bluuebunny/crossref_metadata_2025 huggingface dataset:
The covariance of two type II counters
Authors: Berman Mark
Published: July 2016

Consider three independent Poisson processes of point events of rates λ1, λ 2 and λ12. Suppose there are two electronic counters, the first recording events from the first and third Poisson processes, and the second recording events from the second and third Poisson processes. Both counters are subject to dead times of Type II, i.e. where all events, whether recorded or not, cause dead times. The covariance of the numbers of events recorded by the two counters in a suitable time period is derived. This extends earlier work of Cox and Isham (1977) and Kingman (1977) who considered the same problem for Type I counters, i.e. where only recorded events cause dead times.
On the ruin probability of the generalised Ornstein–Uhlenbeck process in the cramér case
Authors: Bankovsky Damien , Klüppelberg Claudia , Maller Ross
Published: July 2016

For a bivariate Lévy process (ξt,ηt)t≥ 0and initial valueV0define the generalised Ornstein–Uhlenbeck (GOU) processVt:=eξt(V0+∫t0e-ξs-dηs),t≥0, and the associated stochastic integral processZt:=∫0te-ξs-dηs,t≥0. LetTz:=inf{t>0:Vt<0|V0=z} and ψ(z):=P(Tz<∞) forz≥0 be the ruin time and infinite horizon ruin probability of the GOU process. Our results extend previous work of Nyrhinen (2001) and others to give asymptotic estimates for ψ(z) and the distribution ofTzasz→∞, under very general, easily checkable, assumptions, when ξ satisfies a Cramér condition.
Checkpointing for the RESTART problem in Markov networks
Authors: Lipsky Lester , Doran Derek , Gokhale Swapna
Published: July 2016

We apply the known formulae of the RESTART problem to Markov models of software (and many other) systems, and derive new equations. We show how checkpoints might be included, with their resultant performance under RESTART. The result is a complete procedure for finding the mean, variance, and tail behavior of the job completion time as a function of the failure rate. We also provide a detailed example.
The size of a major epidemic of a vector-borne disease
Authors: Daley Daryl J. , Swift Randall J.
Published: July 2016

Based on a simple model due to Dietz, it is shown that the size of a major epidemic of a vector-borne disease with basic reproduction ratio R
0>1 is dominated by the size of a standard SIR (susceptible–infected–removed) epidemic with direct host-to-host transmission of disease and the same R
0. Further bounds and numerical illustrations are provided, broadly spanning situations where the size of the epidemic is short of infecting almost all those susceptible. The total size is moderately sensitive to changes in the population parameters that contribute to R
0, so that the fluctuating behaviour in ‘annual’ epidemics is not surprising.
On exponential limit laws for hitting times of rare sets for Harris chains and processes
Authors: Glynn Peter W.
Published: July 2016

This paper provides a simple proof for the fact that the hitting time to an infrequently visited subset for a one-dependent regenerative process converges weakly to an exponential distribution. Special cases are positive recurrent Harris chains and Harris processes. The paper further extends this class of limit theorems to ‘rewards’ that are cumulated to the hitting time of such a rare set.
What is typical?
Authors: Last Günter , Thorisson Hermann
Published: July 2016

Let ξ be a random measure on a locally compact second countable topological group, and letXbe a random element in a measurable space on which the group acts. In the compact case we give a natural definition of the concept that the origin is a typical location forXin the mass of ξ, and prove that when this holds, the same is true on sets placed uniformly at random around the origin. This new result motivates an extension of the concept of typicality to the locally compact case where it coincides with the concept of mass-stationarity. We describe recent developments in Palm theory where these ideas play a central role.
Buffer overflow asymptotics for a buffer handling many traffic sources
Authors: Courcoubetis Costas , Weber Richard
Published: July 2016

As a model for an ATM switch we consider the overflow frequency of a queue that is served at a constant rate and in which the arrival process is the superposition of N traffic streams. We consider an asymptotic as N → ∞ in which the service rate Nc and buffer size Nb also increase linearly in N. In this regime, the frequency of buffer overflow is approximately exp(–NI(c, b)), where I(c, b) is given by the solution to an optimization problem posed in terms of time-dependent logarithmic moment generating functions. Experimental results for Gaussian and Markov modulated fluid source models show that this asymptotic provides a better estimate of the frequency of buffer overflow than ones based on large buffer asymptotics.
On the generating functions of a random walk on the non-negative integers
Authors: Dette Holger
Published: July 2016

In the random walk whose state space is a subset of the non-negative integers explicit representations for the generating functions of then-step transition and the first return probabilities are obtained. These representations involve the Stieltjes transform of the spectral measure of the process and the corresponding orthogonal polynomials. Several examples are given in order to illustrate the application of the results.
On effects of discretization on estimators of drift parameters for diffusion processes
Authors: Kloeden P. E. , Platen E. , Schurz H. , Sørensen M.
Published: July 2016

In this paper statistical properties of estimators of drift parameters for diffusion processes are studied by modern numerical methods for stochastic differential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diffusions. A review is given of the necessary theory for parameter estimation for diffusion processes and for simulation of diffusion processes. Three examples are studied.
On wald-type optimal stopping for Brownian motion
Authors: Graversen S. E. , Peškir G.
Published: July 2016

The solution is presented to all optimal stopping problems of the form supτ
E(G(|Β τ
|) – cτ), where is standard Brownian motion and the supremum is taken over all stopping times τ for B with finite expectation, while the map G : ℝ+ → ℝ satisfies for some being given and fixed. The optimal stopping time is shown to be the hitting time by the reflecting Brownian motion of the set of all (approximate) maximum points of the map . The method of proof relies upon Wald's identity for Brownian motion and simple real analysis arguments. A simple proof of the Dubins–Jacka–Schwarz–Shepp–Shiryaev (square root of two) maximal inequality for randomly stopped Brownian motion is given as an application.
Properties of calculated predictions of grade sizes and the associated integer valued vectors
Authors: Guerry M. A.
Published: July 2016

In a system modelled by a time-discrete deterministic model, predictions of the distribution of the members over the different classes do not result automatically in an integer valued vector. In this paper, for a constant size system, we discuss how to associate with the calculated vector an integer valued vector. Furthermore we examine whether the evolution of the calculated vectors on the one hand, and the evolution of the associated integer valued vectors on the other hand, have the same properties.
Variants of the Choquet–Deny theorem with applications
Authors: Fosam E. B. , Shanbhag D. N.
Published: July 2016

A characterization of the exponential distribution based on a relevation-type equation and its discrete version are extended to the case of multidimensional spaces via variants of the Choquet–Deny theorem. Comments on some recent results in the literature are made.
Simplified bounds on the tails of compound distributions
Authors: Willmot G. E. , Lin Xiaodong
Published: July 2016

Upper and lower bounds are derived for the tail probabilities of compound distributions using simple properties of the claim size distribution. General bounds are then obtained for various classes of claim size distributions. Some examples are given.
Braess's paradox in a queueing network with state-dependent routing
Authors: Calvert Bruce , Solomon Wiremu , Ziedins Ilze
Published: July 2016

We consider initially two parallel routes, each of two queues in tandem, with arriving customers choosing the route giving them the shortest expected time in the system, given the queue lengths at the customer's time of arrival. All interarrival and service times are exponential.

We then augment this network to obtain a Wheatstone bridge, in which customers may cross from one route to the other between queues, again choosing the route giving the shortest expected time in the system, given the queue lengths ahead of them.

We find that Braess's paradox can occur: namely in equilibrium the expected transit time in the augmented network, for some service rates, can be greater than in the initial network.
The rate of occurrence of failures
Authors: Yeh Lam
Published: July 2016

In this article, we assume that the state of a system forms a continuous-time Markov chain or a higher-dimensional Markov process after introducing some supplementary variables. A formula for evaluating the rate of occurrence of failures for the system is derived. As an application of the theory, a maintenance model for a two-component system is also studied.
Explicit solution of an optimal stopping problem: the burn-in of conditionally exponential components
Authors: Costantini C. , Spizzichino F.
Published: July 2016

We consider the problem of the optimal duration of a burn-in experiment for n identical units with conditionally exponential life-times of unknown parameter Λ. The problem is formulated as an optimal stopping problem for a suitably defined two-dimensional continuous-time Markov process. By exploiting monotonicity properties of the statistical model and of the costs we prove that the optimal stopping region is monotone (according to an appropriate definition) and derive a set of equations that uniquely determine it and that can be easily solved recursively. The optimal stopping region varies monotonically with the costs. For the class of problems corresponding to a prior distribution on Λ of type gamma it is shown how the optimal stopping region varies with respect to the prior distribution and with respect to n.
Limit theorems and estimation theory for branching processes with an increasing random number of ancestors
Authors: Dion J. P. , Yanev N. M.
Published: July 2016

This paper deals with a Bienaymé-Galton-Watson process having a random number of ancestors. Its asymptotic properties are studied when both the number of ancestors and the number of generations tend to infinity. This yields consistent and asymptotically normal estimators of the mean and the offspring distribution of the process. By exhibiting a connection with the BGW process with immigration, all results can be transported to the immigration case, under an appropriate sampling scheme. A key feature of independent interest is a new limit theorem for sums of a random number of random variables, which extends the Gnedenko and Fahim (1969) transfer theorem.
On the rate of convergence of Borovkov's multidimensional ergodic Markov chain
Authors: Tanikawa Akio
Published: July 2016

This paper is concerned with the multidimensional Markov chain {X(n)} = {X(x, n)}, considered by Borovkov [1], [2], [3], with the form X(n + 1) = ?(n) + ?(?(n), n), where the distribution of depends only on x. Sufficient conditions on moments of |?(x)| are established for the Markov chain {X(n)} to have rate of convergence results (i.e. geometric ergodicity or sub-geometric rates for a variety of rate functions).
On the time a Markov chain spends in a lumped state
Authors: Norberg Tommy
Published: July 2016

The sojourn time that a Markov chain spends in a subset E of its state space has a distribution that depends on the hitting distribution on E and the probabilities (resp. rates in the continuous-time case) that govern the transitions within E. In this note we characterise the set of all hitting distributions for which the sojourn time distribution is geometric (resp. exponential).
Fixation in bisexual models with variable population sizes
Authors: Möhle M.
Published: July 2016

A general exchangeable bisexual model with variable population sizes is introduced. First the forward process, i.e. the number of certain descending pairs, is studied. For the bisexual Wright-Fisher model fixation of the descendants occurs, i.e. their proportion tends to 0 or 1 almost surely.

The main part of this article deals with necessary and sufficient conditions for ultimate homozygosity, i.e. the proportion of an arbitrarily chosen allelic type tends to 0 or 1 almost surely. The results are applied to a bisexual Wright-Fisher model and to a bisexual Moran model.
Large deviations for departures from a shared buffer
Authors: O'connell Neil
Published: July 2016

In this paper we describe how the joint large deviation properties of traffic streams are altered when the traffic passes through a shared buffer according to a FCFS service policy with stochastic service capacity. We also consider the stationary case, proving large deviation principles for the state of the system in equilibrium and for departures from an equilibrium system.
Coherent structures and unimodality
Authors: Sabnis S. V. , Nair Mini R.
Published: July 2016

This paper is concerned with the preservation of unimodality under coherent structures of independent components having a common life distribution function. This result in a way generalizes a result of Alam [1], as Alam's result indirectly also deals with preservation of unimodality for (n – i + 1)-out-of-n systems of independent and identically distributed components. The usefulness of this property of coherent systems in obtaining sharper upper bounds on the reliability of the concerned system has been illustrated below for a bridge structure with components having a gamma life distribution function.
A geometric invariant in weak lumpability of finite Markov chains
Authors: Ledoux James
Published: July 2016

We consider weak lumpability of finite homogeneous Markov chains, which is when a lumped Markov chain with respect to a partition of the initial state space is also a homogeneous Markov chain. We show that weak lumpability is equivalent to the existence of a direct sum of polyhedral cones that is positively invariant by the transition probability matrix of the original chain. It allows us, in a unified way, to derive new results on lumpability of reducible Markov chains and to obtain spectral properties associated with lumpability.
On the rate of convergence for extremes of mean square differentiable stationary normal processes
Authors: Kratz Marie F. , Rootzén Holger
Published: July 2016

Let ξ (t); t ≧ 0 be a normalized continuous mean square differentiable stationary normal process with covariance function r(t). Further, let

and set . We give bounds which are roughly of order Τ –δ
for the rate of convergence of the distribution of the maximum and of the number of upcrossings of a high level by ξ (t) in the interval [0, T]. The results assume that r(t) and r′(t) decay polynomially at infinity and that r
″ (t) is suitably bounded. For the number of upcrossings it is in addition assumed that r(t) is non-negative.
Asymptotic stationarity of queueing processes
Authors: Szczotka Władysław
Published: July 2016

We show that if an input process ζ to a queue is asymptotic stationary in some sense, satisfies a condition AB and some other natural conditions, then the output processes (w, ζ) and (w, q,ζ) are asymptotic stationary in the same sense. Here, w and q are the waiting time and queue length processes, respectively.
Convergence rate for the distributions of GI/M/1/n and M/GI/1/n as n tends to infinity
Authors: Simonot F.
Published: July 2016

In this note, we compare the arrival and time stationary distributions of the number of customers in the GI/M/1/n and GI/M/1 queueing systems. We show that, if the inter-arrival c.d.f. H is non-lattice with mean value λ –

1
, and if the traffic intensity ρ = λμ –

1 is strictly less than one, then the convergence rate of the stationary distributions of GI/M/1/n to the corresponding stationary distributions of GI/M/1 is geometric. More-over, the convergence rate can be characterized by the number ω, the unique solution in (0, 1) of the equation . A similar result is established for the M/GI/1/n and M/GI/1 queueing systems.
Darwin's log: a toy model of speciation and extinction
Authors: Aldous David
Published: July 2016

Stochastic models for the origin and extinction of species have been rather neglected in applied probability. As an alternative to modelling speciation and extinction as intrinsically random, I shall describe and show simulations of a rulebased model. This involves mathematical representations of notions such as genetic type of species, environmental niche, fitness of a species in a niche, and adaptation. There are underlying random mechanisms for changes of niche sizes and for disconnection and reconnection of geographical regions, and these ultimately drive the evolution of species.

Other approaches to mathematical modelling of evolution are briefly mentioned.
Limit distributions for the Bernoulli meander
Authors: Takács Lajos
Published: July 2016

This paper is concerned with the distibutions and the moments of the area and the local time of a random walk, called the Bernoulli meander. The limit behavior of the distributions and the moments is determined in the case where the number of steps in the random walk tends to infinity. The results of this paper yield explicit formulas for the distributions and the moments of the area and the local time for the Brownian meander.
Persistence of a critical super-2 process
Authors: Gorostiza Luis G. , Hochberg Kenneth J. , Wakolbinger Anton
Published: July 2016

It is shown that the critical two-level (2, d, 1, 1)-superprocess is persistent in dimensions d greater than 4. This complements the extinction result of Wu (1994) and implies that the critical dimension is 4.
A discrete-time proof of Neveu's exchange formula
Authors: Konstantopoulos Takis , Zazanis Michael
Published: July 2016

Neveu's exchange formula relates the Palm probabilities with respect to two jointly stationary simple point processes. We give a new proof of the exchange formula by using a simple result from discrete time stationary stochastic processes.
The value of an Asian option
Authors: Rogers L. C. G. , Shi Z.
Published: July 2016

This paper approaches the problem of computing the price of an Asian option in two different ways. Firstly, exploiting a scaling property, we reduce the problem to the problem of solving a parabolic PDE in two variables. Secondly, we provide a lower bound which is so accurate that it is essentially the true price.
On the asymptotic joint distribution of the sum and maximum of stationary normal random variables
Authors: Ho Hwai-Chung , Hsing Tailen
Published: July 2016

Let X
1, X
2, ·· ·be stationary normal random variables with ρn
= cov(X
0, Xn
). The asymptotic joint distribution of and is derived under the condition ρn
log n → γ
[0,∞). It is seen that the two statistics are asymptotically independent only if γ = 0.
Asymptotic inference for an Ising lattice
Authors: Pickard D. K.
Published: July 2016

Kaufmann's exact characterization of the partition function for the classical Ising model is used to obtain limit theorems for the sample correlation between nearest neighbours in the non-critical case. This provides a basis for the asymptotic testing and estimation (by confidence intervals) of the correlation between nearest neighbours.
Limit theorems for processes of randomly displaced regular events
Authors: Collings P. S.
Published: July 2016

Two types of limit theorems are proved for processes of randomly displaced regular events. Firstly, as the displacements tend to infinity, the counting process is shown to converge weakly to a Poisson process and secondly, as the interval between events tends to zero, convergence of the finite-dimensional distributions of the associated storage process to a diffusion is proved.
On a two-dimensional binary process
Authors: Galbraith R. F. , Walley D.
Published: July 2016

We consider a two-dimensional stochastic process of binary variables x
ij
defined so that the conditional distribution of x
ij
given its predecessors depends only on x
i–1j
and x
ij–1 Properties of the equilibrium distribution, when this exists, are investigated using three different representations of the process and explicit results are given in some special cases.
Subfair discounted red-and-black game with a house limit
Authors: Chen Robert
Published: July 2016

A subfair discounted red-and-black game with a house limit is introduced and the optimality of the modified bold strategy is shown.
On a single-server queue with group arrivals
Authors: Cohen J. W.
Published: July 2016

The queueing system GI/G/1 with group arrivals and individual service of the customers is considered. For the stable situation the limiting distribution of the waiting time distribution of the kth served customer for k → ∞ is derived by using the theory of regenerative processes. It is assumed that the group sizes are i.i.d. variables of which the distribution is aperiodic. The relation between this limiting distribution and the stationary distribution of the virtual waiting time is derived.
Some remarks on a counter process
Authors: Takács Lajos
Published: July 2016

In 1968 Lampard determined some limit distributions for a counter process in which the input and output are independent Poisson processes. In 1974 Phatarfod dealt with the generalizations of Lampard's formulas for the case when the input and output form a bivariate Poisson process; however, his reasoning is erroneous. The object of this paper is to determine the correct limit distributions for the generalized process.
An extension of Erlang's loss formula
Authors: Wolff Ronald W. , Wrightson Charles W.
Published: July 2016

A two-server loss system is considered with N classes of Poisson arrivals, where the service distribution function and server preferences are arrival-class dependent. The stationary state probabilities are derived and found to be independent of the form of the service distributions.
The inter-arrival times of accepted customers in an M/G/1 queue with finite capacity
Authors: Gilchrist Robert
Published: July 2016

This note draws attention to a practical problem of interest to physicists, namely the distribution of the inter-arrival times of those customers accepted by the M/D/1/K + 1 queue. The form of the distribution is found for all K and the generalisation to M/G/1/K + 1 is discussed. The approach follows the contour integral method described by Cohen.
A note on the cost of carrier-borne, right-shift, epidemic models
Authors: Kryscio Richard J. , Saunders Roy
Published: July 2016

We establish a sufficient condition for which the expected area under the trajectory of the carrier process is directly proportional to the expected number of removed carriers in the class of carrier-borne, right-shift, epidemic models studied by Severo (1969a). This result generalizes the previous work of Downton (1972) and Jerwood (1974) for some special cases of these models. We use the result to compute expected costs in the carrier-borne model due to Downton (1968) when it is unlikely that all the susceptibles will be infected. We conclude by showing that for the special case considered by Weiss (1965) this treatment of the expected cost is reasonable for populations with a large initial number of susceptibles.
La formule de crofton pour les sections Épaisses
Authors: Matheron Georges
Published: July 2016

The classical Crofton formula is generalized to the case of thick sections, and the corresponding corrective formulae are given.
On a characterization of the exponential distribution by order statistics
Authors: Ahsanullah M.
Published: July 2016

Let X
1
, X2, …, Xn
be a random sample of size n from a population with probability density function f(x), x >0, and let X
1,n < X
2,n < … < Xn,n
be the associated order statistics. A characterization of the exponential distribution is shown by considering the identical distribution of the random variables nX
1,n and (n − i + 1)(X
1,n
−; X
i–1,n
) for one i and one n with 2 ≦ i ≦ n.
On a recurrence of Steck
Authors: Breth M.
Published: July 2016

This paper contains a simple proof of Steck's recurrence equation for determining the power of the standard one-sided Kolmogorov test for a distribution function.
Non-uniform breakage-mechanism branching processes and degradation of long-chain polymers
Authors: Olson William H.
Published: July 2016

We consider the problem of number and weight distributions for breakage-mechanism branching processes whose break distributions are general. We derive a recursive relation between the expected empirical distribution after (n + 1) breaks and after n breaks, making use of length-biased sampling. Using this relation and the strong law of large numbers, we derive integro-differential equations for the asymptotic expected empirical distribution and its associated weight distribution. The mean of the asymptotic number distribution is derived using the integro-differential equation. We then provide approximate solutions to these equations and the moments of these approximations. Finally we apply these results to the case where the break distribution is a symmetric beta distribution.
A generalization of a result of Erdös and Rényi
Authors: Kaplan Norman
Published: July 2016

Consider a finite urn scheme with balls uniformly distributed amongst the urns. Using a technique of Karlin, the asymptotic behavior of the number of throws necessary to obtain at least m (m ≧ 1) balls in a given fraction of the urns is studied.
Estimators for distances
Authors: Gilbert E. N.
Published: July 2016

This paper compares estimators of u = |P
2 — P
1| based on partial knowledge about points P
1, P
2 which are chosen independently from the same probability distribution in the plane. Typical estimators are the conditional expectations E(u | r
1, r
2), E(u | s), E(u | rM
), E(u | θ) where ri
is the distance from Pi
to the origin 0, s = r
1 + r
2, rM
= Max{r
1, r
2}, θ = ∠ P
10P
2. These expectations are the least squares estimators, given the conditioned variables. Simpler linear estimators are also considered.
On up- and downcrossings
Authors: Cohen J. W.
Published: July 2016

For the sample functions of the stationary virtual waiting-time process vt
of the GI/G/1 queueing system some properties of the number of up- and downcrossings of level v by the vt
-process during a busy cycle are investigated. It turns out that the simple fact that this number of upcrossings is equal to that of downcrossings leads in a rather easy way to basic relations for the waiting-time distributions. This approach to the study of the vt
-process of the GI/G/1 system seems to be applicable to many other types of stochastic processes. As another example of this approach the infinite dam with non-constant release rate is briefly discussed.
Two-stage output procedure of a finite dam
Authors: Zuckerman Dror
Published: July 2016

The input of water into a finite dam is a Wiener process with positive drift. Water may be released at either of two possible rates 0 or M. At any time the output rate can be increased from 0 to M with cost KM, (K ≧ 0), or decreased from M to 0 with zero cost. There is a reward of A monetary units for each unit of output, (A > 0). We will consider the problem of specifying an optimal control output policy under the following optimal criteria:

(a)

Minimum total long-run average cost per unit time.

(b)

Minimum expected total discounted cost.
A note on Gaussian processes and zero-memory non-linear transformations
Authors: Masry Elias
Published: July 2016

We consider the problem of the distinguishability between Gaussian processes observed through non-linear systems. For a broad class 𝒜 of zero-memory time-dependent non-linear transformations, the following uniqueness relationship is obtained. If Xi
, i = 1, 2 are two normalized jointly Gaussian processes, then they are indistinguishable if and only if their transformed processes Yi
= A[Xi
], i = 1, 2, A ∈ 𝒜, are indistinguishable. Some interesting examples of functions in 𝒜 are presented.
An approximation to a two-dimensional binary process
Authors: Welberry T. R. , Miller G. H.
Published: July 2016

We further consider a two-dimensional stochastic process of binary variables previously discussed by Galbraith and Walley (1976). A method of estimating properties of the equilibrium distribution is described which considerably extends the region of probability space that can satisfactorily be explored.
On the reversibility of the input and output processes for a general birth-and-death queueing model
Authors: Natvig Bent
Published: July 2016

In this paper we consider the general birth-and-death queueing model of Natvig (1975). Define the input and output processes by the steady-state behaviour of respectively successive input and output intervals. Ignoring balking customers, two cases are considered. In the first case we treat a lost customer neither as an input nor as an output, then secondly as both. For both cases we show the input and output processes to be reverse processes. One mistake and two erroneous comments in Natvig (1975) are also corrected.
An upper bound for multi-channel queues
Authors: Wolff Ronald W.
Published: July 2016

Moments of the delay distribution and other measures of performance for a multi-channel queue are shown to be bounded above by corresponding quantities for one or a collection of single-channel queues.
A local limit theorem for the critical age-dependent branching process
Authors: Quigg David L.
Published: July 2016

Let Z(t) denote the number of particles alive at time t in a critical age-dependent branching process. It is proved that, for k ≧ 1, there exists a constant Ak
> 0 such that t
2
P(Z(t) = k)→Ak
as t→∞.
On the age distribution of a Markov chain
Authors: Pakes Anthony G.
Published: July 2016

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain,obtained by restarting the original chain at a fixed state after each absorption. The limiting age,A(j), is the weak limit of the timegivenXn=j(n → ∞).

A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems forA(J) (J →∞) are given for these examples.
The asymptotic behaviour of a divergent linear birth and death process
Authors: Haigh John
Published: July 2016

A recent paper in Advances in Applied Probability (Siegel (1976)) considered the duration of the time Tmn
for a linear birth and death process to grow from a (large) initial size m to a larger size n. The main aim was to show that, when the birth rate exceeds the death rate, Tmn
is close to its mean value, log n/m, with high probability. This paper establishes this result using much simpler techniques.
An upper bound for the mean of Yaglom's limit
Authors: Evans Lawrence S.
Published: July 2016

For a single-type Galton—Watson branching process with mean less than one and finite second moment, we establish an upper bound for the mean of the associated Yaglom limit. This bound is attained if and only if the generating function of the process is linear.
On the kinetics of multitype polymer reactions
Authors: Mellein B. , Vicente J. L.
Published: July 2016

A multitype polymer reaction process is considered. Its dynamics is described by means of the evolution in time of the average number of reactions (of any kind) and the mean number of unreacted chain segments (of any length), assuming that the initial chain molecule consists of n unreacted units. The asymptotic behaviour (n →∞) of the variance of the extent of reaction is also studied.
On characterizations of the geometric distribution by independence of functions of order statistics
Authors: Srivastava R. C.
Published: July 2016

Let X1, · ··, X
n
, n ≧ 2 be i.i.d. random variables having a geometric distribution, and let Y
1 ≦ Y
2 ≦ · ·· ≦ Y

n
denote the corresponding order statistics. Define Rn = Yn – Y
1 and Z
n
= Σ
j=2

n
(Υ
j
– Y1). Then it is well known that (i) Y, and Rn
are independent and (ii) Y
1 and Zn
are independent. In this paper, we show that a very weak form of each of these independence properties is a characterizing property of the geometric distribution in the class of discrete distributions.
On the existence of quasi-stationary distributions in denumerable R-transient Markov chains
Authors: Kijima Masaaki
Published: July 2016

Let {Xn, n= 0, 1, 2, ···} be a transient Markov chain which, when restricted to the state space 𝒩+= {1, 2, ···}, is governed by an irreducible, aperiodic and strictly substochastic matrix𝐏= (pij), and letpij(n) =P∈Xn=j, Xk∈ 𝒩+fork= 0, 1, ···,n|X0=i],i, j𝒩+. The prime concern of this paper is conditions for the existence of the limits,qijsay, ofasn →∞. Ifthe distribution (qij) is called the quasi-stationary distribution of {Xn} and has considerable practical importance. It will be shown that, under some conditions, if a non-negative non-trivial vectorx= (xi) satisfyingrxT=xT𝐏andexists, whereris the convergence norm of𝐏, i.e.r=R–1andand T denotes transpose, then it is unique, positive elementwise, andqij(n) necessarily converge toxjasn →∞.Unlike existing results in the literature, our results can be applied even to theR-null andR-transient cases. Finally, an application to a left-continuous random walk whose governing substochastic matrix isR-transient is discussed to demonstrate the usefulness of our results.
Lajos Takács - an appreciation
Authors: Galambos J. , Gani J.
Published: July 2016

Lajos Takács was born in Maglód, Hungary, on 21 August 1924. It was in this small town, 16 miles from Budapest, that he attended elementary school. He showed an early fascination with numbers, and worked on personal projects such as predicting the weight of the apricot harvest from 12 trees in the garden of the family home, by counting the buds on one tree in the spring.
A note on the background of several Bonferroni–Galambos-type inequalities
Authors: Móri Tamás F. , Székely Gábor J.
Published: July 2016

LetA1,A2, · ··,Anbe events on a probability space. Denote bySkthekth binomial moment of the number Mnof thoseA's which occur. Sharp lower and upper bounds ofSmwill be given in terms ofSkandSl.The same method can be applied for proving Bonferroni–Galambos-type inequalities.
The slow server problem: a queue with stalling
Authors: Rubinovitch Michael
Published: July 2016

A queue with Poisson arrivals and two different exponential servers is considered. It is assumed that customers are allowed to stall, i.e., to wait for a busy fast server at times when the slow server is free. A stochastic analysis of the queue is given, steady-state probabilities are computed, and policies for overall optimization are characterized and computed. The issue of individual customer's optimization versus overall optimization is also discussed.
Asymptotic distribution of the coefficient of cross-association based on a simple Markov-chain model
Authors: Vrbik Jan
Published: July 2016

The asymptotic distribution of the ‘coefficient of cross-association' between two sequences of nominal data is investigated, under the assumption that the sequences have been generated by means of a specific (n − 1)-parameter Markov-chain model, frequently used by geologists. A practical approximate formula for calculating the standard deviation of this distribution is presented, and the region of its applicability examined.
A one dimensional random space-filling problem
Authors: Mullooly John P.
Published: July 2016

Consider an interval of the real line (0, x), x > 0; and place in it a random subinterval S(x) defined by the random variables Xx
and Yx
, the position of the center of S(x) and the length of S(x). The set (0, x)– S(x) consists of two intervals of length δ and η. Let a > 0 be a fixed constant. If δ ≦ a, then a random interval S(δ) defined by Xδ, Yδ
is placed in the interval of length δ. If δ < a, the placement of the second interval is not made. The same is done for the interval of length η. Continue to place non-intersecting random subintervals in (0, x), and require that the lengths of all the random subintervals be ≦ a. The process terminates after a finite number of steps when all the segments of (0, x) uncovered by random subintervals are of length < a. At this stage, we say that (0, x) is saturated. Define N(a, x) as the number of random subintervals that have been placed when the process terminates. We are interested in the asymptotic behavior of the moments of N(a, x), for large x.
Limit laws for kth order statistics from conditionally mixing arrays of random variables
Authors: Dziubdziela W.
Published: July 2016

We present sufficient conditions for the weak convergence of the distributions of thekth order statistics from a conditionally mixing array of random variables to limit laws which are represented in terms of a mixed compound Poisson distribution.
A full-information best-choice problem with finite memory
Authors: Tamaki Mitsushi
Published: July 2016

n i.i.d. random variables with known continuous distribution are observed sequentially with the objective of selecting the largest. This paper considers the finite-memory case which, at each stage, allows a solicitation of anyone of the last m observations as well as of the present one. If the (k – t)th observation with value x is solicited at the k th stage, the probability of successful solicitation is p
1(x) or p
2(x) according to whether t = 0 or 1 ≦ t ≦ m. The optimal procedure is shown to be characterized by the double sequences of decision numbers. A simple algorithm for calculating the decision numbers and the probability of selecting the largest is obtained in a special case.
A prediction problem of the branching random walk
Authors: Révész P.
Published: July 2016

The location of the ‘favourite’ point at time T (T = 1, 2,…) of a supercritical branching random walk at ℤd is investigated.
On the foundations of multivariate heavy-tail analysis
Authors: Resnick Sidney
Published: July 2016

Univariate heavy-tailed analysis rests on the analytic notion of regularly varying functions. For multivariate heavy-tailed analysis, reliance on functions is awkward because multivariate distribution functions are not natural objects for many purposes and are difficult to manipulate. An approach based on vague convergence of measures makes the differences between univariate and multivariate analysis evaporate. We survey the foundations of the subject and discuss statistical attempts to assess dependence of large values. An exploratory technique is applied to exchange rate return data and shows clear differences in the dependence structure of large values for the Japanese Yen versus German Mark compared with the French Franc versus the German Mark.
David Vere-Jones's influence on statistical education
Authors: Phillips Brian
Published: July 2016

David Vere-Jones's interest in mathematical and statistical education was triggered in the early 1960s during his visits to Russia when he met leading mathematicians who were deeply involved in these areas as well as in their own research. Through the 1970s and 1980s David's involvement in education grew in New Zealand where he became influential in the developments in mathematics and statistics teaching, especially at school level. By the early 1990s he had become very involved in the Education Committee of the International Statistical Institute (ISI) and had a key role in the Third International Conference on Teaching Statistics (ICOTS-3) in Dunedin, 1990. This was soon followed by the pivotal role he played in the establishment of the International Association for Statistical Education (IASE). This paper describes the formative influences underlying David's involvement in statistical education and his influence on the growth of the IASE.
On the statistics of the linked stress release model
Authors: Bebbington Mark , Harte David S.
Published: July 2016

The paper reviews the formulation of the linked stress release model for large scale seismicity together with aspects of its application. Using data from Taiwan for illustrative purposes, models can be selected and verified using tools that include Akaike's information criterion (AIC), numerical analysis, residual point processes and Monte Carlo simulation.
A study of the role of the transmission mechanism in macroparasite aggregation
Authors: Herbert Julian , Isham Valerie
Published: July 2016

The dynamics of host-macroparasite infections pose considerable challenges for stochastic modelling because of the need to take into account a large number of relevant factors and many nonlinear interactions between them. This paper focuses attention on the infection transmission process and the effects of specific modelling assumptions about the mechanisms involved. Some dramatically simplified linear models are considered; they are based on multidimensional linear birth and death processes, and are designed to illuminate qualitative effects of interest. Both single and compound infections are allowed. It is shown that such simple models can generate and increase dispersion of parasite counts, even among homogeneous hosts.
Queues with time-dependent arrival rates. III — A mild rush hour
Authors: Newell G. F.
Published: July 2016

The arrival rate of customers to a service facility is assumed to have the formλ(t) =λ(0) —βt2for some constantβ.Diffusion approximations show that forλ(0) sufficiently close to the service rateμ, the mean queue length at time 0 is proportional toβ–1/5. A dimensionless form of the diffusion equation is evaluated numerically from which queue lengths can be evaluated as a function of time for allλ(0) andβ.Particular attention is given to those situations in which neither deterministic queueing theory nor equilibrium stochastic queueing theory apply.
A solution to the fixed-cycle traffic light problem for compound Poisson arrivals
Authors: McNeil Donald R.
Published: July 2016

A basic problem in the mathematical theory of traffic flow is to obtain the expected delay to a vehicle at a fixed-cycle traffic light controlling an intersection. The factors affecting this expected delay are the cycle time, the fraction of time the traffic light is effectively red, the number of vehicles which can pass through when the signal is green, and the nature of the (random) arrival process for the vehicles. Once a solution is known, it is possible to fix the cycle time and the effective red period in such a way that the expected delay is minimized.
Stationary distributions of open Markov processes in discrete time with application to hospital planning
Authors: Staff P. J. , Vagholkar M. K.
Published: July 2016

Suppose that a new hospital is to be planned or an existing one reorganized according to the modern Progressive Patient Care System (Hartley et al. (1968)). In this system, there are three types of wards, viz.
Brownian motion with quadratic killing and some implications
Authors: Wenocur Michael L.
Published: July 2016

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.
The discrete uhlenbeck–ornstein process
Authors: Renshaw Eric
Published: July 2016

A correlated random walk is studied in which, at each stage, the velocity changes according to a first-order process. Motion is considered both with and without friction, the former situation being the discrete analogy of the Uhlenbeck–Ornstein process. Exact and limiting expressions are developed for the cumulant structures.
Orthant probabilities and Gaussian Markov processes
Authors: Moran P. A. P.
Published: July 2016

A method is given for calculating the probability that n terms of a Gaussian first-order autoregressive series lie between any specified limits where n may be as large as 20 or more.
Stochastic processes directed by randomized time
Authors: Lee Mei-Ling Ting , Whitmore G. Alex
Published: July 2016

The paper investigates stochastic processes directed by a randomized time process. A new family of directing processes called Hougaard processes is introduced. Monotonicity properties preserved under subordination, and dependence among processes directed by a common randomized time are studied. Results for processes subordinated to Poisson and stable processes are presented. Potential applications to shock models and threshold models are also discussed. Only Markov processes are considered.
Marked point processes as limits of Markovian arrival streams
Authors: Asmussen Søren , Koole Ger
Published: July 2016

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.
Estimating the critical values of stochastic growth models
Authors: Buttell L. , Cox J. T. , Durrett R.
Published: July 2016

Interacting particle systems provide an attractive framework for modelling the growth and spread of biological populations and diseases. One problem with their use in applications is that in most cases the existing information about their critical values and equilibrium densities is too crude to be useful. In this paper we describe a method for estimating these quantities that does not require very much computer time and produces fairly accurate results.
A Diffusion Approximation for Markov Renewal Processes
Authors: Clark Steven P. , Kiessler Peter C.
Published: July 2016

For a Markov renewal process where the time parameter is discrete, we present a novel method for calculating the asymptotic variance. Our approach is based on the key renewal theorem and is applicable even when the state space of the Markov chain is countably infinite.
Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process
Authors: Kyprianou A. E. , Palmowski Z.
Published: July 2016

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.