covariance of the driving noise sufficient for the string to have finite expected energy and calculate this energy as a function of time. We show that these same conditions on the spatial covariance of the driving noise are also sufficient to guarantee that the energy of the string has a version which is continuous almost surely.
Rigorous lower bounds for the critical infection rate in the diffusive contact process
Authors: Sudbury Aidan
Published: July 2016

The contact process is an interacting particle system which models a spatially restricted infection. In the basic contact process the infection can only spread to an uninfected neighbour, but the diffusive contact process allows an infected individual to move to an uninfected site. If the infection rate is too low, the process will die out. If the individual can move (or diffuse), the disease can spread with a lower infection rate. An idea of the relationship between these rates is obtained by obtaining rigorous lower bounds for the critical infection rate for various values of the diffusion rate. In this paper we also improve the lower bound for the critical infection rate for the basic contact process from 1.539 to 1.5517.
Poisson approximation of multivariate Poisson mixtures
Authors: Roos Bero
Published: July 2016

We show how good multivariate Poisson mixtures can be approximated by multivariate Poisson distributions and related finite signed measures. Upper bounds for the total variation distance with applications to risk theory and generalized negative multinomial distributions are given. Furthermore, it turns out that the ideas used in this paper also lead to improvements in the Poisson approximation of generalized multinomial distributions.
Average cost under the P
M
λ, τ policy in a finite dam with compound Poisson inputs
Authors: Bae Jongho , Kim Sunggon , Lee Eui Yong
Published: July 2016

We consider the policy in a finite dam in which the input of water is formed by a compound Poisson process and the rate of water release is changed instantaneously from a to M and from M to a (M > a) at the moments when the level of water exceeds λ and downcrosses τ (λ > τ) respectively. After assigning costs to the changes of release rate, a reward to each unit of output, and a cost related to the level of water in the reservoir, we determine the long-run average cost per unit time.
Carousel system performance
Authors: Park Byung Chun , Park Jae Young , Foley Robert D.
Published: July 2016

We analyse the performance of a material handling system consisting of two carousels and one picker. We derive expressions for the system throughput and picker utilization. We work out the throughput and picker utilization for two different pick-time distributions: deterministic, which might be a reasonable model when a robot is picking, and exponential, which might be a reasonable model when a person is picking.
Power-law correlations and other models with long-range dependence on a lattice
Authors: Ma Chunsheng
Published: July 2016

This paper introduces long-range dependence for a stationary random field on a plane lattice, derives an exact power-law correlation model and other models with long-range dependence on the lattice, and explores the close connection between short-range dependent correlation functions and absolutely summable double sequences.
Family trees of continuous-time birth-and-death processes
Authors: Müller Johannes , Möhle Martin
Published: July 2016

We consider a stochastic graph generated by a continuous-time birth-and-death process with exponentially distributed waiting times. The vertices are the living particles, directed edges go from mothers to daughters. The size and the structure of the connected components are investigated. Furthermore, the number of connected components is determined.
The classification of matrix GI/M/1-type Markov chains with a tree structure and its applications to queueing
Authors: He Qi-Ming
Published: July 2016

In this paper, we study the classification of matrix GI/M/1-type Markov chains with a tree structure. We show that the Perron–Frobenius eigenvalue of a Jacobian matrix provides information for classifying these Markov chains. A fixed-point approach is utilized. A queueing application is presented to show the usefulness of the classification method developed in this paper.