bluuebunny/crossref_metadata_2025 huggingface dataset:
A stochastic process related to language change
Authors: Brainerd Barron
Published: April 2006

The purpose of this note is two-fold. First, to introduce the mathematical reader to a group of problems in the study of language change which has received little attention from mathematicians and probabilists. Secondly, to introduce a birth and death process, arising naturally out of this group of problems, which has received little attention in the literature. This process can be solved using the standard methods and the solution is exhibited here.
Optimal reinsurance
Authors: Dayananda P. W. A.
Published: April 2006

The fundamental principle underlying insurance is that the expected value of claims is equal to the premium. This was established by Bernoulli [4] in 1738. Subsequent work on the development of ‘risk theory’ led to research concerning the probability of ‘ruin’ of an insurance company. A critical study of these investigations was made by Borch [3] in a recent paper.
Queueing with alternating priorities, treated as random walk on the lattice in the plane
Authors: Yadin Micha
Published: April 2006

The queueing system with alternating priorities has been discussed recently in numerous articles (Maxwell [1], Avi Itzhak, Maxwell, Miller [2], Neuts, Yadin [3], and others). Consider a system which consists of several queueing units. Each unit is independent in the sense that it has its own waiting line which is generated by an independent Poisson stream of customers. However, all of the units are served by a single server who allocates his time between the units.
Limit theorems for the single server queue with traffic intensity one
Authors: Prabhu N. U.
Published: April 2006

We consider a single server queueing system with inter-arrival times {un,n≧ 1}, and service times {υn,n≧ 1} and the queue-discipline, ‘first-come, first-served’. It is assumed that {un} and {υn} are two independent renewal processes, and 0 >E(un) =a< ∞, 0 <E(υn) =b< ∞. The traffic intensity isPρb/a(0 > ρ > ∞). This paper is concerned with the case ρ = 1, where it is known that the various queueing processes such as the queue-lengthQ(t) and waiting timeW(t) diverge to + ∞ in distribution ast→ ∞. Borovkov [1], [2] and Brody [3] have obtained limit distributions forQ(t) andW(t) with appropriate location and scale parameters in the casesP≧ 1. Here we investigate random variables related to the busy and idle periods in the system. To explain our approach, we consider the random variablesXn=υn–un(n≧ 1). LetS0≡ 0,Sn=X1+X2+ ··· +Xn(n≧ 1), and define the sequence {Nk,k≧ 0} as
On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals
Authors: Kyprianou E. K.
Published: April 2006

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with meanλt, and the service time has distributiondB(t), 0 <t< ∞. LetW(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at timetwould have to wait before beginning his service. We also let the random variabledenote the first busy period initiated by a waiting timeuat timet= 0.
First passage to a general threshold for a process corresponding to sampling at Poisson times
Authors: Belkin Barry
Published: April 2006

The problem of computing the distribution of the time of first passage to a constant threshold for special classes of stochastic processes has been the subject of considerable study. For example, Baxter and Donsker (1957) have considered the problem for processes with stationary, independent increments, Darling and Siegert (1953) for continuous Markov processes, Mehr and McFadden (1965) for Gauss-Markov processes, and Stone (1969) for semi-Markov processes. The results, however, generally express the first passage distribution in terms of transforms which can be inverted only in a relatively few special cases, such as in the classical case of the Weiner process and for certain stable and compound Poisson processes. For linear threshold functions and processes with non-negative interchangeable increments the first passage problem has been studied by Takács (1957) (an explicit result was obtained by Pyke (1959) in the special case of a Poisson process). Again in the case of a linear threshold, an explicit form for the first passage distribution was found by Slepian (1961) for the Weiner process. For the Ornstein-Uhlenbeck process and certainU-shaped thresholds the problem has recently been studied by Daniels (1969).
On a characterization of the Poisson process
Authors: Srivastava R. C.
Published: April 2006

In a recent paper Rényi (1964) has obtained the following characterization of the Poisson process.
Selective interaction of two independent recurrent processes
Authors: Hoopen M. Ten , Reuver H. A.
Published: April 2006

Considered are two mutually independent recurrent processes each consisting of a time series of unitary stimuli. The durations of the intervals between the stimuli in each series are independent of each other and identically distributed with probability density functions φ (t) and ψ (t). Every stimulus of the ψ (t) process annihilates the next stimulus of the φ (t) process. The probability density function of the intervals of the transformed φ (t) process is derived for the case where either the φ (t) or the ψ (t) process is Poisson.
On the theory of birth, death and diffusion processes
Authors: Davis A. W.
Published: April 2006

Several authors have recently discussed the asymptotic properties of stochastic populations which diffuse randomly throughout a given region. Sevast'yanov ([8], [9]) has investigated the extinction probability of a Markovian population in a compact region with an absorbing boundary, his analysis being in terms of “generation times”. Adke and Moyal have considered the spatial dispersion of a population which multiplies according to a simple time-dependent birth-and-death process and undergoes Gaussian diffusion on the real line ([2] and [3]) or on a finite interval with reflecting boundaries [1]. A serious limitation in Adke and Moyal's asymptotic results is that they are conditional upon a finite number of survivors. Moyal [7] has also obtained some basic formulae for a Markovian population diffusing through a general space.
Asymptotic properties of the periodogram of a discrete stationary process
Authors: Olshen Richard A.
Published: April 2006

Suppose x1,…, xN are indefinitely many observations on a stochastic process which is weakly stationary with spectral density f(λ), – π ≦ λ ≦ π. An asymptotically unbiased, and to that extent plausible, estimate of 4rf(λ)is the periodogram

Yet the periodograms of processes which possess spectral densities are notoriously subject to erratic behavior.
Some probability distributions for neutron transport in a half-space
Authors: Mullikin T. W.
Published: April 2006

We will consider a problem which arises in the theories of neutron transport and radiative (photon) transfer. We will consider, in a particular case, the random motion of a particle in a scattering medium and determine asymptotics for the probability distribution of the random variable for the number of collisions before exit from the medium.
An Extension of the Hájek-Rényi inequality for the case without moment conditions
Authors: Heyde C. C.
Published: April 2006

In the paper [1], Hájek and Rényi established an inequality which they formulated in the following way: X1,X2,··· are independent random variables and . For each k, EXk = 0 and , while is a non-increasing sequence of positive numbers. Then, for any ε > 0 and any positive integers n and m (n < m),

The well-known Kolmogorov inequality is the particular case ck = 1, all k, and n = 1 of (1). It is the object of the present note to produce an extended version of (1) where no moment conditions need be satisfied. This provides a useful general bound and illuminates the role of certain standard techniques in the study of the almost sure behaviour of sums of independent random variables.
A theory of dams with continuous input and a general release rule
Authors: Moran P.A.P.
Published: April 2006

Consider an infinite capacity dam in which the input and release occur continuously in time. Write X(t) for the total input up to time t starting from X(0) = 0 at t = 0. Let Z(t) be the content of the dam at time t and R(u) (0 ≦ u < ∞) a release function such that in any interval of time (t, t + dt), the amount of water released is R(Z(t))dt + o(t) for any bounded realisation of the process {Z(t)}. Thus R(u) can be regarded as a “rate of release”.
A Note on the First Emptiness Problem of a Finite Dam with Poisson Type Inputs
Authors: Phatarfod R. M.
Published: April 2006

This paper is concerned with the problem of first emptiness in a continuous time dam model formulated by Gani and Prabhu (1959) based on Moran's (1954) discrete time dam model. Briefly the dam model is as follows: The dam is of finite capacity K, whose content 0 ≦ Z(t) ≦ K is defined in continuous time t (0 ≦ t < ∞) by the equation

where ηδt is the time the dam is empty in (t, t + δt). X(t) represents the input into the dam during time t, a Poisson process with parameter λ, such that in a small interval of time δt, the quantity δX(t) = 0 or h (< K) may be added to the dam content; min{Z(t) + δX(t),K} indicates that there will be an overflow whenever Z(t) + δX(t) > K, leaving only the amount K in the dam, and (1-η)δt represents a continuous release occurring at a steady unit rate except when z(t) = 0, when there is no release.
On a stochastic model of an epidemic
Authors: Ridler-Rowe C. J.
Published: April 2006

The epidemic model considered here, first given by Bartlett (see for example [2]), provides for the immigration of new susceptibles and infectives, as well as describing the spread of infection to susceptibles already present and the removal of infectives. The epidemic curve, relating the numbers of susceptibles and infectives, has been studied for certain cases by Bartlett [1], Kendall [6] and others, and provides a motivation for the results given here. With the aid of criteria given by Reuter [8], [9], the main question considered is the asymptotic behaviour of the mean duration of the epidemic. The behaviour of the limits of the transition probabilities pij(t) as t → ∞ is also investigated.
Recurrent composite events
Authors: Leslie R.T.
Published: April 2006

On a sequence of Bernoulli trials, the definition of a recurrent event ε involves the occurrence of a unique pattern of successes (S) and failures (F), the final element of which is the result of the nth trial. Success runs are the best known of such recurrent events, but Feller (1959, §13.8) mentions more complicated patterns, among which two types may be distinguished. The simpler involves a single more complex pattern such as SSFFSS; the second type involves a set of alternative events defining ε, which is said to occur when any one of the alternatives occurs at trial number n. Thus if ε stands for “either a success run of length r or a failure run of length ρ”, there are two alternatives in the set; the problem is elementary because the component events are “non-overlapping”.
The spectrum of intervals of a geometric branching poisson process
Authors: Gilles D. C. , Lewis P. A. W.
Published: April 2006

A summation formula is given for computing the spectrum of intervals of a geometric branching Poisson process from the cumulant generating function of the associated counting process.
Assembly-like queues
Authors: Harrison J. Michael
Published: April 2006

A queueing theoretic model of an assembly operation is introduced. The model, consisting of K ≧ 2 renewal input processes and a single server, is a multiple input generalization of the GI/G/1 queue. The server requires one input item of each type k = 1,…, K for each of his services. It is shown that the model is inherently unstable in the following sense. The associated vector waiting time process Wn cannot converge in distribution to a non-defective limit, regardless of how well balanced the input and service processes may be. Limit theorems are developed for appropriately normalized versions of Wn under the various possible load conditions. Another waiting time process, equivalent to that in a single-server queue whose input is the minimum of K renewal processes, is also identified. It is shown to converge in distribution to a particular limit under certain load conditions.
Optimal issuing policies under stochastic field lives
Authors: Brown Mark , Solomon Herbert
Published: April 2006

We consider a stockpile in which each item has a stochastic field life with known distribution, this distribution changing in time in a known deterministic fashion. The problem is to order the items for issuance to the field with the objective of maximizing the total field life. Under monotone likelihood ratio assumptions on the field life distributions, and convexity and concavity assumptions on the change in time of the distributions, we show that LIFO and FIFO policies possess a strong optimality property.
Expectations and variances of stopping variables in sequential selection processes
Authors: Henke M.
Published: April 2006

A sequential stochastic decision process with independent random variables is considered in which the decision maker selects a chance with a certain probability at each time period or at random times. If the decision maker has selected m chances, the process has to be stopped. The expectation and the variance of the stopping variable are determined for a finite and an infinite decision horizon.
Some non-stationary point processes with stationary forward recurrence time distribution
Authors: Rudemo Mats
Published: April 2006

Examples are given of point processes that are non-stationary but have stationary forward recurrence time distributions. They are obtained by modification of stationary Poisson and renewal processes.
Stationary measures for multitype branching processes
Authors: Hoppe Fred
Published: April 2006

The multitype Galton-Watson process is considered both with and without immigration. Proofs are given for the existence of invariant measures and their uniqueness is examined by functional equation methods. Theorem 2.1 proves the uniqueness, under certain conditions, of solutions of a multidimensional Schröder equation. Regular variation is shown to play a role in the multitype theory.
Extremal processes and random measures
Authors: Shorrock R. W.
Published: April 2006

Discrete time extremal processes with a continuous underlying c.d.f. are random measures which can be viewed as two-dimensional Poisson processes and this representation is used to obtain the conditional law of the sequence of states the process passes through (upper record values) given the sequence of holding times in states (inter-record times). In addition the Gamma processes (which lead to the Ferguson Dirichlet processes) and a random measure that arises in sampling from a biological population are discussed as two-dimensional Poisson processes.
Orderliness, intensities and Palm-Khinchin equations for multivariate point processes
Authors: Daley D. J. , Milne R. K.
Published: April 2006

Simple definitions and derivations of elementary properties are given for the various intensities and Palm-Khinchin functions associated with a multivariate point process.
On a characterization of information improvement
Authors: Autar Ram
Published: April 2006

The object of this paper is to characterize an information theoretic measure associated with three probability distributions known as ‘Information Improvement’ through a functional equation which arises by considering the additive property of the measure. This measure has been extensively used in economic analysis.
Malthusian behaviour of the critical and subcritical age-dependent branching processes with arbitrary state space
Authors: Saunders D. I.
Published: April 2006

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.
Random record models
Authors: Gaver Donald P.
Published: April 2006

We study record times, mainly, and sizes in the following context. LetXndenote the size of thenth event occurring in a point stochastic pacing process 𝒫. TheXnis i.i.d. and 𝒫 is, variously, Poisson, negative binomial, renewal and Furry. Explicit distributions of first-record times are found, domains of attraction studied and the asymptotic lognormality of thenth-record time is shown for Poisson 𝒫.
Some remarks on a result of Blomqvist
Authors: Speed T. P.
Published: April 2006

An alternative proof of a result of Blomqvist on the “heavy-traffic” problem for the two-barrier random walks using weak convergence methods is indicated.
Physical nearest-neighbour models and non-linear time-series: III Non-zero means and sub-critical temperatures
Authors: Bartlett M. S.
Published: April 2006

The product moment equations previously derived are first discussed for the infinite one- and two-dimensional Ising (or autologistic) model in the case of non-zero mean, as a prelude to an examination of the probability structure in the higher-dimensional (and nominally zero mean) casebelowthe ‘critical temperature’. Of two simple possible models, A and B, both consistent with the division of the product momentpinto ergodic, and long-range non-ergodic, components, such thatρ=r(1 –m2) +m2, whereris the intrinsic correlation coefficient, it is shown that the second model B appears appropriate to the three-dimensional ‘spherical model’, but the first model A to the Ising model. Model A is defined byxi= yi+M, whereM= +mor –m, andE{yi} = 0; and Model B by
Quasi-stationary distributions for Markov chains on a general state space
Authors: Tweedie Richard. L.
Published: April 2006

The quasi-stationary behaviour of a Markov chain which is φ-irreducible when restricted to a subspace of a general state space is investigated. It is shown that previous work on the case where the subspace is finite or countably infinite can be extended to general chains, and the existence of certain quasi-stationary limits as honest distributions is equivalent to the restricted chain being R-positive with the unique R-invariant measure satisfying a certain finiteness condition.
Transience and recurrence of an interesting Markov chain
Authors: Letac Gérard
Published: April 2006

This note studies the natural extension to the countable case of a chain considered by Hendricks (1972) and gives necessary and sufficient conditions for transience, null recurrence and positive recurrence.
Some limit theorems for a class of network problems as related to finite Markov chains
Authors: Nakamura Masao
Published: April 2006

This paper is concerned with a class of dynamic network flow problems in which the amount of flow leaving node i in one time period for node j is the fraction pij of the total amount of flow which arrived at node i during the previous time period. The fraction pij whose sum over j equals unity may be interpreted as the transition probability of a finite Markov chain in that the unit flow in state i will move to state j with probability pij during the next period of time. The conservation equations for this class of flows are derived, and the limiting behavior of the flows in the network as related to the properties of the fractions Pij are discussed.
Analytic solution of a finite dam governed by a general input
Authors: Srinivasan S. K.
Published: April 2006

A stochastic model of a finite dam in which the epochs of input form a renewal process is considered. It is assumed that the quantities of input at different epochs and the inter-input times are two independent families of random variables whose characteristic functions are rational functions. The release rate is equal to unity. An imbedding equation is set up for the probability frequency governing the water level in the first wet period and the resulting equation is solved by Laplace transform technique. Explicit expressions relating to the moments of the random variables representing the number of occasions in which the dam becomes empty as well as the total duration of the dry period in any arbitrary interval of time are indicated for negative exponentially distributed inter-input times.
Crossing problems for non-constant thresholds and certain non-Markov processes
Authors: Belkin Barry , Stone Lawrence D. , Snyder Martin Avery
Published: April 2006

A result in Stone, Belkin, and Snyder ((1970)J.Math. Anal. Appl.30,448–470) gave a method for finding the Laplace-Stieltjes transform of the distribution of certain non-negative, homogeneous, additive functionals of a Markov process with stationary transition measure. By considering certain two dimensional Markov processes and applying this result, a method is obtained for finding time above a threshold and first passage distributions for a one dimensional process either when (1) the process is Markovian and the threshold is possibly non-constant, or (2) the threshold is constant and the process is the indefinite integral of a Markov process. Specific process-threshold combinations are considered in several examples including the case of a linear threshold for a Wiener process and a for compound Poisson process with exponential (either one-sided or bilateral) after-jump distribution. In addition, the first passage distribution to a constant threshold is computed for an integrated Poisson sampling process.
Physical nearest-neighbour models and non-linear time-series. II Further discussion of approximate solutions and exact equations
Authors: Bartlett M. S.
Published: April 2006

The approximate two- and three-dimensional solutions for spatial correlations, using the non-linear time-series approach for nearest-neighbour systems developed in my previous paper, are further discussed.

Orthogonal expansions for the correlation functions are also developed which determine with this approach, though so far only in principle, the exact solutions.
On the first-exit time problem for temporally homogeneous Markov processes
Authors: Tuckwell Henry C.
Published: April 2006

Using an integral equation of Darling and Siegert in conjunction with the backward Kolmogorov equation for the transition probability density function, recurrence relations are derived for the moments of the time of first exit of a temporally homogeneous Markov process from a set in the phase space. The results, which are similar to those for diffusion processes, are used to find the expectation of the time between impulses of a Stein model neuron.
Geometrical measures of the smoothness of random functions
Authors: Switzer Paul
Published: April 2006

For stationary isotropic random functions on a Euclidean space, we characterize and compare the mean values of certain geometric measures of the smoothness of realizations. In particular we examine mean properties of the contours and gradients of the random function, and the effect of local averaging on smoothness in special cases.
Optimal replacement with semi-Markov shock models
Authors: Feldman Richard M.
Published: April 2006

Consider a system that is subject to a sequence of randomly occurring shocks; each shock causes some damage of random magnitude to the system. Any of the shocks might cause the system to fail, and the probability of such a failure is a function of the sum of the magnitudes of damage caused from all previous shocks.

The purpose of this paper is to derive the optimal replacement rule for such a system whose cumulative damage process is a semi-Markov process. This allows for both the time between shocks and the damage due to the next shock to be dependent on the present cumulative damage level.

Only policies within the class of control-limit policies will be considered; namely, policies with which no action is taken if the damage is below a fixed level, and a replacement is made if the damage is above that.

An example will be given illustrating the use of the optimal replacement rule.
The rates of growth of the Galton-Watson process in varying environment
Authors: Foster James H. , Goettge Robert T.
Published: April 2006

It is shown that a Galton-Watson process in varying environment, unlike a classical Galton-Watson process, can have an infinite (though at most countable) number of distinct rates of growth.
On the distribution of attributes in organizations
Authors: Störmer Horand
Published: April 2006

The paper deals with a stochastic organization model described by a generalized Poisson input process and given probabilities of entering individuals obtaining certain attributes in this organization. The distribution of attributes at any given instant is derived. Examples illustrate the application of obtained results.
The periodicity of endemic malaria
Authors: Radcliffe J.
Published: April 2006

In many endemic areas the level of infection of malaria exhibits a regular cycle. A stochastic model is formulated to describe endemic malaria and is used to justify a set of deterministic equations. The model incorporates the properties peculiar to malaria, i.e., in endemic areas malaria is not usually fatal and an individual can be and usually is reinfected. The theory of small oscillations is applied to the deterministic equations, and under the assumption of stable populations of hosts and vectors an expression is obtained for the period of oscillation of the infection.
Notes on “Estimation theory for growth and immigration rates in a multiplicative process”
Authors: Heyde C. C. , Seneta E.
Published: April 2006

Some minor corrections to Heyde and Seneta (1972) are made, and new convergence rate results given. Estimation by recurrence methods is discussed, as announced earlier.
Uniform coupling of non-homogeneous Markov chains
Authors: Griffeath David
Published: April 2006

The Markov-Dobrushin condition for (weak) ergodicity of non-homogeneous discrete-time Markov chains, and an analogous criterion for continuous chains, are derived by means of coupling techniques.
Limiting diffusions for the conditioned M/G/1 queue
Authors: Kennedy Douglas P.
Published: April 2006

The virtual waiting time process, W(t), in the M/G/1 queue is investigated under the condition that the initial busy period terminates but has not done so by time n ≥ t. It is demonstrated that, as n → ∞, W(t), suitably scaled and normed, converges to the unsigned Brownian excursion process or a modification of that process depending whether ρ ≠ 1 or ρ = 1, where ρ is the traffic intensity.
On the distribution of records in a finite sequence of observations, with an application to a road traffic problem
Authors: Haghighi-Talab Dariush , Wright Christopher
Published: April 2006

Probability distributions relating to the number and positions of lower records in a finite sequence of observations are obtained by methods involving the permutation of ranks. Some new results, and results of earlier authors, are expressed in a form thought to be useful for direct practical applications.

Expressions are also given for the distribution of the length of a ‘record run’, which consists of the observations following and including a given record either up to the next record, or, if a further record does not occur, up to the end of the sequence of observations. This quantity, whose distribution is closely related to that of the inter-record time, has a direct and simple application to road traffic leaving a signal which has been confirmed by experimental observation.
On a model for storage and search
Authors: Burville P. J. , Kingman J. F. C.
Published: April 2006

Books are arranged on a shelf. After use, a book is returned to the left hand end of the shelf. Expressions are derived for the probability that, in equilibrium, a given book is in a given position on the shelf, in terms of the frequencies with which the different books are demanded.
Estimation theory for growth and immigration rates in a multiplicative process
Authors: Heyde C. C. , Seneta E.
Published: April 2006

This paper deals with the simple Galton-Watson process with immigration, {Xn} with offspring probability generating function (p.g.f.)F(s) and immigration p.g.f.B(s), under the basic assumption that the process is subcritical (0 <m≡F'(1–) < 1), and that 0 <λ≡B'(1–) < ∞, 0 <B(0) < 1, together with various other moment assumptions as needed. Estimation theory for the ratesmandλon the basis of a single terminated realization of the process {Xn} is developed, in that (strongly) consistent estimators for bothmandλare obtained, together with associated central limit theorems in relation tomandμ≡λ(1–m)–1Following this, historical antecedents are analysed, and some examples of application of the estimation theory are discussed, with particular reference to the continuous-time branching process with immigration. The paper also contains a strong law for martingales; and discusses relation of the above theory to that of a first order autoregressive process.
On the inverse of the first passage time probability problem
Authors: Capocelli R. M. , Ricciardi L. M.
Published: April 2006

Since the pioneering work of Siegert (1951), the problem of determining the first passage time distribution for a preassigned continuous and time homogeneous Markov process described by a diffusion equation has been deeply analyzed and satisfactorily solved. Here we discuss the “inverse problem” — of applicative interest — consisting in deciding whether a given function can be considered as the first passage time probability density function for some continuous and homogeneous Markov diffusion process. A constructive criterion is proposed, and some examples are provided. One of these leads to a singular diffusion equation representing a dynamical model for the genesis of the lognormal distribution.
Conditional Poisson processes
Authors: Serfozo Richard F.
Published: April 2006

A conditional Poisson process (often called a double stochastic Poisson process) is characterized as a random time transformation of a Poisson process with unit intensity. This characterization is used to exhibit the jump times and sizes of these processes, and to study their limiting behavior. A conditional Poisson process, whose intensity is a function of a Markov process, is discussed. Results similar to those presented can be obtained for any process with conditional stationary independent increments.
Processes with conditional stationary independent increments
Authors: Serfozo Richard F.
Published: April 2006

We study a class of processes which are essentially processes with stationary independent increments whose basic parameters are allowed to vary randomly over time. These processes are equivalent to random time transformations of processes with stationary independent increments where the time process is independent of the original process. Several limiting theorems are presented including weak and strong laws of large numbers and a functional central limit theorem.
Inequalities with application in economic risk analysis
Authors: Agnew Robert A.
Published: April 2006

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.
A characterization of the exponential distribution
Authors: Ahsanullah M. , Rahman M.
Published: April 2006

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.
The dispersion of a neutral allele considered as a branching process
Authors: Crump Kenny S. , Gillespie John H.
Published: April 2006

The spatial dispersion of a neutral allele is described using the theory of multitype branching processes in which the types represent colonies between which individuals can migrate. Each mutant individual averages less than one offspring, so the mutant population faces certain extinction. Expressions are given for the first two moments of the total number of individuals to visit specified colonies in one, two and three dimensions. Data fromDrosophilapopulations are used to show the improbability of the same neutral allele occurring at widely separated localities.
Excursions above a fixed level by n-dimensional random fields
Authors: Adler Robert J.
Published: April 2006

For an n-dimensional random field X(t) we define the excursion set A of X(t) by A = [t ∊ S: X(t) ≧ u] for real u and compact S ⊂ Rn. We obtain a generalisation of the number of upcrossings of a one-dimensional stochastic process to random fields via a characteristic of the set A related to the Euler characteristic of differential topology. When X(t) is a homogeneous Gaussian field satisfying certain regularity conditions we obtain an explicit formula for the mean value of this characteristic.
Stochastic pursuit-evasion games with information lag. II. Observation with error
Authors: Schreuer Mynda
Published: April 2006

A discrete-time, scalar, pursuit-evasion game is considered in which an evader, moving according to a stationary stochastic process, is continually being tracked by a pursuer. The pursuer's observations of the evader's positions are subject to observation error and time lag. It is shown that when the pursuer's strategy is linear in his information, the evader is able to infer the values of the observation errors (subject to the same time lag) and it is assumed that he utilises this information linearly in his evasion strategy. The payoff is taken to be the mean-square distance between pursuer and evader, the latter being subject to a quadratic constraint on his movement. The upper value and the corresponding strategies are found for all values of the varianceθof the observation error. The lower value problem is solved completely only for variancesθbelow a certain critical value and partially for variances above this critical value.
The convergence of a position-dependent branching process used as an approximation to a model describing the spread of an epidemic
Authors: Radcliffe J.
Published: April 2006

A supercritical position-dependent Markov branching process has been used as an approximation to a model describing the initial geographical spread of a measles epidemic (Bartlett (1956)). Let α be its Malthusian parameter, ß its velocity of propagation, Z(A, t) the number of individuals in the set A at time t, and A√(ßt) = [√(ßt) r: r ∈ A]. The mean square convergence of the random variable W(A, t)= e–αtZ(A√(ßt), t) to a limit variable W(A) is established.
The size of the region occupied by one type in an invasion process
Authors: Sudbury Aidan
Published: April 2006

Particles are situated on a rectangular lattice and proceed to invade each other's territory. When they are equally competitive this creates larger and larger blocks of one type as time goes by. It is shown that the expected size of such blocks is equal to the expected range of a simple random walk.
On exponential bounds for the waiting-time distribution function inGI/G/1
Authors: Bergmann R. , Stoyan D.
Published: April 2006

Exponential bounds for the stationary waiting-time distribution of the typeae–θtare considered. These bounds are obtained by the use of Kingman's method of ‘integral inequalities’. Approximations of Θ andaare given which are useful especially if the service and/or inter-arrival time distribution functions are NBUE or NWUE.
The capacity of an intersection with non-stationary traffic
Authors: Wegmann Helmut
Published: April 2006

The average number of vehicles being able to enter an intersection per time unit from a minor road with a stop or yield sign — the capacity of the intersection — depends on the density of the traffic stream on the major road. In case the time-process of the major road traffic at the intersection is a non-homogeneous Poisson process with a periodic intensity function the capacity is calculated and compared with the capacity in the homogeneous case.
On the waiting-time and busy period distributions for a general birth-and-death queueing model
Authors: Natvig Bent
Published: April 2006

A general birth-and-death queueing model is considered with N waiting positions (0 ≦ N ≦ ∞), s servers (1 ≦ s ≦ ∞) and a first-come-first-served queueing discipline. The first and second order moments of the steady-state waiting-time (excluding service) for a non-lost arriving customer are given. By setting the busy period equal to the time where at least one service is in progress, we obtain the first and second order moments of the length of a busy period and also the distribution of the number served during it, given an arbitrary number of customers present originally. Using a direct approach all expressions are given in explicit forms which, although being far from elegant, are suitable for evaluation on a computer.
An absorption probability for the Ornstein-Uhlenbeck process
Authors: Dirkse John P.
Published: April 2006

An asymptotic expression for an absorption probability for the Ornstein-Uhlenbeck process is presented along with an application of the result to a problem in optional stopping. The relation of this result to the asymptotic behavior of a weighted Kolmogorov-Smirnov statistic is also discussed. Sweet and Hardin (1970) derive an exact solution (not in closed form) for this same problem.
On the reversible counters system of Lampard
Authors: Phatarfod R. M.
Published: April 2006

The paper considers some properties of the reversible counters system of Lampard [1]. Lampard considers the properties of the process imbedded at the epochs when the counter receiving pulses from the ‘down’ process reaches zero. We consider here the time-dependent and stationary distributions of the original process. Their derivation gives rise to some interesting functional equations, besides being useful for studying queues with Markov-dependent inter-arrival or service times.
Count distributions, orderliness and invariance of Poisson cluster processes
Authors: Ammann Larry P. , Thall Peter F.
Published: April 2006

The probability generating functional (p.g.fl.) of a non-homogeneous Poisson cluster process is characterized in Ammann and Thall (1977) via a decomposition of the KLM measure of the process. This p.g.fl. representation is utilized in the present article to show that the family 𝒟 of Poisson cluster processes with a.s. finite clusters is invariant under a class of cluster transformations. Explicit expressions for the finite-dimensional count distributions, product moment measures, and the distribution of clusters are derived in terms of the KLM measure. It is also shown that an element of 𝒟 has no multiple events iff the points of each cluster are a.s. distinct.
Rates of Poisson convergence for U-statistics
Authors: Brown T. C. , Silverman B. W.
Published: April 2006

Poisson limit theorems for U-statistics are studied. A general rate of convergence is obtained; this rate is improved for the special case where the U-statistic arises from the consideration of distances between uniformly distributed points in a well-behaved plane region.
On estimation of the integrals of certain functions of spectral density
Authors: Taniguchi Masanobu
Published: April 2006

Let g(x) be the spectral density of a Gaussian stationary process. Then, for each continuous function ψ (x) we shall give an estimator of whose asymptotic variance is O(n–1), where Φ(·) is an appropriate known function. Also we shall investigate the asymptotic properties of its estimator.
A martingale approach to central limit theorems for exchangeable random variables
Authors: Weber N. C.
Published: April 2006

In this paper it will be shown that by an appropriate choice of σ-fields, martingale methods can be used to obtain simple proofs of many of the central limit theorems known for triangular arrays of exchangeable random variables.
Normal approximations for binary lattice systems
Authors: Kryscio Richard J. , Saunders Roy , Funk Gerald M.
Published: April 2006

Consider an array of binary random variables distributed over anm1(n) bym2(n) rectangular lattice and letY1(n) denote the number of pairs of variablesd, units apart and both equal to 1. We show that if the binary variables are independent and identically distributed, then under certain conditionsY(n) = (Y1(n), · ··,Yr(n)) is asymptotically multivariate normal fornlarge and r finite. This result is extended to versions of a model which provide clustering (repulsion) alternatives to randomness and have clustering (repulsion) parameter values nearly equal to 0. Statistical applications of these results are discussed.
On the Kesten–Spitzer algorithm for minimal hitting time in controlled Markov chains
Authors: Bhattacharjee Manish C. , Basu Sujit K.
Published: April 2006

For a Markov chain with optional transitions, except for those to an arbitrary fixed state accessible from all others, Kesten and Spitzer proved the existence of a control policy which minimized the expected time to reach the fixed state and for constructing an optimal policy, proposed an algorithm which works in certain cases. For the algorithm to work they gave a sufficient condition which breaks down if there are countably many states and the minimal hitting time is bounded. We propose a modified algorithm which is shown to work under a weaker sufficient condition. In the bounded case with countably many states, the proposed sufficient condition is not necessary but a similar condition is. In the unbounded case as well as when the state space is finite, the proposed condition is shown to be both necessary and sufficient for the original Kesten–Spitzer algorithm to work. A new iterative algorithm which can be used in all cases is given.
Failure distributions of shock models
Authors: Gottlieb Gary
Published: April 2006

A single device shock model is studied. The device is subject to some damage process. Under the assumption that as the cumulative damage increases, the probability that any additional damage will cause failure increases, we find sufficient conditions on the shocking process so that the life distribution will be increasing failure rate.
Subdivided populations: the characteristic values and rate of loss of alleles
Authors: Gladstien Keith
Published: April 2006

The characteristic values of certain matrices (underlying multiple allele models in population genetics) are investigated and related to the asymptotic behavior of Pr(at leastkdistinct alleles in the population at timet).
Dams with additive inputs revisited
Authors: Pegg P. A. , Phatarfod R. M.
Published: April 2006

In this paper we consider a finite dam with unit release. Ratio identities are derived for the various random walks involved, and from them are derived the p.g.f. of the time of first emptiness with and without overflow and the complete time-dependent distribution of the dam content.
An application of the Jordan canonical form to the epidemic problem
Authors: Gilbert Linda P. , Johnson A. M.
Published: April 2006

The stochastic version of the simple epidemic is considered in this paper. A method which has computational advantages is developed for computing the probability distribution of the process. The method is based on the Jordan canonical form of the matrix representing the system of differential-difference equations for the simple epidemic. It is also shown that the method can be applied to the general epidemic as well as any other right-shift process.
An autoregressive model for multilag Markov chains
Authors: Pegram G. G. S.
Published: April 2006

By assembling the transition matrix of a finite discrete Markov chain from overlays of matrices which are defined only by the serial correlation coefficients and marginal distribution of the chain to be modelled, a considerable saving is made in the number of parameters required to define a multilag Markov chain. This parsimony is achieved without detriment to the marginal distribution or serial correlation structure of the modelled chain. Applications to daily precipitation sequences and reservoir reliability are outlined to demonstrate the model's versatility.
On the asymptotic geometrical behaviour of percolation processes
Authors: Schürger Klaus
Published: April 2006

In this paper the global behaviour of percolation processes on thed-dimensional square lattice is studied. Using techniques of Richardson (1973) we prove, under weak moment assumptions on the time coordinate distribution, the following result. There exists a norm N(·) onRdsuch that, for all 0 <ε< 1, we have that almost surely for all sufficiently largettheN-ball of radius (1 –ε)tis contained inη, (the set of all sites occupied by timet) andη, is contained in theN-ball of radius (1 +ε)t. Richardson (1973) derived the corresponding ‘in probability' result for a class of spread processes onRd, satisfying certain conditions.
On the quantogram of Kendall and Kent
Authors: Csörgő Sándor
Published: April 2006

The ‘snake' theorem, i.e., the weak convergence of the complex quantogram, is studied. The moment assumption is weakened by a strong approximation result of the author.
Place probabilities in normal order statistics models for horse races
Authors: Henery R. J.
Published: April 2006

Independent observationsX0,X1…,XN+1are drawn from each ofNpopulations whose distribution functionsF(x – θi) have meansθi, 0 ≦i<N, and we wish to calculate the probabilityPk;NthatX0is thekth largest observation. For normal populations an approximation is given forPK;Nbased on a Taylor series expansion in theθ's. IfF(x) has an increasing failure rate, as is the case for the normal, an upper bound can be given for the ‘win' probabilityP1;NSome moment relations for normal order statistics are also given.
Record values and the exponential distribution
Authors: Dallas A. C.
Published: April 2006

Let X1, X2, · ·· be i.i.d. and Yk = max{X1, X2, · ··, Xk}. Then Xj is a record value of the sequence if Yj > Yj-1). Denoting record value times by Ln, n = 1, 2, ···, we prove that the independence of XLj − XLi and XLi with Lj > characterizes the exponential distribution.
Asymptotic analysis and local lattice central limit theorems
Authors: Moran P.A.P.
Published: April 2006

Methods of evaluating the coefficients of high powers of functions defined by power series with positive coefficients are considered. Such methods, which were originally used by Laplace, can, for example, be used to obtain asymptotic formulae for Stirling numbers. They are equivalent to using local lattice central limit theorems. An alternative method using direct numerical integration on a contour integral giving the required coefficient is described. Exact bounds for the accuracy of this method can often be obtained by considerations of the unimodality of discrete distributions. The results are illustrated using convolutions of the rectangular and logarithmic distributions.
Convex majorization with an application to the length of critical paths
Authors: Meilijson Isaac , Nádas Arthur
Published: April 2006

1.

(Y) for all non-negative, non-decreasing convex functions φ (X is convexly smaller than Y) if and only if, for all .

2.

Let H be the Hardy–Littlewood maximal function HY(x) = E(Y – X | Y > x). Then HY(Y) is the smallest random variable exceeding stochastically all random variables convexly smaller than Y.

3.

Let X1X2 · ·· Xn be random variables with given marginal distributions, let I1,I2, ···, Ik be arbitrary non-empty subsets of {1,2, ···, n} and let M = max (M is the completion time of a PERT network with paths Ij, and delay times Xi.) The paper introduces a computation of the convex supremum of M in the class of all joint distributions of the Xi's with specified marginals, and of the ‘bottleneck probability' of each path.