bluuebunny/crossref_metadata_2025 huggingface dataset:
Tree Polymers in the Infinite Volume Limit at Critical Strong Disorder
Authors: Johnson Torrey , Waymire Edward C.
Published: July 2016

The almost-sure existence of a polymer probability in the infinite volume limit is readily obtained under general conditions of weak disorder from standard theory on multiplicative cascades or branching random walks. However, speculations in the case of strong disorder have been mixed. In this note existence of an infinite volume probability is established at critical strong disorder for which one has convergence in probability. Some calculations in support of a specific formula for the almost-sure asymptotic variance of the polymer path under strong disorder are also provided.
Summary Statistics for Endpoint-Conditioned Continuous-Time Markov Chains
Authors: Hobolth Asger , Jensen Jens Ledet
Published: July 2016

Continuous-time Markov chains are a widely used modelling tool. Applications include DNA sequence evolution, ion channel gating behaviour, and mathematical finance. We consider the problem of calculating properties of summary statistics (e.g. mean time spent in a state, mean number of jumps between two states, and the distribution of the total number of jumps) for discretely observed continuous-time Markov chains. Three alternative methods for calculating properties of summary statistics are described and the pros and cons of the methods are discussed. The methods are based on (i) an eigenvalue decomposition of the rate matrix, (ii) the uniformization method, and (iii) integrals of matrix exponentials. In particular, we develop a framework that allows for analyses of rather general summary statistics using the uniformization method.
Limiting distributions for minimum relative entropy calibration
Authors: Kruk Łukasz
Published: July 2016

We consider minimum relative entropy calibration of a given prior distribution to a finite set of moment constraints. We show that the calibration algorithm is stable (in the Prokhorov metric) under a perturbation of the prior and the calibrated distributions converge in variation to the measure from which the moments have been taken as more constraints are added. These facts are used to explain the limiting properties of the minimum relative entropy Monte Carlo calibration algorithm.
Asymptotic behavior of tail and local probabilities for sums of subexponential random variables
Authors: Ng Kai W. , Tang Qihe
Published: July 2016

Let {X

k
, k ≥ 1} be a sequence of independently and identically distributed random variables with common subexponential distribution function concentrated on (−∞, ∞), and let τ be a nonnegative and integer-valued random variable with a finite mean and which is independent of the sequence {X

k
, k ≥ 1}. This paper investigates asymptotic behavior of the tail probabilities P(· > x) and the local probabilities P(x < · ≤ x + h) of the quantities and for n ≥ 1, and their randomized versions X
(τ), S

τ
and S
(τ), where X
0 = 0 by convention and h > 0 is arbitrarily fixed.
On the distribution of the search cost for the move-to-front rule with random weights
Authors: Barrera Javiera , Paroissin Christian
Published: July 2016

Consider a countable list of files updated according to the move-to-front rule. Files have independent random weights, which are used to construct request probabilities. Exact and asymptotic formulae for the Laplace transform of the stationary search cost are given for i.i.d. weights. Similar expressions are derived for the first two moments. Some results are extended to the case of independent weights.
A note on the waiting time in M[X]/G/1 queueing systems with a removable server
Authors: Loris-Teghem Jacqueline
Published: July 2016

For the M[X]/G/1 queueing model with a general exhaustive-service vacation policy, it has been proved that the Laplace-Stieltjes transform (LST) of the steady-state distribution function of the waiting time of a customer arriving while the server is active is the product of the corresponding LST in the bulk arrival model with unremovable server and another LST. The expression given for the latter, however, is valid only under the assumption that the number of groups arriving in an inactive phase is independent of the sizes of the groups. We here give an expression which holds in the general case. For the N-policy case, we also give an expression for the LST of the steady-state distribution function of the waiting time of a customer arriving while the server is inactive.
The compound Poisson immigration process subject to binomial catastrophes
Authors: Economou Antonis
Published: July 2016

Recently, several authors have studied the transient and the equilibrium behaviour of stochastic population processes with total catastrophes. These models are reasonable for modelling populations that are exposed to extreme disastrous phenomena. However, under mild disastrous conditions, the appropriate model is a stochastic process subject to binomial catastrophes. In the present paper we consider a special such model in which a population evolves according to a compound Poisson process and catastrophes occur according to a renewal process. Every individual of the population survives after a catastrophe with probabilityp, independently of anything else, i.e. the population size is reduced according to a binomial distribution. We study the equilibrium distribution of this process and we derive an algorithmic procedure for its approximate computation. Bounds on the error of this approximation are also included.
A bold strategy is not always optimal in the presence of inflation
Authors: Chen Robert W. , Shepp Larry A. , Zame Alan
Published: July 2016

A gambler, with an initial fortune less than 1, wants to buy a house which sells today for 1. Due to inflation, the price of the house tomorrow will be 1 + α, where α is a nonnegative constant, and will continue to go up at this rate, becoming (1 + α)
n
on the nth day. Once each day, he can stake any amount of fortune in his possession, but no more than he possesses, on a primitive casino. It is well known that, in a subfair primitive casino without the presence of inflation, the gambler should play boldly. The presence of inflation would motivate the gambler to recognize the time value of his fortune and to try to reach his goal as quickly as possible; intuitively, we would conjecture that the gambler should again play boldly. However, in this note we will show that, unexpectedly, bold play is not necessarily optimal.
Note on Zeifman's bounds on the rate of convergence for birth-death processes
Authors: Mitrophanov A. Yu.
Published: July 2016

It is shown that the method of deriving bounds on the rate of convergence for birth–death processes developed by Zeifman can be effectively applied to stochastic models of chemical reactions.
Hoare's Selection Algorithm: A Markov Chain Approach
Authors: Grübel Rudolf
Published: July 2016

We obtain bounds for the distribution of the number of comparisons needed by Hoare's randomized selection algorithm FIND and give a new proof for Grübel and Rösler's (1996) result on the convergence of this distribution. Our approach is based on the construction and analysis of a suitable associated Markov chain. Some numerical results for the quantiles of the limit distributions are included, leading for example to the statement that, for a set S with n elements and n large, FIND will need with probability 0.9 about 4.72 x n comparisons to find the median of S.
Characterization of the Output Rate Process for a Markovian Storage Model
Authors: Aalto Samuli
Published: July 2016

We consider storage models where the input rate and the demand are modulated by a Markov jump process. One particular example from teletraffic theory is a fluid model of a multiplexer loaded by exponential on-off sources. Although the storage level process has been widely studied, little attention has been paid to the output rate process. We will show that, under certain assumptions, there exists another Markov jump process that modulates the output rate. The modulating process is explicitly constructed. It turns out to be a modification of a GI/G/1 queueing process
On the Inverse of Erlang's Function
Authors: Berezner S. A. , Krzesinski A. E. , Taylor P. G.
Published: July 2016

Erlang's functionB(λ,C) gives the blocking probability that occurs when Poisson traffic of intensity λ is offered to a link consisting ofCcircuits. However, when dimensioning a telecommunications network, it is more convenient to use the inverseC(λ,B) of Erlang's function, which gives the number of circuits needed to carry Poisson traffic λ with blocking probability at mostB. This paper derives simple bounds forC(λ,B). These bounds are close to each other and the upper bound provides an accurate linear approximation toC(λ,B) which is asymptotically exact in the limit as λ approaches infinity withBfixed
Multitype processes with reproduction-dependent immigration
Authors: Rahimov Ibrahim
Published: July 2016

The multitype discrete time indecomposable branching process with immigration is considered. Using a martingale approach a limit theorem is proved for such processes when the totality of immigrating individuals at a given time depends on evolution of the processes generating by previously immigrated individuals. Corollaries of the limit theorem are obtained for the cases of finite and infinite second moments of offspring distribution in critical processes.
A canonical representation for aggregated Markov processes
Authors: Larget Bret
Published: July 2016

A deterministic function of a Markov process is called an aggregated Markov process. We give necessary and sufficient conditions for the equivalence of continuous-time aggregated Markov processes. For both discrete- and continuous-time, we show that any aggregated Markov process which satisfies mild regularity conditions can be directly converted to a canonical representation which is unique for each class of equivalent models, and furthermore, is a minimal parameterization of all that can be identified about the underlying Markov process. Hidden Markov models on finite state spaces may be framed as aggregated Markov processes by expanding the state space and thus also have canonical representations.
Geometric inequalities for the eigenvalues of concentrated Markov chains
Authors: François Olivier
Published: July 2016

This article describes new estimates for the second largest eigenvalue in absolute value of reversible and ergodic Markov chains on finite state spaces. These estimates apply when the stationary distribution assigns a probability higher than 0.702 to some given state of the chain. Geometric tools are used. The bounds mainly involve the isoperimetric constant of the chain, and hence generalize famous results obtained for the second eigenvalue. Comparison estimates are also established, using the isoperimetric constant of a reference chain. These results apply to the Metropolis-Hastings algorithm in order to solve minimization problems, when the probability of obtaining the solution from the algorithm can be chosen beforehand. For these dynamics, robust bounds are obtained at moderate levels of concentration.
Chaoticity on path space for a queueing network with selection of the shortest queue among several
Authors: Graham Carl
Published: July 2016

We consider a network with N infinite-buffer queues with service rates λ, and global task arrival rate Nν. Each task is allocated L queues among N with uniform probability and joins the least loaded one, ties being resolved uniformly. We prove Q-chaoticity on path space for chaotic initial conditions and in equilibrium: any fixed finite subnetwork behaves in the limit N goes to infinity as an i.i.d. system of queues of law Q. The law Q is characterized as the unique solution for a non-linear martingale problem; if the initial conditions are q-chaotic, then Q
0 = q, and in equilibrium Q
0 = q
ρ is the globally attractive stable point of the Kolmogorov equation corresponding to the martingale problem. This result is equivalent to a law of large numbers on path space with limit Q, and implies a functional law of large numbers with limit (Q

t
)
t≥0. The significant improvement in buffer utilization, due to the resource pooling coming from the choices, is precisely quantified at the limit.
Linear estimation of self-similar processes via Lamperti's transformation
Authors: Nuzman Carl J. , Poor H. Vincent
Published: July 2016

Lamperti's transformation, an isometry between self-similar and stationary processes, is used to solve some problems of linear estimation of continuous-time, self-similar processes. These problems include causal whitening and innovations representations on the positive real line, as well as prediction from certain finite and semi-infinite intervals. The method is applied to the specific case of fractional Brownian motion (FBM), yielding alternate derivations of known prediction results, along with some novel whitening and interpolation formulae. Some associated insights into the problem of discrete prediction are also explored. Closed-form expressions for the spectra and spectral factorization of the stationary processes associated with the FBM are obtained as part of this development.
Multicomponent lifetime distributions in the presence of ageing
Authors: Fan Juanjuan , Ghurye S. G. , Levine Richard A.
Published: July 2016

Lifetime distributions for multicomponent systems are developed through the interplay of ageing and stress shocks to the system. The ageing process is explicitly modeled by an exponential function with rate affected by the magnitude of stresses from a compound Poisson process shock model. Applications of these life distributions and associated failure rates towards the study of multicomponent system survival are discussed. In particular, we illustrate the behavior of these survival functions in relevant subsets of the parameter space.
Weak convergence of high-speed network traffic models
Authors: Resnick Sidney , van den Berg Eric
Published: July 2016

We consider a network traffic model consisting of an infinite number of sources linked to a server. Sources initiate transmissions to the server at Poisson time points. The duration of each transmission has a heavy-tailed distribution. We show that suitable scalings of the traffic process converge to a totally skewed stable Lévy motion in Skorohod space, equipped with the Skorohod M
1 topology. This allows us to prove a heavy-traffic theorem for a single-server fluid model.
On normalization constants for closed queueing networks with finite local buffers
Authors: Tetzlaff Ulrich A. W.
Published: July 2016

We present new closed form solutions for partition functions used to normalize the steady-state flow balance equations of certain Markovian type queueing networks. The results focus on single class closed product form networks with state space constraints at the queueing stations. They are achieved by combining the partition function of the open network, having finite local buffers with a delta function in order to fix the number of customers in the system.
A minimax optimal stop rule in reliability checking
Authors: Gouweleeuw José M.
Published: July 2016

Consider a machine, which may or may not have a defect, and the probability q that this machine is defective is unknown. In order to determine whether the machine is defective, it is tested. On each test, the defect is found with probability p, if it has not been found yet. Performing n tests costs c

n
dollars and there is a fine of 1 dollar if there is a defect and it is not found on the tests. When should we stop testing, in order to minimize the cost?

This problem is treated in a minimax setting: we try to find a strategy that works well, even for ‘bad’ q's. It turns out that the minimax optimal stop rule can be unexpectedly complicated. For example, if p = 1/2 and c

n
= c
n = 0.25n, then the optimal rule is to start by performing one test. If a defect is found we stop, otherwise we perform a second test. If a defect is found, then again we stop, else toss a coin and stop if this shows heads. If we still have not stopped, a third and last test is performed.
Keeping a satellite aloft: two finite fuel stochastic control models
Authors: Jacka S. D.
Published: July 2016

We consider two models for the control of a satellite–in the first, fuel is expended in a linear fashion to move a satellite following a diffusion–where the aim is to keep the satellite above a critical level for as long as possible (or indeed to reach a higher, ‘safe’ level). Under suitable assumptions for the drift and diffusion coefficients, it is shown that the stochastic maximum of the time to fall below the critical level is attained by a policy which imposes a reflecting boundary at the critical level until the fuel is exhausted and jumps the satellite directly to the safe level if this is ever possible. In the second model, there is a nonlinear response to the expenditure of fuel, and no safe level. It is shown that the optimal policy for maximizing the expected discounted time for the satellite to crash is similar, in that equal packets of fuel are used to jump the satellite upwards each time it reaches the critical level.
On the existence of submultiplicative moments for the stationary distributions of some Markovian random walks
Authors: Sgibnev M. S.
Published: July 2016

This paper is concerned with submultiplicative moments for the stationary distributions π of some Markov chains taking values in ℝ+ or ℝ which are closely related to the random walks generated by sequences of independent identically distributed random variables. Necessary and sufficient conditions are given for ∫φ(x)π(dx) < ∞, where φ(x) is a submultiplicative function, i.e. φ(0) = 1 and φ(x+y) ≤ φ(x)φ(y) for all x, y.
Existence and positivity of the limit in processes with a branching structure
Authors: Quine M. P. , Szczotka W.
Published: July 2016

We define a stochastic process {X

n
} based on partial sums of a sequence of integer-valued random variables (K
0,K
1,…). The process can be represented as an urn model, which is a natural generalization of a gambling model used in the first published exposition of the criticality theorem of the classical branching process. A special case of the process is also of interest in the context of a self-annihilating branching process. Our main result is that when (K
1,K
2,…) are independent and identically distributed, with mean a ∊ (1,∞), there exist constants {c

n
} with c

n+1/c

n
→ a as n → ∞ such that X

n
/c

n
converges almost surely to a finite random variable which is positive on the event {X

n
↛ 0}. The result is extended to the case of exchangeable summands.
On waiting times to populate an environment and a question of statistical inference
Authors: Bruss F. Thomas , Slavtchova-Bojkova M.
Published: July 2016

The mathematical model we consider here is the classical Bienaymé–Galton–Watson branching process modified with immigration in the state zero.

We study properties of the waiting time to explosion of the supercritical modified process, i.e. that time until all beginning cycles which die out have disappeared. We then derive the expected total progeny of a cycle and show how higher moments can be computed. With a view to applications the main goal is to show that any statistical inference from observed cycle lengths or estimates of total progeny on the fertility rate of the process must be treated with care. As an example we discuss population experiments with trout.
A probabilistic proof of Stein's factors
Authors: Xia Aihua
Published: July 2016

We provide a probabilistic proof of the Stein's factors based on properties of birth and death Markov chains, solving a tantalizing puzzle in using Markov chain knowledge to view the celebrated Stein–Chen method for Poisson approximations. This work complements the work of Barbour (1988) for the case of Poisson random variable approximation.
Ornstein–Uhlenbeck type processes with non-normal distribution
Authors: Jensen Jens Ledet , Pedersen Jan
Published: July 2016

We analyse a class of diffusion models that (i) allows an explicit expression for the likelihood function of discrete time observation, (ii) allows the possibility of heavy-tailed observations, and (iii) allows an analysis of the tails of the increments. The class simply consists of transformed Ornstein–Uhlenbeck processes and is of relevance for heavy-tailed time series. We also treat the question of the existence of an equivalent martingale measure for the class of models considered.
Weak convergence to the coalescent in neutral population models
Authors: Möhle M.
Published: July 2016

For a large class of neutral population models the asymptotics of the ancestral structure of a sample of n individuals (or genes) is studied, if the total population size becomes large. Under certain conditions and under a well-known time-scaling, which can be expressed in terms of the coalescence probabilities, weak convergence in D

E
([0,∞)) to the coalescent holds. Further the convergence behaviour of the jump chain of the ancestral process is studied. The results are used to approximate probabilities which are of certain interest in applications, for example hitting probabilities.
Setups in polling models: does it make sense to set up if no work is waiting?
Authors: Cooper Robert B. , Niu Shun-Chen , Srinivasan Mandyam M.
Published: July 2016

We compare two versions of a symmetric two-queue polling model with switchover times and setup times. The SI version has State-Independent setups, according to which the server sets up at the polled queue whether or not work is waiting there; and the SD version has State-Dependent setups, according to which the server sets up only when work is waiting at the polled queue. Naive intuition would lead one to believe that the SD version should perform better than the SI version. We characterize the difference in the expected waiting times of these two versions, and we uncover some surprising facts. In particular, we show that, regardless of the server utilization or the service-time distribution, the SD version performs (i) the same as, (ii) worse than, or (iii) better than its SI counterpart if the switchover and setup times are, respectively, (i) both constants, (ii) variable (i.e. non-deterministic) and constant, or (iii) constant and variable. Only (iii) is consistent with naive intuition.
On overshoots and hitting times for random walks
Authors: Bertoin Jean
Published: July 2016

Consider an oscillating integer valued random walk up to the first hitting time of some fixed integer x > 0. Suppose there is a fee to be paid each time the random walk crosses the level x, and that the amount corresponds to the overshoot. We determine the distribution of the sum of these fees in terms of the renewal functions of the ascending and descending ladder heights. The proof is based on the observation that some path transformation of the random walk enables us to translate the problem in terms of the intersection of certain regenerative sets.
Analytical solution of finite capacity M/D/1 queues
Authors: Brun Olivier , Garcia Jean-Marie
Published: July 2016

Although the M/D/1/N queueing model is well solved from a computational point of view, there is no known analytical expression of the queue length distribution. In this paper, we derive closed-form formulae for the distribution of the number of customers in the system in the finite-capacity M/D/1 queue. We also give an explicit solution for the mean queue length and the average waiting time.
The structured coalescent process with weak migration
Authors: Notohara Morihiro
Published: July 2016

The aim of this paper is to study genealogical processes in a geographically structured population with weak migration. The coalescence time for sampled genes from different colonies diverges to infinity as the migration rates among colonies are close to zero. We investigate the moment generating functions of the coalescence time, the number of segregating sites and the number of allele types in sampled genes when there is low migration. Employing a perturbation method, we obtain a system of recurrence relations for the approximate solutions of these moment generating functions and solve them in some cases.
Monotone Markov processes with respect to the reversed hazard rate ordering: an application to reliability
Authors: Bloch-Mercier Sophie
Published: July 2016

We consider a repairable system with a finite state space which evolves in time according to a Markov process as long as it is working. We assume that this system is getting worse and worse while running: if the up-states are ranked according to their degree of increasing degradation, this is expressed by the fact that the Markov process is assumed to be monotone with respect to the reversed hazard rate and to have an upper triangular generator. We study this kind of process and apply the results to derive some properties of the stationary availability of the system. Namely, we show that, if the duration of the repair is independent of its completeness degree, then the more complete the repair, the higher the stationary availability, where the completeness degree of the repair is measured with the reversed hazard rate ordering.
Analysis of the busy period for the M/M/c queue: an algorithmic approach
Authors: Artalejo J. R. , Lopez-Herrero M. J.
Published: July 2016

This paper presents an algorithmic analysis of the busy period for the M/M/c queueing system. By setting the busy period equal to the time interval during which at least one server is busy, we develop a first step analysis which gives the Laplace-Stieltjes transform of the busy period as the solution of a finite system of linear equations. This approach is useful in obtaining a suitable recursive procedure for computing the moments of the length of a busy period and the number of customers served during it. The maximum entropy formalism is then used to analyse what is the influence of a given set of moments on the distribution of the busy period and to estimate the true busy period distribution. Our study supplements a recent work of Daley and Servi (1998) and other studies where the busy period of a multiserver queue follows a different definition, i.e., a busy period is the time interval during which all servers are engaged.
On the closure of the IFR(2) and NBU(2) classes
Authors: Franco Manuel , Ruiz José M. , Carmen Ruiz M.
Published: July 2016

In this note, we give some preservation results for the classes IFR(2), NBU(2) and their dual classes under the formation of special coherent systems. Further, we show with examples that the relationships among these aging classes and others are strictly one-way implications.
Parallel fluid queues with constant inflows and simultaneous random reductions
Authors: Kella Offer , Miyazawa Masakiyo
Published: July 2016

We consider I fluid queues in parallel. Each fluid queue has a deterministic inflow with a constant rate. At a random instant subject to a Poisson process, random amounts of fluids are simultaneously reduced. The requested amounts for the reduction are subject to a general I-dimensional distribution. The queues with inventories that are smaller than the requests are emptied. Stochastic upper bounds are considered for the stationary distribution of the joint buffer contents. Our major interest is in finding exponential product-form bounds, which turn out to have the appropriate decay rates with respect to certain linear combinations of buffer contents.
On the stability of a batch clearing system with Poisson arrivals and subadditive service times
Authors: Aldous David , Miyazawa Masakiyo , Rolski Tomasz
Published: July 2016

We study a service system in which, in each service period, the server performs the current set B of tasks as a batch, taking time s(B), where the function s(·) is subadditive. A natural definition of ‘traffic intensity under congestion’ in this setting is ρ := lim
t→∞
t
-1Es (all tasks arriving during time [0,t]). We show that ρ > 1 and a finite mean of individual service times are necessary and sufficient to imply stability of the system. A key observation is that the numbers of arrivals during successive service periods form a Markov chain {A

n
}, enabling us to apply classical regenerative techniques and to express the stationary distribution of the process in terms of the stationary distribution of {A

n
}.
On the existence of the stable birth-type distribution in a general branching process cell cycle model with unequal cell division
Authors: Alexandersson Marina
Published: July 2016

We use multi-type branching process theory to construct a cell population model, general enough to include a large class of such models, and we use an abstract version of the Perron-Frobenius theorem to prove the existence of the stable birth-type distribution. The generality of the model implies that a stable birth-size distribution exists in most size-structured cell cycle models. By adding the assumption of a critical size that each cell has to pass before division, called the nonoverlapping case, we get an explicit analytical expression for the stable birth-type distribution.
On the convergence to stationarity of birth-death processes
Authors: Coolen-Schrijner Pauline , Van Doorn Erik A.
Published: July 2016

Taking up a recent proposal by Stadje and Parthasarathy in the setting of the many-server Poisson queue, we consider the integral ∫0∞[limu→∞E(X(u))-E(X(t))]dtas a measure of the speed of convergence towards stationarity of the process {X(t),t≥0}, and evaluate the integral explicitly in terms of the parameters of the process in the case that {X(t),t≥0} is an ergodic birth-death process on {0,1,….} starting in 0. We also discuss the discrete-time counterpart of this result, and examine some specific examples.
A note on stochastic comparisons of excess lifetimes of renewal processes
Authors: Belzunce Félix , Ortega Eva M. , Ruiz José M.
Published: July 2016

In this paper we provide new results about stochastic comparisons of the excess lifetime at different times of a renewal process when the interarrival times belong to several ageing classes. We also provide a preservation result for the new better than used in the Laplace transform order ageing class for series systems.
On hitting times for compound Poisson dams with exponential jumps and linear release rate
Authors: Kella Offer , Stadje Wolfgang
Published: July 2016

For a compound Poisson dam with exponential jumps and linear release rate (shot-noise process), we compute the Laplace-Stieltjes transform (LST) and the mean of the hitting time of some positive level given that the process starts from some given positive level. The solution for the LST is in terms of confluent hypergeometric functions of the first and second kinds (Kummer functions).
Rates of convergence for products of random stochastic 2 × 2 matrices
Authors: Neininger Ralph
Published: July 2016

Products of independent identically distributed random stochastic 2 × 2 matrices are known to converge in distribution under a trivial condition. Rates for this convergence are estimated in terms of the minimal L

p
-metrics and the Kolmogoroff metric and applications to convergence rates of related interval splitting procedures are discussed.
On the uniform ergodicity of Markov processes of order 2
Authors: Herkenrath Ulrich
Published: July 2016

We study the uniform ergodicity of Markov processes (Z

n
, n ≥ 1) of order 2 with a general state space (Z, 𝒵). Markov processes of order higher than 1 were defined in the literature long ago, but scarcely treated in detail. We take as the basis for our considerations the natural transition probability Q of such a process. A Markov process of order 2 is transformed into one of order 1 by combining two consecutive variables Z
2n–1 and Z
2n
into one variable Y

n
with values in the Cartesian product space (Z × Z, 𝒵 ⊗ 𝒵). Thus, a Markov process (Y

n
, n ≥ 1) of order 1 with transition probability R is generated. Uniform ergodicity for the process (Z

n
, n ≥ 1) is defined in terms of the same property for (Y

n
, n ≥ 1). We give some conditions on the transition probability Q which transfer to R and thus ensure the uniform ergodicity of (Z

n
, n ≥ 1). We apply the general results to study the uniform ergodicity of Markov processes of order 2 which arise in some nonlinear time series models and as sequences of smoothed values in sequential smoothing procedures of Markovian observations. As for the time series models, Markovian noise sequences are covered.
Haplotypes: the joint distribution of alleles at linked loci
Authors: Fearnhead Paul
Published: July 2016

We prove a result concerning the joint distribution of alleles at linked loci on a chromosome drawn from the population at stationarity. For a neutral locus, the allele is a draw from the stationary distribution of the mutation process. Furthermore, this allele is independent of the alleles at different loci on any chromosomes in the population.
Buffer content of a leaky-bucket system with long-range dependent input traffic
Authors: González-Arévalo Bárbara , Samorodnitsky Gennady
Published: July 2016

The leaky bucket is a flow control mechanism that is designed to reduce the effect of the inevitable variability in the input stream into a node of a communication network. In this paper we study what happens when an input stream with heavy-tailed work sessions arrives to a server protected by such a leaky bucket. Heavy-tailed sessions produce long-range dependence in the input stream. Previous studies of single server fluid queues without flow control suggested that such long-range dependence can have a dramatic effect on the system performance. By concentrating on the expected time till overflow of a large finite buffer we show that leaky-bucket flow control does make the system overflow less often, but long-range dependence still makes its presence felt.
Exact overflow asymptotics for queues with many Gaussian inputs
Authors: Dębicki Krzysztof , Mandjes Michel
Published: July 2016

In this paper we consider a queue fed by a large number of independent continuous-time Gaussian processes with stationary increments. After scaling the buffer exceedance threshold and the (constant) service capacity by the number of sources, we present asymptotically exact results for the probability that the buffer threshold is exceeded. We consider both the stationary overflow probability and the (transient) probability of overflow at a finite time horizon. We give detailed results for the practically important cases in which the inputs are fractional Brownian motion processes or integrated Gaussian processes.
Asymptotic inference for nearly unstable INAR(1) models
Authors: Ispány M. , Pap G. , van Zuijlen M. C. A.
Published: July 2016

A sequence of first-order integer-valued autoregressive (INAR(1)) processes is investigated, where the autoregressive-type coefficient converges to 1. It is shown that the limiting distribution of the conditional least squares estimator for this coefficient is normal and the rate of convergence is n
3/2. Nearly critical Galton–Watson processes with unobservable immigration are also discussed.
Statistical properties of a system reliability estimator using the Littlewood software reliability model
Authors: Agustin Marcus A.
Published: July 2016

This paper considers a competing risks system withppieces of software where each piece follows the model by Littlewood (1980) described as follows. The failure rate of a piece of software relies on the residual number of bugs remaining in the software where each bug produces failures at varying rates. In effect, bugs with higher failure rates tend to be observed earlier in the testing period. Tasks are assigned to the system and the task completion times as well as the software failure times are assumed to be independent of each other. The system is observed over a fixed testing period and the system reliability upon test termination is examined. An estimator of the system reliability is presented and its asymptotic properties as well as finite-sample properties are obtained.
A note on quasi-stationary distributions of birth–death processes and the SIS logistic epidemic
Authors: Clancy Damian , Pollett Philip K.
Published: July 2016

For Markov processes on the positive integers with the origin as an absorbing state, Ferrari, Kesten, Martínez and Picco studied the existence of quasi-stationary and limiting conditional distributions by characterizing quasi-stationary distributions as fixed points of a transformation Φ on the space of probability distributions on {1, 2, …}. In the case of a birth–death process, the components of Φ(ν) can be written down explicitly for any given distributionν. Using this explicit representation, we will show that Φ preserves likelihood ratio ordering between distributions. A conjecture of Kryscio and Lefèvre concerning the quasi-stationary distribution of the SIS logistic epidemic follows as a corollary.
Coalescent theory for seed bank models
Authors: Kaj Ingemar , Krone Stephen M. , Lascoux Martin
Published: July 2016

We study the genealogical structure of samples from a population for which any given generation is made up of direct descendants from several previous generations. These occur in nature when there are seed banks or egg banks allowing an individual to leave offspring several generations in the future. We show how this temporal structure in the reproduction mechanism causes a decrease in the coalescence rate. We also investigate the effects of age-dependent neutral mutations. Our main result gives weak convergence of the scaled ancestral process, with the usual diffusion scaling, to a coalescent process which is equivalent to a time-changed version of Kingman's coalescent.
Approximations of small jumps of Lévy processes with a view towards simulation
Authors: Asmussen Søren , Rosiński Jan
Published: July 2016

Let X = (X(t):t ≥ 0) be a Lévy process and X
∊ the compensated sum of jumps not exceeding ∊ in absolute value, σ2(∊) = var(X
∊(1)). In simulation, X - X
∊ is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when X
∊/σ(∊) can be approximated by another Brownian term. A necessary and sufficient condition in terms of σ(∊) is given, and it is shown that when the condition fails, the behaviour of X
∊/σ(∊) can be quite intricate. This condition is also related to the decay of terms in series expansions. We further discuss error rates in terms of Berry-Esseen bounds and Edgeworth approximations.
Locally contracting iterated functions and stability of Markov chains
Authors: Jarner S. F. , Tweedie R. L.
Published: July 2016

We consider Markov chains in the context of iterated random functions and show the existence and uniqueness of an invariant distribution under a local contraction condition combined with a drift condition, extending results of Diaconis and Freedman. From these we deduce various other topological stability properties of the chains. Our conditions are typically satisfied by, for example, queueing and storage models where the global Lipschitz condition used by Diaconis and Freedman normally fails.
A lower bound using Hamilton-type circuits and its applications
Authors: Chen John T.
Published: July 2016

This paper presents a degree-two probability lower bound for the union of an arbitrary set of events in an arbitrary probability space. The bound is designed in terms of the first-degree Bonferroni summation and pairwise joint probabilities of events, which are represented as weights of edges in a Hamilton-type circuit in a connected graph. The proposed lower bound strengthens the Dawson–Sankoff lower bound in the same way that Hunter and Worsley's degree-two upper bound improves the degree-two Bonferroni-type optimal upper bound. It can be applied to statistical inference in time series and outlier diagnoses as well as the study of dose response curves.
Long-memory continuous-time correlation models
Authors: Ma Chunsheng
Published: July 2016

This paper introduces a rather general class of stationary continuous-time processes with long memory by randomizing the time-scale of short-memory processes. In particular, by randomizing the time-scale of continuous-time autoregressive and moving-average processes, many power-law decay and slow decay correlation functions are obtained.
Recurrence properties of autoregressive processes with super-heavy-tailed innovations
Authors: Zeevi Assaf , Glynn Peter W.
Published: July 2016

This paper studies recurrence properties of autoregressive (AR) processes with ‘super-heavy-tailed’ innovations. Specifically, we study the case where the innovations are distributed, roughly speaking, as log-Pareto random variables (i.e. the tail decay is essentially a logarithm raised to some power). We show that these processes exhibit interesting and somewhat surprising behaviour. In particular, we show that AR(1) processes, with the usual root assumption that is necessary for stability, can exhibit null-recurrent as well as transient dynamics when the innovations follow a log-Cauchy-type distribution. In this regime, the recurrence classification of the process depends, somewhat surprisingly, on the value of the constant pre-multiplier of this distribution. More generally, for log-Pareto innovations, we provide a positive-recurrence/null-recurrence/transience classification of the corresponding AR processes.
Telegraph processes with random velocities
Authors: Stadje W. , Zacks S.
Published: July 2016

We study a one-dimensional telegraph process (Mt)t≥0describing the position of a particle moving at constant speed between Poisson times at which new velocities are chosen randomly. The exact distribution ofMtand its first two moments are derived. We characterize the level hitting times ofMtin terms of integro-differential equations which can be solved in special cases.
Large deviations for supercritical multitype branching processes
Authors: Jones Owen Dafydd
Published: July 2016

Large deviation results are obtained for the normed limit of a supercritical multitype branching process. Starting from a single individual of type i, let L[i] be the normed limit of the branching process, and let be the minimum possible population size at generation k. If is bounded in k (bounded minimum growth), then we show that P(L[i] ≤ x) = P(L[i] = 0) + x

α

F
*[i](x) + o(x

α
) as x → 0. If grows exponentially in k (exponential minimum growth), then we show that −log P(L[i] ≤ x) = x
−β/(1−β) G*[i](x) + o (x
−β/(1−β)) as x → 0. If the maximum family size is bounded, then −log P(L[i] > x) = x

δ/(δ−1)
H
*[i](x) + o(x

δ/(δ−1)) as x → ∞. Here α, β and δ are constants obtained from combinations of the minimum, maximum and mean growth rates, and F
*, G
* and H
* are multiplicatively periodic functions.
Change-point models and conditionally pure birth processes: an inequality on the stochastic intensity
Authors: De Santis Emilio , Spizzichino Fabio
Published: July 2016

We analyse several aspects of a class of simple counting processes that can emerge in some fields of applications where a change point occurs. In particular, under simple conditions we prove a significant inequality for the stochastic intensity.
The collision branching process
Authors: Chen Anyue , Pollett Phil , Zhang Hanjun , Li Junping
Published: July 2016

We consider a branching model, which we call the collision branching process (CBP), that accounts for the effect of collisions, or interactions, between particles or individuals. We establish that there is a unique CBP, and derive necessary and sufficient conditions for it to be nonexplosive. We review results on extinction probabilities, and obtain explicit expressions for the probability of explosion and the expected hitting times. The upwardly skip-free case is studied in some detail.
The use of the variogram in construction of stationary time series models
Authors: Ma Chunsheng
Published: July 2016

This paper studies a class of stationary covariance models, in both the discrete- and the continuous-time domains, which possess a simple functional form γ(τ + τ0)+γ(τ − τ0)− 2γ(τ), where τ
0 is a fixed lag andγ(τ) is an intrinsically stationary variogram, and include the fractional Gaussian noise of Kolmogorov (1940) and a stochastic volatility model of Barndorff-Nielsen and Shephard (2001), (2002) as special cases. Properties of the class, and interesting special cases with long memory, are studied. We also characterize the covariance function via the variogram.
Scale functions of Lévy processes and busy periods of finite-capacity M/GI/1 queues
Authors: Dube Parijat , Guillemin Fabrice , Mazumdar Ravi R.
Published: July 2016

In this paper we use the exit time theory for Lévy processes to derive new closed-form results for the busy period distribution of finite-capacity fluid M/G/1 queues. Based on this result, we then obtain the busy period distribution for finite-capacity queues with on–off inputs when the off times are exponentially distributed.
On the multifractal spectrum of the branching measure on a Galton–Watson tree
Authors: Mörters Peter , Shieh Narn-Rueih
Published: July 2016

Suppose that μ is the branching measure on the boundary of a supercritical Galton–Watson tree with offspring distribution N such that E[N log N] < ∞ and P{N = 1} > 0. We determine the multifractal spectrum of μ using a method different from that proposed by Shieh and Taylor, which is flawed.
The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1
Authors: Hwang Gang Uk , Choi Bong Dae , Kim Jae-Kyoon
Published: July 2016

We consider a discrete-time queueing system with the discrete autoregressive process of order 1 (DAR(1)) as an input process and obtain the actual waiting time distribution and the virtual waiting time distribution. As shown in the analysis, our approach provides a natural numerical algorithm to compute the waiting time distributions, based on the theory of the GI/G/1 queue, and consequently we can easily investigate the effect of the parameters of the DAR(1) on the waiting time distributions. We also derive a simple approximation of the asymptotic decay rate of the tail probabilities for the virtual waiting time in the heavy traffic case.
Probability distribution of the number of deceits in collective robotics
Authors: Murciano Antonio , Zamora Javier , Lopez-Sanchez Jesus , Rodriguez-Santamaria Emilia
Published: July 2016

The benefit obtained by a selfish robot by cheating in a real multirobotic system can be represented by the random variable X

n,q
: the number of cheating interactions needed before all the members in a cooperative team of robots, playing a TIT FOR TAT strategy, recognize the selfish robot. Stability of cooperation depends on the ratio between the benefit obtained by selfish and cooperative robots. In this paper, we establish the probability model for X

n,q
. If the values of the parameters n and q are known, then this model allows us to make predictions about the stability of cooperation. Moreover, if these parameters are modifiable, it is possible to tune them to guarantee the viability of cooperation.
Moderate deviation for super-Brownian Motion with super-Brownian immigration
Authors: Hong Wen-Ming
Published: July 2016

Moderate deviation principles are established in dimensionsd≥ 3 for super-Brownian motion with random immigration, where the immigration rate is governed by the trajectory of another super-Brownian motion. It fills in the gap between the central limit theorem and large deviation principles for this model which were obtained by Hong and Li (1999) and Hong (2001).
Selecting the last success in Markov-dependent trials
Authors: Hsiau Shoou-Ren , Yang Jiing-Ru
Published: July 2016

In a sequence of Markov-dependent trials, the optimal strategy which maximizes the probability of stopping on the last success is considered. Both homogeneous Markov chains and nonhomogeneous Markov chains are studied. For the homogeneous case, the analysis is divided into two parts and both parts are realized completely. For the nonhomogeneous case, we prove a result which contains the result of Bruss (2000) under an independence structure.