bluuebunny/crossref_metadata_2025 huggingface dataset:
Applications of Borovkov's Renovation Theory to Non-Stationary Stochastic Recursive Sequences and Their Control
Authors: Altman Eitan , Hordijk Arie
Published: April 2006

We investigate in this paper the stability of non-stationary stochastic processes, arising typically in applications of control. The setting is known as stochastic recursive sequences, which allows us to construct on one probability space stochastic processes that correspond to different initial states and even different control policies. It does not require any Markovian assumptions. A natural criterion for stability for such processes is that the influence of the initial state disappears after some finite time; in other words, starting from different initial states, the process will couple after some finite time to the same limiting (not necessarily stationary nor ergodic) stochastic process. We investigate this as well as other types of coupling, and present conditions for them to occur uniformly in some class of control policies. We then use the coupling results to establish new theoretical aspects in the theory of non-Markovian control.
Products of Irreducible Random Matrices in the (Max, +) Algebra
Authors: Mairesse Jean
Published: April 2006

We consider the recursive equationx(n+ 1)=A(n)⊗x(n), wherex(n +1) andx(n) are ℝk-valued vectors andA(n) is an irreducible random matrix of sizek × k.The matrix-vector multiplication in the (max, +) algebra is defined by (A(n)⊗x(n))= maxj(Aij(n) +xj(n)). This type of equation can be used to represent the evolution of stochastic event graphs which include cyclic Jackson networks, some manufacturing models and models with general blocking (such as Kanban). Let us assume that the sequence {A(n),n∈ ℕ} is i.i.d. or, more generally, stationary and ergodic. The main result of the paper states that the system couples in finite time with a unique stationary regime if and only if there exists a set of matricessuch thatand the matriceshave a unique periodic regime.
Random mosaics with cells of general topology
Authors: Cowan Richard , Tsang Albert K. L.
Published: April 2006

This paper considers a structure, named a ‘random partition process’, which is a generalisation of a random tessellation. The cells, possibly multi-part and with holes, have a general topology summarised by the Euler characteristic. Vertices of all orders are allowed. Using the tools of ergodic theory, all of the formulae, from the traditional theory of random tessellations with convex cells, are generalised. Some motivating examples are given.
On projected thick sections of stationary point processes and Boolean models
Authors: Kiderlen Markus
Published: April 2006

For a stationary point process X of convex particles in ℝd the projected thick section process X(L) on a q-dimensional linear subspace L is considered. Formulae connecting geometric functionals, e.g. the quermass densities of X and X(L), are presented. They generalize the classical results of Miles (1976) and Davy (1976) which hold only in the isotropic case.
Further random tessellations with the classic Poisson polygon distributions
Authors: Miles Roger E. , Mackisack Margaret S.
Published: April 2006

It is well-known that Poisson lines in the plane, with orientation distribution Θ on [0, π), generate a (random) tessellation M0 of (random) convex polygons whose characteristics (area, perimeter, etc.) conform, in an ergodic sense, to a certain class {DΘ} of distributions.
Estimation of oriented direction distribution of a planar body
Authors: Rataj Jan
Published: April 2006

Methods of estimation of the oriented direction distribution (i.e. the distribution of unit outer normals over the boundary) of a planar set from the convex ring are proposed. The methods are based on an estimation of the area dilation of the investigated set by chosen test sets.
Extremal problems for geometric probabilities involving convex bodies
Authors: Bauer Christina , Schneider Rolf
Published: April 2006

The theory of geometric probabilities is concerned with randomly generated geometric objects. The aim is to compute probabilities of certain geometric events or distributions of random variables defined in a geometric way. Very often the computation even of simple expectations is too difficult, and one has to be satisfied with establishing estimates and, if possible, sharp inequalities. In geometric probabilities, convex sets play a prominent role, since often the convexity assumptions simplify the situation considerably. Extremal problems for geometric probabilities involving convex bodies can sometimes be attacked successfully by using suitable integral-geometric transformations and then applying classical inequalities from the geometry of convex bodies, or known methods for obtaining such inequalities. Examples of such results are the topic of this paper.
Inequalities for the anisotropic Poisson polytope
Authors: Mecke J.
Published: April 2006

The typical cell of a stationary Poisson hyperplane tessellation in thed-dimensional Euclidean space is called the Poisson polytope, and the cell containing the origin is called the Poisson 0-polytope. The intention of the paper is to show that the cells of the anisotropic tessellations are in some sense larger than those of the isotropic tessellations. Under the condition of equal intensities, it is proved that the moments of ordern= 1, 2, … for the volume of the Poisson 0-polytope in the anisotropic case are not smaller than the corresponding moments in the isotropic case. Similar results are derived for the Poisson polytope. Finally, generalizations are mentioned.
Limit laws for a stochastic process and random recursion arising in probabilistic modelling
Authors: Rachev Svetlozar T. , Samorodnitsky Gennady
Published: April 2006

We study certain stochastic processes arising in probabilistic modelling. We discuss the limit behavior of these processes and estimate the rate of convergence to the limit.
Measure generation by Euler functionals
Authors: Ambartzumian R. V.
Published: April 2006

Guided by analogy with Euler's spherical excess formula, we define a finite-additive functional on bounded convex polygons in ℝ2 (the Euler functional). Under certain smoothness assumptions, we find some sufficient conditions when this functional can be extended to a planar signed measure. A dual reformulation of these conditions leads to signed measures in the space of lines in ℝ2. In this way we obtain two sets of conditions which ensure that a segment function corresponds to a signed measure in the space of lines. The latter conditions are also necessary.
A central limit theorem for the isotropic random sphere
Authors: Brown Jason J.
Published: April 2006

Let be a real-valued, homogeneous, and isotropic random field indexed in . When restricted to those indices with , the Euclidean length of , equal to r (a positive constant), then the random field resides on the surface of a sphere of radius r. Using a modified stratified spherical sampling plan (Brown (1993)) on the sphere, define to be a realization of the random process and to be the cardinality of . Without specifying the dependence structure of nor the marginal distribution of the , conditions for asymptotic normality of the standardized sample mean, , are given. The conditions on and are motivated by the ideas and results for dependent stationary sequences.
On steady-state queue size distributions of the discrete-time GI/G/1 queue
Authors: Yang Tao , Chaudhry M. L.
Published: April 2006

In this paper, we present results for the steady-state system length distributions of the discrete-time GI/G/1 queue. We examine the system at customer arrival epochs (customer departure epochs) and use the residual service time (residual interarrival time) as the supplementary variable. The embedded Markov chain is of GI/M/1 type if the embedding points are arrival epochs and is of M/G/1 type if the embedding points are departure epochs. Using the matrix analytic method, we identify the necessary and sufficient condition for both Markov chains to be positive recurrent. For the GI/M/1 type chain, we derive a matrix-geometric solution for its steady-state distribution and for the M/G/1 type chain, we develop a simple linear transformation that relates it to the GI/M/1 type chain and leads to a simple analytic solution for its steady-state distribution. We also show that the steady-state system length distribution at an arbitrary point in time can be obtained by a simple linear transformation of the matrix-geometric solution for the GI/M/1 type chain. A number of applications of the model to communication systems and numerical examples are also discussed.
Distributions of projective invariants and model-based machine vision
Authors: Mardia K. V. , Goodall Colin , Walder Alistair
Published: April 2006

In machine vision, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, thecross-ratiois the simplest statistic which is invariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means. The case of identical means, which has appeared previously in the literature, is derived as a particular case. Various alternative forms of the cross-ratio density are obtained, e.g. under the Casey arccos transformation, and under an arctan transformation from the real projective line of cross-ratios to the unit circle. The cross-ratio distributions are novel to the probability literature; surprisingly various types of Cauchy distribution appear. To gain some analytical insight into the distribution, a simplelinear-ratiois also introduced. We also give some results for the projective invariants of five coplanar points. We discuss the general moment properties of the cross-ratio, and consider some inference problems, including maximum likelihood estimation of the parameters.
Continuous-time threshold AR(1) processes
Authors: Stramer O. , Brockwell P. J. , Tweedie R. L.
Published: April 2006

A threshold AR(1) process with boundary width 2δ> 0 was defined by Brockwell and Hyndman [5] in terms of the unique strong solution of a stochastic differential equation whose coefficients are piecewise linear and Lipschitz. The positive boundary-width is a convenient mathematical device to smooth out the coefficient changes at the boundary and hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equation from which the process is derived. In this paper we give a direct definition of a threshold AR(1) process withδ= 0 in terms of the weak solution of a certain stochastic differential equation. Two characterizations of the distributions of the process are investigated. Both express the characteristic function of the transition probability distribution as an explicit functional of standard Brownian motion. It is shown that the joint distributions of this solution withδ= 0 are the weak limits asδ↓ 0 of the distributions of the solution withδ> 0. The sense in which an approximating sequence of processes used by Brockwell and Hyndman [5] converges to this weak solution is also investigated. Some numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the process obtained from the Cameron–Martin–Girsanov formula and results of Engelbert and Schmidt [9]. We also derive the stationary distribution (under appropriate assumptions) and investigate stability of these processes.
Passage-time moments for continuous non-negative stochastic processes and applications
Authors: Menshikov M. , Williams R. J.
Published: April 2006

We give criteria for the finiteness or infiniteness of the passage-time moments for continuous non-negative stochastic processes in terms of sub/supermartingale inequalities for powers of these processes. We apply these results to one-dimensional diffusions and also reflected Brownian motion in a wedge. The discrete-time analogue of this problem was studied previously by Lamperti and more recently by Aspandiiarov, Iasnogorodski and Menshikov [2]. Our results are continuous analogues of those in [2], but our proofs are direct and do not rely on approximation by discrete-time processes.
Limit theorems for a general critical branching process
Authors: Durham Stephen D.
Published: April 2006

A general branching process begins with an initial object born at time 0. The initial object lives a random length of time and, during its life-time, has offspring which reproduce and die as independent probabilistic copies of the parent. Number and times of births to a parent are random and, once an object is born, its behavior is assumed to be independent of all other objects, independent of total population size and independent of absolute time. The life span of a parent and the number and times its offspring arrive may be interdependent. Multiple births are allowed. The process continues as long as there are objects alive.
Expenditure patterns for risky R and D projects
Authors: Kamien Morton I. , Schwartz Nancy L.
Published: April 2006

In this paper we derive the time pattern of optimal planned expenditures for R and D projects in which the total effort required to complete the research satisfactorily is not known. Theorem 1 summarizes our results under alternative assumptions regarding the behavior of the function governing the supposed probability of completing the R and D satisfactorily with incremental effort. In addition, an easily employed necessary condition for the R and D project to be worthwhile is provided in Theorem 2.
Group representations and applied probability
Authors: Hannan E. J.
Published: April 2006

This account is meant to serve as an introduction to certain mathematical ideas which can be of considerable importance in applied probability and in statistical theory. Since a wide field is to be covered in a short space, the account has to depend on heuristic arguments and a sympathetic understanding from those familiar with the technical details of the subject is called for. Constant reference to the sources of the underlying mathematics isnotcalled for, as it will be clear that the author is not claiming these results as his own. The references at the end of the paper, which are arranged so as to aid those readers who wish to proceed further, will also indicate the sources of the mathematical and statistical ideas used. We assume some knowledge of the groups, though most concepts are defined below. By beginning from the finite dimensional case and motivating later concepts in terms of this we hope to make the later, rather technical sections, more accessible.
On time changes in a digraph
Authors: Bhargava T. N.
Published: April 2006

The object of this paper is to present a probabilistic model for analyzing changes through time in a binary dyadic relation on a finite set of points. The total relation on the set takes the form of an aggregate of directed binary dyadic relations between ordered pairs of points belonging to the set; equivalently, the total relation on the set can be represented by means of a digraph or an incidence matrix isomorphic with the total relation. Such a relation, changing in its structure as time proceeds, is a reasonable mathematical model, for example, for the evolution of inter-relationships of the members of a social or any other group. A group of this kind is organized for a specific activity involving some sort of “communication” from one member to the other, and may be observed at successive discrete points in time generating statistics on the evolutionary process. (For a detailed treatment, see [3].) As a matter of fact, under suitable assumptions, the model presented here has potentialities for application in those situations which can be represented mathematically in terms of a finite set of points and an all-or-none relationship between ordered pairs of these points. Some of the other examples are communication networks, ecology, animal sociology, and management sciences (see [5]).
A note on the equilibrium distribution of levels in a semi-infinite reservoir subject to Markovian inputs and unit withdrawals
Authors: Odoom S. , Lloyd E. H.
Published: April 2006

A well-known theorem (Moran, 1959) relating to the equilibrium condition of a semi-infinite “Moran reservoir” with stationary independent inputs and unit output, gives (a) the probability of emptiness, and (b) the generating function of the distribution of levels, in terms of the input distribution. A further theorem (Prabhu, 1958) points out that, in a finite reservoir, the ratio of the probabilities of any two comparable levels is independent of the size of the reservoir, and and is in fact the same as the corresponding ratio for the semi-infinite reservoir. In the present note the theorems are extended to deal with the case of Markovian inputs.
Sequential decisions in the control of a space-ship (terminal cost proportional to magnitude of miss distance)
Authors: Feder Paul I. , Stroud Thomas
Published: April 2006

This is an extension of the paper [1] by Bather and Chernoff, where approximations are obtained to the solution of a sequential control problem involving the decision of when to use fuel to alter course during a space-ship's approach to a target. It is desired to come as close as possible to the target. If fuel is used too early, it may be used incorrectly due to imprecise information about the location of the target; however if the fuel is used too late it may be ineffective in reducing the miss distance.
A method for studying the integral functionals of stochastic processes with applications: I. Markov chain case
Authors: Puri Prem S.
Published: April 2006

The subject of this paper is the study of the distribution of integrals of the type

where {X(t); t ≧ 0} is some appropriately defined continuous-time parameter stochastic process, and f is a suitable non-negative function of its arguments. This subject has also sometimes been labelled as “the occupation time or the sojourn time problem” in literature. These integrals arise in several domains of applications such as in the theory of inventories and storage (see Moran [14], Naddor [15]), in the study of the cost of the flow-stopping incident involved in the automobile traffic jams (see Gaver [8], Daley [3], Daley and Jacobs [4]). The author encountered such integrals while studying certain stochastic models suitable for the study of response time distributions arising in various live situations. In fact in [19], it was shown that such a distribution is equivalent to the study of an integral of the type (1). Again, in the study of response of host to injection of virulent bacteria, Y(t) with f(X(t), t) = bX(t), with b > 0, could be regarded as a measure of the total amount of toxins produced by the bacteria during (0, t), assuming a constant toxin-excretion rate per bacterium. Here X(t) denotes the number of live bacteria at time t, the growth of which is governed by a birth and death process (see Puri [16], [17] and [18]).
Quasi-stationary distributions in semi-Markov processes
Authors: Cheong C. K.
Published: April 2006

Our main concern in this paper is the convergence, as t → ∞, of the quantities
i, j ∈ E; where Pij(t) is the transition probability of a semi-Markov process whose state space E is irreducible but not closed (i.e., escape from E is possible), and rj is the probability of eventual escape from E conditional on the initial state being i. The theorems proved here generalize some results of Seneta and Vere-Jones ([8] and [11]) for Markov processes.
Composable Markov processes
Authors: Schweder Tore
Published: April 2006

Many phenomena studied in the social sciences and elsewhere are complexes of more or less independent characteristics which develop simultaneously. Such phenomena may often be realistically described by time-continuous finite Markov processes. In order to define such a model which will take care of all the relevant a priori information, there ought to be a way of defining a Markov process as a vector of components representing the various characteristics constituting the phenomenon such that the dependences between the characteristics are represented by explicit requirements on the Markov process, preferably on its infinitesimal generator.
An invariance property of Poisson processes
Authors: Brown Mark
Published: April 2006

In this paper we shall investigate point processes generated by random variables of the form 〈gi(Ti]), i=± 1, ± 2, … 〉, where 〈Ti, i= ± 1, … 〉 is the set of arrival times from a (not necessarily homogeneous) Poisson process or mixture of Poisson processes, and 〈gi, i = ± 1, … 〉 is an independently and identically distributed (i.i.d.) or interchangeable sequence of random functions, independent of 〈Ti〉.
Two theorems on solutions of differential-difference equations and applications to epidemic theory
Authors: Severo Norman C.
Published: April 2006

We present two theorems that provide simple iterative solutions of special systems of differential-difference equations. We show as examples of the theorems the simple stochastic epidemic (cf. Bailey, 1957, p. 39, and Bailey, 1963) and the general stochastic epidemic (cf. Bailey, 1957; Gani, 1965; and Siskind, 1965), in each of which we let the initial distribution of the number of uninfected susceptibles and the number of infectives be arbitrary but assume the total population size bounded. In all of the references cited above the methods of solution involve solving a corresponding partial differential equation, whereas we deal directly with the original system of ordinary differential-difference equations. Furthermore in the cited references the authors begin at time t = 0 with a population having a fixed number of uninfected susceptibles and a fixed number of infectives. For the simple stochastic epidemic with arbitrary initial distribution we provide solutions not obtainable by the results given by Bailey (1957 or 1963). For the general stochastic epidemic, if we use the results of Gani or Siskind, then the solution of the problem having an arbitrary initial distribution would involve additional steps that would sum proportionally-weighted conditional results.
Queues with semi-Markovian arrivals
Authors: Çinlar Erhan
Published: April 2006

A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the nth interarrival time has a distribution function which may depend on the types of the nth and the n–1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.
Dynamic programming, pattern recognition and location of faults in complex systems
Authors: Bellman Richard
Published: April 2006

In a previous paper devoted to an application of dynamic programming to pattern recognition [1], we pointed out that some identification problems could be regarded as generalized trajectory processes. The functional equation technique [2] could then be employed to obtain an analytic formulation of the determination of optimal search techniques. In many cases, however, (for example, in chess or checkers), a straightforward use of the functional equation is impossible because of dimensionality difficulties. In circumventing these obstacles to effective computational solution, we employed a decomposition technique which we called “stratification” [1, 3]. In this paper, we present a different way of avoiding the dimensionality problem, based upon the concept of “extended state variable”. To indicate the utility of the concept, we shall apply it to the problem of finding a fault in a complex system.
The Effect of an age replacement to a standby redundant system
Authors: Mine Hisashi , Asakura Tatsuyuki
Published: April 2006

A multiple unit standby redundant system with preventive maintenance is presented. The Laplace transform of the probability density function of the time to the first emptiness for the system and its mean time are derived. As a special case, a two-unit standby redundant system with preventive maintenance is investigated in detail.
Limiting diffusion approximations for the many server queue and the repairman problem
Authors: Iglehart Donald L.
Published: April 2006

We shall consider a many server (multiple channels in parallel) queueing process in which customers arrive at the queue according to a Poisson process. The service times are assumed to be independent and exponentially distributed. As usual, the service times are independent of the arrival process. We assume that no server is idle if there is a customer waiting, but that otherwise the service discipline is arbitrary. If there are n servers and we let Xn(t) denote the number of customers waiting or being served at time t, then it is well known that Xn(t) is a birth and death process with stationary transition probabilities. A very comprehensive analysis of this many server queue from the point of view of birth and death processes has been carried out by Karlin and McGregor [5].
On a stochastic process occurring in queueing systems
Authors: Bhat U. N.
Published: April 2006

Transition distribution functions (d.f.) of the stochastic processu+t−X(t), whereX(t) has a compound Poisson distribution, are used to derive explicit results for the transition d.f.s of the waiting time processes in the queueing systemsM/G/1 andGI/M/1.
On the maximum of a stationary independent increment process
Authors: Ross Sheldon M.
Published: April 2006

A stationary independent increment process is the continuous time analogue of the discrete random walk, and, as such, has a wide variety of applications. In this paper we consider M(t), the maximum value that such a process attains by time t. By using renewal theoretic methods we obtain results about M(t). In particular we show that if μ, the mean drift of the process, is positive, then M(t)/t converges to μ, and E[M(t + h) – M(t)] → hμ.
Clone-selection and optimal rates of mutation
Authors: Eshel Ilan
Published: April 2006

The paper employs methods of multitype branching processes to evaluate the probability of survival of mutable clones under environmental conditions which are unfavorable to the original parent of the clone. When other factors are taken to be constant, the long-term survival probability of a clone is implicitly demonstrated as a function of the intrinsic rate of mutation carried by this clone. The existence of a mutation rate which maximizes clone survival probability is shown and the effects of environmental deterioration on this optimal rate are studied. Finally, rigorous quantitative results are obtained for the classical situation of a Poisson distribution of offspring numbers. These results are then applied to the biological problem of indirect selection (Eshel (1972)).
A generalised incomplete beta function and its application to multi-line stock control
Authors: Gittins J. C. , Maher M. J.
Published: April 2006

The distribution function for the negative binomial distribution is known to be an incomplete beta function. Here, some of the properties of the family of distribution functions for multivariate negative binomial distributions are explored. These properties are then used in deriving the expected cost per unit time for a multi-line joint-reordering system with Poisson demands. Policies are considered for which the quantity of any particular line in stock is the same at the beginning of every cycle. A method which gives good approximations to the optimal values of these quantities is described.
Some remarks on probability inequalities for sums of bounded convex random variables
Authors: Goldstein M.
Published: April 2006

Let X1, X2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi, i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.
An extension of watanabe's theorem of characterization of poisson processes over the positive real half line
Authors: Bremaud P.
Published: April 2006

We give an elementary proof of the martingale characterization theorem for Poisson processes over the positive real half line. This theorem is due to Watanabe [8] in the case where the mean measure associated to the Poisson process is the Lebesgue measure.
On a stochastic differential equation modeling of prey-predator evolution
Authors: Gard T. C. , Kannan D.
Published: April 2006

We study a stochastic differential equation model of prey-predator evolution. To keep in line with a known deterministic model we include the social and interaction terms with the drift in our model, and the randomness arises as fluctuations in the ecosystem. The notion of equilibrium population level and special types of fluctuations force us to work with degenerate elliptic operators. We consider the propagation of the population system in a sufficiently large but bounded domain. This enables us to look at not only the population extinction, but also the explosion beyond a certain level. Both extinction and explosion are possible; and, when they are not, we show that the population asymptotically reaches the equilibrium level. We show that the extinction, explosion and saturation probabilities satisfy, as functions of the initial population size, an integral equation arising out of a Dirichlet problem for a non-degenerate elliptic equation; and these probabilities are also smooth solutions of the Dirichlet problems. They are also used to express the solution of another Dirichlet problem for a degenerate elliptic equation.
On a quickest detection problem with variable monitoring
Authors: Balmer D. W.
Published: April 2006

The problem of detecting the arrival of a ‘disorder' in a process observed through a monitoring facility which may operate in ‘slow’ or ‘fast’ mode, is formulated as an optimal stopping problem. It is shown that in all circumstances where there may exist an optimal policy specifying the mode of observation and the time of stopping there is a unique policy satisfying certain necessary conditions of optimality; the various circumstances and control policies are specified.
On a general random exchange model
Authors: Helland Inge S. , Nilsen Trygve S.
Published: April 2006

Two independent i.i.d. sequences of random variables {Un} and {Dn} generate a Markov process {Xn} by Xn = max(Xn–1 – Dn, Un), n = 1, 2, …. ‘Exchange’ is defined as the event [Un > Xn–1 – Dn]. Conditions for existence of a limiting distribution for {Xn} are established, and normalization is discussed when no limiting distribution exists. Finally the process {Xn at the k th exchange; k = l, 2, …} and the time between consecutive exchanges are considered.
Stepping stone models for population growth
Authors: Renshaw E.
Published: April 2006

A population is composed of an infinite number of colonies situated at the integer points of a single co-ordinate axis. Each colony develops according to a simple birth and death process and migration is allowed between nearest neighbours. An approximate solution is obtained for the probability structure of the population size, and exact results are derived for the process when immigration is introduced into a single colony from outside the system.
On time-dependent linear transformations of non-stationary stochastic processes
Authors: Tong H.
Published: April 2006

We have introduced a definition of a slowly changing time-dependent linear transformation on a class of non-stationary stochastic processes and have studied the “spectral” relationship between the input and output processes. In addition, using this definition, we have extended one important property of coherency and the concept of residual variance bound pertinent to the theory of stationary stochastic processes to the non-stationary case.
An example of a perturbation-type extremum controller
Authors: Titterington D. M.
Published: April 2006

Controls of the perturbation type are evolved for the purpose of monitoring the random movement of a response curve using noise-corrupted observations of the response. Dynamic programming methods are used to obtain approximations to the optimum control system when the response is quadratic. Further results are briefly described.
Limit theorems for sums of chain-dependent processes
Authors: O'Brien G. L.
Published: April 2006

Chain-dependent processes, also called sequences of random variables defined on a Markov chain, are shown to satisfy the strong law of large numbers. A central limit theorem and a law of the iterated logarithm are given for the case when the underlying Markov chain satisfies Doeblin's hypothesis. The proofs are obtained by showing independence of the initial distribution of the chain and by then restricting attention to the stationary case.
Limiting results for carrier-borne epidemics
Authors: Saunders Roy
Published: April 2006

In this article we examine the behavior of Mm(τ), the number of susceptibles remaining uninfected at time τ in a carrier-borne epidemic as the initial number of susceptibles m tends to infinity for fixed initial number of carriers n. We show that as m → ∞ the proportion of susceptibles process {Mm(τ)/m : ≧ 0} converges weakly to a limiting process. The limiting process has sample functions almost surely of the form exp (– h(τ)), where the form of h(τ) is determined by the removal times of the carriers.
Necessary conditions for Markovian processes on a lattice
Authors: Moran P. A. P.
Published: April 2006

Stationary processes which are defined on the points of a square lattice and are Markovian in various senses are considered. It is shown that a certain assumption of linearity of regression forces the spectral distribution to be of a certain explicit form, and that given this form Gaussian processes of this kind are easily constructed. Certain non-Gaussian processes satisfying the various Markovian properties are also constructed and the difference from nearest-neighbour systems emphasized. It is conjectured, but not proved, that the assumption of linearity of regression also implies Gaussianity.
Shot noise generated by a semi-Markov process
Authors: Smith Woollcott
Published: April 2006

In this note a model for shot noise generated by a semi-Markov process is developed. The moments of the shot noise process are found, and some applications of this model are briefly indicated.
Convergence of critical Galton-Watson branching processes
Authors: Lindvall Torgny
Published: April 2006

The purpose of this note is to extend results on critical Galton-Watson branching processes, due to Feller and Lamperti: if the number of individuals at time zero tends to infinity, the process converges, with two different normalizations, to certain diffusion processes in the functional form, discussed in, e.g., Billingsley.
A formal approach to queueing processes in the steady state and their applications
Authors: Miyazawa Masakiyo
Published: April 2006

Using the theory of point processes, we give a formal treatment to queueing processes in the steady state. Based on this result, we obtain invariance relations between several quantities in G/G/c queues. As applications, the finiteness of their moments is discussed for G/G′/c queues. The basic results and notations used in this paper are contained in the author's previous paper (Miyazawa (1977)).
Some comments concerning a curious singularity
Authors: Feigin Paul D.
Published: April 2006

Consider the maximum likelihood estimation of θ based on continuous observation of the process X, which satisfies dXt = θXtdt + dWt. Feigin (1976) showed that, when suitably normalized, the maximum likelihood estimate is asymptotically normally distributed when the true value of θ ≠ 0. The claim that this asymptotic normality also holds for θ = 0 is shown to be false. The parallel discrete-time model is mentioned and the ramifications of these singularities to martingale central limit theory is discussed.
Uniform conditional stochastic order
Authors: Whitt Ward
Published: April 2006

One probability measure is less than or equal to another in the sense of UCSO (uniform conditional stochastic order) if a standard form of stochastic order holds for each pair of conditional probability measures obtained by conditioning on appropriate subsets. UCSO can be applied to the comparison of lifetime distributions or the comparison of decisions under uncertainty when there may be reductions in the set of possible outcomes. When densities or probability mass functions exist on the real line, then the main version of UCSO is shown to be equivalent to the MLR (monotone likelihood ratio) property. UCSO is shown to be preserved by some standard probability operations and not by others.
Reliability analysis of complex repairable systems by means of marked point processes
Authors: Franken Peter , Streller Arnfried
Published: April 2006

Starting from the theory of point processes the concept of a process with an embedded marked point process is defined. It is shown that the known formula expressing the relation between the stationary and synchronous version of a regenerative process remains valid without the assumption of independence of cycles. General formulae for stationary availability and interval reliability of complex systems with repair are also obtained. In this way generalizations of Keilson's results for Markovian systems and Ross's results for systems with separately maintained elements are presented. The formulae are applied to a two-unit parallel system with a single repair facility.
Convergence of successive approximations to a stochastic fixed point
Authors: Sanghvi Arun P.
Published: April 2006

This paper describes some sufficient conditions that ensure the convergence of successive random applications of a family of mappings {Γα : α ∈ A} on a compact metric space (X, d) to a stochastic fixed point. The results are similar in spirit to a recent result of Yahav (1975).
Single-shelf library-type Markov chains with infinitely many books
Authors: Nelson Peter R.
Published: April 2006

In a single-shelf library having infinitely many books B1, B2, …, the probability of selecting each book is assumed known. Books are removed one at a time and replaced in position k prior to the next removal. Books are moved either to the right or the left as is necessary to vacate position k. Those arrangements of books where after some finite position all the books are in natural order (book i occupies position i) are considered as states in an infinite Markov chain. When k > 1, we show that the chain can never be positive recurrent. When k = 1, we find the limits of ratios of one-step transition probabilities; and when k = 1 and the chain is transient, we find the Martin exit boundary.
Modelling yeast cell growth using stochastic branching processes
Authors: Green P. J.
Published: April 2006

This paper aims to demonstrate that the general Crump–Mode–Jagers branching process may be used in a natural way to model the asymmetric growth of budding yeast cells. The models obtained are generalisations of the deterministic model proposed by Hartwell and Unger (1977): all the results that are derived in that paper may be obtained using branching-process methods, but such methods also apply when account is taken of the biologically obvious fact that the various phases of the cell growth are of random rather than fixed duration. In their full generality, branching processes involve more parameters than can be estimated by experiment, but we present below a special case in which this problem is not likely to arise.

A recent paper, Lord and Wheals (1980), discusses more of the biological background than is appropriate here. In the present paper, we show how certain statistical procedures for our model may be developed.
Stability in a non-homogeneous Markov chain model in manpower systems
Authors: Vassiliou P.-C. G.
Published: April 2006

Necessary and sufficient conditions for stability, imposed firstly on the initial structure and the sequence of recruitment, and secondly on the initial structure and the sequence of expansion are provided in forms of two theorems. Also the limiting behaviour of the expected relative grade sizes is studied if we drop the conditions for stability imposed on the initial structure and keep the same sequence of expansion. Finally we examine the limiting behaviour of the expected grade sizes if we drop the assumption of a continuously expanding system.
On competition between two species
Authors: Ridler-Rowe C. J.
Published: April 2006

A stochastic model for the competition between two species is considered and a limit theorem is obtained for the size of the surviving species. A limit theorem is also obtained for the probability that a given species should survive the other.
Birth and death processes with random environments in continuous time
Authors: Cogburn Robert , Torrez William C.
Published: April 2006

A generalization to continuous time is given for a discrete-time model of a birth and death process in a random environment. Some important properties of this process in the continuous-time setting are stated and proved including instability and extinction conditions, and when suitable absorbing barriers have been defined, methods are given for the calculation of extinction probabilities and the expected duration of the process.
The effect of variability in the GI/G/s queue
Authors: Whitt Ward
Published: April 2006

In 1969 H. and D. Stoyan showed that the stationary waiting-time distribution in a GI/G/1 queue increases in the ordering determined by the expected value of all non-decreasing convex functions when the interarrival-time and service-time distributions become more variable, as expressed in the ordering determined by the expected value of all convex functions. Ross (1978) and Wolff (1977) showed by counterexample that this conclusion does not extend to all GI/G/s queues. Here it is shown that this conclusion does hold for all GI/G/s queues for several other measures of congestion which coincide with the waiting time in single-server systems. One such alternate measure of congestion is the clearing time, the time required after the arrival epoch of the nth customer for the system to serve all customers in the system at that time, excluding the nth customer. The stochastic comparisons also imply an ordering for the expected waiting times in M/G/s queues.
Distances on circles, toruses and spheres
Authors: Silverman B. W.
Published: April 2006

Families of heavily dissociated random variables are defined and discussed. These include families of the form g(Yi, Yj) for some suitable function g of two arguments and independent uniformly distributed random variables Y1, Y2, ··· on the circle, the torus or the sphere. The weak convergence of the empirical distribution process is discussed. The particular case of distances between pairs of observations on the circle is considered in greater detail.
The transient state probabilities for a queueing model where potential customers are discouraged by queue length
Authors: Van Doorn Erik A.
Published: April 2006

Exact expressions are derived for the transition probabilities of the birth-death process with parameters and which serves as a queueing model where potential customers are discouraged by queue length.
Evolutionarily stable strategies with two types of player
Authors: Taylor Peter D.
Published: April 2006

A definition of ESS (evolutionarily stable strategy) is suggested for games in which there are two types of player, each with its own set of strategies, and the fitness of any strategy depends upon the strategy mix, of both types, in the population as a whole. We check that the standard ESS results hold for this definition and discuss a mate-desertion model which has appeared in the literature in which the two types are male and female.
The optimality equation in average cost denumerable state semi-Markov decision problems, recurrency conditions and algorithms
Authors: Federgruen A. , Tijms H. C.
Published: April 2006

This paper is concerned with the optimality equation for the average costs in a denumerable state semi-Markov decision model. It will be shown that under each of a number of recurrency conditions on the transition probability matrices associated with the stationary policies, the optimality equation has a bounded solution. This solution indeed yields a stationary policy which is optimal for a strong version of the average cost optimality criterion. Besides the existence of a bounded solution to the optimality equation, we will show that both the value-iteration method and the policy-iteration method can be used to determine such a solution. For the latter method we will prove that the average costs and the relative cost functions of the policies generated converge to a solution of the optimality equation.
A variation of Weiss's carrier-borne epidemic model
Authors: Routleff C.
Published: April 2006

A carrier-borne epidemic model, in which the carrier removal process is influenced by the number of detected infected individuals, is considered. The distribution of the total size of the epidemic is found and expressions for the expected ultimate size of the epidemic and the expected area under the trajectory of carriers are given.
Some classes of conditional inference procedures
Authors: Williams E. J.
Published: April 2006

Data classified into several groups usually depend on structural and accessory parameters, the structural parameter θ being a constant for all the data, while the accessory parameters ϕ i vary between groups.

For the structural parameter, inferences that are free of the accessory parameters can be made when the ϕ i admit sufficient statistics or surrogates Si, by conditioning on the Si or by constructing statistics independent of the Si.

When the Si are functions of the structural parameter θ, θ is said to be endomorphic. Conditional likelihood methods are not unequivocal for endomorphic parameters; instead, inference is based on statistics independent of the Si, and so free of the ϕ i, derived from conditional distribution functions.

With endomorphic parameters, since the surrogates and the statistics independent of them are functions of θ, tests of independence of these sets of statistics provide a means of making inferences about θ.
Time series modelling — an integrated approach
Authors: McNeil Don
Published: April 2006

Some inadequacies of both the traditional (exponential smoothing) and Box-Jenkins approaches to time series forecasting of economic data are investigated. An approach is suggested which integrates these two methodologies. It is based on smoothing the data using straight line segments instead of differencing to obtain stationarity, and forecasting using an autoregressive-moving-average model for the residuals from the most recent linear segment. The efficiency of this approach is calculated theoretically using a series comprising integrated white noise.
Characterizations of the geometric distribution by distributional properties
Authors: Zijlstra Mynt
Published: April 2006

Some new characterizations of the geometric distribution are studied. A generalization of the characterization by the well-known ‘lack-of-memory' property is given together with some closely related characterizations. Furthermore the modified geometric distribution is characterized by a distributional property of the difference of two successive order statistics. The latter result extends work of Puri and Rubin (1970). Finally the geometric distribution is characterized by a conditional distribution property of the difference of two arbitrary order statistics, which generalizes a result by Arnold (1980). Some of the results given answer open questions put in earlier papers.
Asymptotic results for the best-choice problem with a random number of objects
Authors: Yasuda Masami
Published: April 2006

This paper considers the best-choice problem with a random number of objects having a known distribution. The optimality equation of the problem reduces to an integral equation by a scaling limit. The equation is explicitly solved under conditions on the distribution, which relate to the condition for an OLA policy to be optimal in Markov decision processes. This technique is then applied to three different versions of the problem and an exact value for the asymptotic optimal strategy is found.