bluuebunny/crossref_metadata_2025 huggingface dataset:
Possible velocities for a simple epidemic
Authors: Mollison Denis
Published: April 2006

It is here shown that for a deterministic simple epidemic in which the spatial distribution of the contacts made by an infectious individual is negative exponential, propagation of the epidemic as a travelling wave is only possible for velocities at or above a certain minimal velocity; and that there exists exactly one waveform for each such velocity. The method employed is to reduce the equation of propagation to a three-dimensional autonomous differential equation, and attack this mainly using topological considerations.

This result is similar to those obtained by Kolmogoroff et al. (1937) for the advance of an advantageous gene, and by Kendall (1965) for the deterministic “general” epidemic, but these were achieved using the considerable simplification of approximating the spatial dependence by a diffusion term in the equation of propagation. Taken together with Mollison (1972), in which it was shown that when the spatial distribution of contacts (V) is more spread than negative exponential (∫eksdV(s) divergent for all k > 0), no finite bound can be set on the velocity of propagation: the present work suggests that the diffusion-term approximations may be reasonable if, as well as only if, V is less spread than negative exponential.
Bivariate stationary point processes, fundamental relations and first recurrence times
Authors: Wisniewski T. K. M.
Published: April 2006

Various types of time and event sampling of a stationary and orderly bivariate point process are considered. Fundamental relations between inter-event intervals and the event counting process are derived. Relations between first forward recurrence times and their moments for different types of sampling are obtained.
Doubly stochastic Poisson processes and process control
Authors: Rudemo Mats
Published: April 2006

Consider a Poisson point process with an intensity parameter forming a Markov chain with continuous time and finite state space. A system of ordinary differential equations is derived for the conditional distribution of the Markov chain given observations of the point process. An estimate of the current intensity, optimal in the least-squares sense, is computed from this distribution. Applications to reliability and replacement theory are given. A special case with two states, corresponding to a process in control and out of control, is discussed at length. Adjustment rules, based on the conditional probability of the out of control state, are studied. Regarded as a function of time, this probability forms a Markov process with the unit interval as state space. For the distribution of this process, integro-differential equations are derived. They are used to compute the average long run cost of adjustment rules.
Criteria for extinction of certain population growth processes with interacting types
Authors: Karlin Samuel , Kaplan Norman
Published: April 2006

A study is made of a series of stochastic growth processes related to multi-type branching models with interaction phenomena among the types with aim to ascertain criteria for extinction or non-extinction of the population. It is established that trends depicting changes of expected sizes of types generally overwhelm any effects of statistical fluctuations such that the conditions for extinction reduce to natural conditions on expected values. Three models are developed. The first two involve special mating systems for certain two sex populations. The last model is a neutralization phenomenon for two types of particles.
On the estimation of a harmonic component in a time series with stationary dependent residuals
Authors: Walker A. M.
Published: April 2006

Let observations (X1, X2, …, Xn) be obtained from a time series {Xt} such that

where the ɛt are independently and identically distributed random variables each having mean zero and finite variance, and the gu(θ) are specified functions of a vector-valued parameter θ. This paper presents a rigorous derivation of the asymptotic distributions of the estimators of A, B, ω and θ obtained by an approximate least-squares method due to Whittle (1952). It is a sequel to a previous paper (Walker (1971)) in which a similar derivation was given for the special case of independent residuals where gu(θ) = 0 for u > 0, the parameter θ thus being absent.
On the length and number of served customers of the busy period of a generalised M/G/1 queue with finite waiting room
Authors: Rosenlund Stig I.
Published: April 2006

Customers arrive in groups to a single server queue with finite waiting room. Two-dimensional distributions for times and numbers of served customers between occurrences of states in the embedded Markov chain are obtained by linear algebra giving systems of equations for joint Laplace-Stieltjes transforms. For M/M/1 a simple recursion relation for the joint transform of the two variables in the title is derived and used to obtained the first and second moments.
The latent roots of certain Markov chains arising in genetics: A new approach, II. Further haploid models
Authors: Cannings C.
Published: April 2006

The method developed for the treatment of the classical drift models of Wright and Moran, and their generalizations, in Cannings (1974) are extended to more complex haploid models. The possibility of subdivision of the population, as for migration models and age-structured models, is incorporated. Models with variable size or reproductive structure determined by another Markov chain are analysed.
Composition and invariance methods for solving some stochastic control problems
Authors: Beneš V. E.
Published: April 2006

This paper considers certain stochastic control problems in which control affects the criterion through the process trajectory. Special analytical methods are developed to solve such problems for certain dynamical systems forced by white noise. It is found that some control problems hitherto approachable only through laborious numerical treatment of the non-linear Bellman-Hamilton-Jacobi partial differential equation can now be solved.
Bounds on the probability of the union and intersection of m events
Authors: Kwerel Seymour M.
Published: April 2006

For dependent probability systems of m events partially specified only by the quantities S1, the sum of the probabilities of the m individual events; S2, the sum of the probabilities of each of the (m) pairs of events and S3 the sum of the probabilities of each of the (m 3) combinations of three events; this paper develops the most stringent upper and lower bounds on P1, the probability of the union of the m events; and on P[m], the probability of the simultaneous occurrence of the m events.
Point processes and random measures
Authors: Grandell Jan
Published: April 2006

The purpose of this paper is to give a short introduction to the theory of point processes and random measures. Our hope is that the paper can be used as a complement to a study of Billingsley's bookConvergence of Probability Measures. The subjects treated are: topological properties of the space of realizations; characterization and convergence of random measures with special attention to simple point processses and diffuse random measures; the relation between the topology used and the Skorohod topology. Some thinning and superposition results are given without proofs.
Integrals on a moving manifold and geometrical probability
Authors: Baddeley Adrian
Published: April 2006

For a manifold which is moving and changing with time, consider some numerical property which at each instant is equal to an integral over the manifold. We derive a general expression for the time rate of change of this integral. Corollaries include a precise general form of Crofton's boundary theorem, de Hoff's interface displacement equations (with some new extensions) and a theorem in fluid mechanics.
A branching model by population size dependence
Authors: Lipow Carla
Published: April 2006

A continuous-time Markov branching process is modified to allow some dependence of offspring generating function on population size. The model involves a given population size M, below which the offspring generating function is supercritical and above which it is subcritical. Immigration is allowed when the population size is 0. The process has a stationary measure, and an expression for its generating function is found. A limit theorem for the stationary measure as M tends to ∞ is then obtained.
On the input and output processes for a general birth-and-death queueing model
Authors: Natvig Bent
Published: April 2006

The steady-state input and output processes are considered for a birth-and-death queueing model withNwaiting positions (0 ≦N≦ ∞),sservers (1 ≦s≦ ∞) and an arbitrary queueing discipline. Let an indexnindicate that the quantity in question depends on the system state but not on timet.The instantaneous arrival rate isλ, the probability of balking (i.e., not trying to obtain service) beingξn. The instantaneous departure rate,μn, of customers having joined the system is the sum of the rate of service completions and the rate of defections before service completion. Three cases are considered. We start by ignoring balking customers; in the first case treating a lost customer neither as an input nor as an output, then secondly as both. Finally, balking and lost customers are considered both as inputs and outputs.
The general bulk queue as a matrix factorisation problem of the Wiener-Hopf type. Part II.
Authors: Dagsvik John
Published: April 2006

In a previous paper (Dagsvik (1975)) the waiting time process of the single server bulk queue is considered and a corresponding waiting time equation is established. In this paper the waiting time equation is solved when the inter-arrival or service time distribution is a linear combination of Erlang distributions. The analysis is essentially based on algebraic arguments.
On the generalization of stable age distributions to Gani-type person-flow models
Authors: Feichtinger Gustav
Published: April 2006

The concept of a stable population, which is of basic importance in demography, is generalized to person-flow models as defined by Gani. Stability and asymptotic stability of manpower systems are investigated in terms of the input process and the transition structure. Extending Bartholomew ((1973), Chapter 3) we obtain a result on the limiting behaviour of Gani-type models with asymptotic exponential input processes. The linear population-dynamic model of Leslie (1945) may be considered as a further case.
The first- and last-birth problems for a multitype age-dependent branching process
Authors: Biggins J. D.
Published: April 2006

If Bn is the time of the first birth in the nth generation in a supercritical irreducible multitype Crump–Mode process then when there are people in every generation Bn/n converges to a constant; if Dn is the time of the last birth in the nth generation then Dn/n also converges to a constant on the survival set. Analogous results hold for the extreme members of the nth generation in a branching random walk.
Functional limit theorems for the queue GI/G/1 in light traffic
Authors: Iglehart Donald L.
Published: April 2006

We consider a single GI/G/1 queueing system in which customer number 0 arrives at time t0 = 0, finds a free server, and experiences a service time v0. The nth customer arrives at time tn and experiences a service time vn. Let the interarrival times tn - tn-1 = un, n ≧ 1, and define the random vectors Xn = (vn-1, un), n ≧ 1. We assume the sequence of random vectors {Xn : n ≧ 1} is independent and identically distributed (i.i.d.). Let E{un} = λ-1 and E{vn} = μ-1, where 0 < λ, μ < ∞. In addition, we shall always assume that E{v02} < ∞ and that the deterministic system in which both vn and un are degenerate is excluded. The natural measure of congestion for this system is the traffic intensity ρ = λ/μ. In this paper we shall restrict our attention to systems in which ρ < 1. Under this condition, which we shall refer to as light traffic, our system is of course stable.
Isotropic random simplices
Authors: Miles R. E.
Published: April 2006

Some obscure, yet fundamental, formulae of integral geometry are re-considered. They are applied to determine all the moments of the random volume of various isotropic random r-dimensional simplices in En (r = 1, …, n).
On the optimal control of a deterministic epidemic
Authors: Morton R. , Wickwire K. H.
Published: April 2006

A control scheme for the immunisation of susceptibles in the Kermack-McKendrick epidemic model for a closed population is proposed. The bounded control appears linearly in both dynamics and integral cost functionals and any optimal policies are of the “bang-bang” type. The approach uses Dynamic Programming and Pontryagin's Maximum Principle and allows one, for certain values of the cost and removal rates, to apply necessary and sufficient conditions for optimality and show that a one-switch candidate is the optimal control. In the remaining cases we are still able to show that an optimal control, if it exists, has at most one switch.
Selecting crosses using information from a diallel cross
Authors: Curnow R. N.
Published: April 2006

Information is available from a diallel cross consisting of all possible crosses among n parental lines, excluding the parental lines themselves and assuming that reciprocal crosses are identical. The lines are assumed to be a random sample from an infinite population of such lines. The general combining abilities of the lines may be more important than the specific combining abilities of the individual crosses. If so, we may decide to rank the lines on the basis of their estimated general combining abilities and make crosses for further testing, or for commercial production, using only lines with high estimated general combining abilities. Existing statistical selection theory can be used to predict the expected genetic superiority of these crosses. For n = 6, 10, and 20, the best five crosses would be expected to result from crossing the lines as follows, where the line number is its rank order on estimated general combining ability: 1 × 2, 1 × 3, 1 × 4, 2 × 3, 1 × 5. Crosses 2×3 and 1 × 5 may not necessarily be expected to be in the stated order, depending on the value of n. In a particular trial, the estimated general combining abilities may suggest that other crosses should be made, but the practical importance of this is doubtful. Note that if only three crosses are to be selected, 1 × 4 is expected to be better than 2 × 3, i.e., the best three crosses are not obtained by crossing the best three lines. This method of selection based on general combining abilities will be referred to as line selection.
Models of clines
Authors: Downham D. Y.
Published: April 2006

The imago of the peppered moth exists in two sharply distinguished forms, speckled and black. Clarke and Sheppard (1966) have shown that the frequency of the black moth decreases from 99% in Liverpool to 10% in North Wales. If these frequencies are in equilibrium, albeit dynamic equilibrium, then such a gradient is called a cline. Haldane (1948) briefly describes several clines, and then defines a model that may represent the life-cycle of certain species. The assumptions defining his model are as follows.
On Kimura's maximum principle in the genetical theory of natural selection
Authors: Edwards A. W. F.
Published: April 2006

In 1958 Kimura introduced his Maximum Principle in the Genetical Theory of Natural Selection, which has since been described in several books (Ewens (1969), Crow and Kimura (1970), Kojima (1970)). I shall use Ewens' notation and treatment which, although it uses the discrete-generation model, is formally equivalent to Kimura's continuous-time treatment.
The validity of the optimum model
Authors: Robertson A.
Published: April 2006

The paper discussed the value of the ‘Optimum Model” in describing the inter-relationship between metric characters and natural fitness. This model represents one extreme of a whole spectrum of models. It can be used to predict changes in the mean of the character under natural selection, but the basic premises underlying such use are not well understood.
Outcrossing and heterozygosity
Authors: Strobeck Curtis
Published: April 2006

A classical question in population genetics is ‘In a species in which there is obligate outcrossing is there an increased amount of heterozygosity?’. This somewhat vague question can have two different meanings: (1) Is the number of alleles maintained at a locus greater? and (2) If there are two alleles at a particular locus is the number of heterozygotes formed greater? It is the last question which is discussed here.
A third note on recent research in geometrical probability
Authors: Little D. V.
Published: April 2006

Recent research on topics related to geometrical probability is reviewed. The survey includes articles on random points, lines, flats, and networks in Euclidean spaces, pattern recognition, random coverage and packing, random search, stereology, and probabilistic aspects of integral geometry.
On the moments of Markov renewal processes
Authors: Hunter Jeffrey J.
Published: April 2006

Recently Kshirsagar and Gupta [5] obtained expressions for the asymptotic values of the first two moments of a Markov renewal process. The method they employed involved formal inversion of matrices of Laplace-Stieltjes transforms. Their method also required the imposition of a non-singularity condition. In this paper we derive the asymptotic values using known renewal theoretic results. This method of approach utilises the fundamental matrix of the imbedded ergodic Markov chain and the theory of generalised matrix inverses. Although our results differ in form from those obtained by Kshirsagar and Gupta [5] we show that they reduce to their results under the added non-singularity condition. As a by-product of the derivation we find explicit expressions for the moments of the first passage time distributions in the associated semi-Markov process, generalising the results of Kemeny and Snell [4] obtained for Markov chains.
Sequential decision processes with essential unobservables
Authors: Whittle P.
Published: April 2006

We consider sequential decision processes in which the posterior distribution of some unobservable must be included among the sufficient statistics. The technique of an earlier paper (Whittle (1964)) is applied to show that the residual loss function is necessarily homogeneous of degree one in this distribution, if certain conventions are adopted. This point leads to considerable simplification. Examples are given of the great variety of problems which can advantageously be formulated in this manner.
Regularly varying functions in the theory of simple branching processes
Authors: Seneta E.
Published: April 2006

It is demonstrated for the non-critical and the explosive cases of the simple Bienaymé-Galton-Watson (B. G. W.) process (with and without immigration) that there exists a natural and intimate connection between regularly varying function theory and the asymptotic structure of the limit laws and corresponding norming constants. A similar fact had been demonstrated in connection with their invariant measures in [22]. This earlier study is complemented here by a similar analysis of the process where immigration occurs only at points of “emptiness” of the B. G. W. process.
Survival probabilities and extinction times for some multitype branching processes
Authors: Pollak Edward
Published: April 2006

This paper deals with the computation of survival probabilities and extinction times for multitype positively regular branching processes. If all of the generating functions of the offspring distributions are of the linear fractional form and have the same denominator, explicit expressions may be obtained for all of their iterates. It is then possible to obtain formulae for survival probabilities and bounds on the mean time to extinction, given extinction, of a line descended from a single individual. If there are two types and the offspring distributions are bivariate Poisson, their generating functions may be bounded by linear fractional generating functions. It is then possible to compute upper and lower bounds on mean times to extinction, given extinction, and this is done for some special cases.
The sampling theory of selectively neutral alleles
Authors: Watterson G. A.
Published: April 2006

In a paper of the same title, Ewens has considered the problem of inferring whether the genotypic frequencies observed in a small sample are consistent with a particular population model, in which all types are selectively neutral. The present paper gives the general theory of such sampling schemes, when the sampling is from either a deterministic, or a stochastically varying, population.

The theory is illustrated on a population model which was introduced by Karlin and McGregor. Various results are proved for both the population, and its sample. It is shown that Ewens' sampling theory applies to this population model, and conjectures as to why different population models may have the same sampling theory are made.
Dynamic programming and gambling models
Authors: Ross Sheldon M.
Published: April 2006

Dynamic programming is used to solve some simple gambling models. In particular we consider the situation where an individual may bet any integral amount not greater than his fortune and he will win this amount with probability p or lose it with probability 1 — p. It is shown that if p ≧ ½ then the timid strategy (always bet one dollar) both maximizes the probability of ever reaching any preassigned fortune, and also stochastically maximizes the time until the bettor becomes broke. Also, if p ≦ ½ then the timid strategy while not stochastically maximizing the playing time does maximize the expected playing time. We also consider the same model but with the additional structure that the bettor need not gamble but may instead elect to work for some period of time. His goal is to minimize the expected time until his fortune reaches some preassigned goal. We show that if p ≦ ½ then (i) always working is optimal, and (ii) among those strategies that only allow working when the bettor is broke it is the bold strategy that is optimal
Poisson process and distribution-free statistics
Authors: Dwass Meyer
Published: April 2006

The well-known connection between the Poisson process and empirical c.d.f.'s is exploited from a new point of view. Distributions of functions of empirical c.d.f.'s for finite sample size n are explicitly described in some new examples, and new qualitative information is obtained for some classical examples.
Markov chains governed by complicated renewal processes
Authors: Gergely T. , Tsukanow I. N. , Yezhow I. I.
Published: April 2006

In this work Markov chains governed by complicated processes are introduced and investigated (Section 1). In Section 2 an ergodic theorem for these processes is formulated, while in Section 3 the sojourn time of the process in a fixed region is studied; in Section 4 some examples are considered. The processes studied are of practical importance in the description of mass service systems and the theory of reliability for which the time intervals between successive demands cannot be assumed to be mutually independent random variables. It is shown that the dependence parameterrof these processes, if it is sufficiently large, allows us to formulate a relationship between the time intervals in question.
Three-terminal communication channels
Authors: Van Der Meulen Edward C.
Published: April 2006

The problem of transmitting information in a specified direction over a communication channel with three terminals is considered. Examples are given of the various ways of sending information. Basic inequalities for average mutual information rates are obtained. A coding theorem and weak converse are proved and a necessary and sufficient condition for a positive capacity is derived. Upper and lower bounds on the capacity are obtained, which coincide for channels with symmetric structure.
Some limit theorems for the total progeny of a branching process
Authors: Pakes A. G.
Published: April 2006

We consider a branching process in which each individual reproduces independently of all others and has probabilityaj(j= 0, 1, · · ·) of giving rise tojprogeny in the following generation. It is assumed, without further comment, that 0 <a0,a0+a1< 1.
First-passage percolation on the square lattice, II
Authors: Wierman John C.
Published: April 2006

Several problems are considered in the theory of first-passage percolation on the two-dimensional integer lattice. The results include: (i) necessary and sufficient conditions for the existence of moments of first-passage times; (ii) determination of an upper bound for the time constant; (iii) an initial result concerning the maximum height of routes for first-passage times; (iv) ergodic theorems for a class of reach processes.
A note on two measures of dependence and mixing sequences
Authors: Peligrad Magda
Published: April 2006

In this note we establish an inequality between the maximal coefficient of correlation and the φ -mixing coefficient which is symmetric in its arguments. Motivated by this inequality, we introduce a mixing coefficient which is the product of two φ -mixing coefficients.

We also study an invariance principle under conditions imposed on this new mixing coefficient. As a consequence of this result it follows that the invariance principle holds when either the direct-time process or its time-reversed process is φ -mixing; when both processes are φ-mixing the invariance principle holds for sequences of L2-integrable random variables under a mixing rate weaker than that used by Ibragimov.
A counterexample to a conjecture of J. M. Hammersley and D. J. A. Welsh concerning first-passage percolation
Authors: Van Den Berg J.
Published: April 2006

Consider first-passage percolation on the square lattice. Hammersley and Welsh, who introduced the subject in 1965, conjectured that the expected minimum travel time from (0, 0) to (n, 0) along paths contained in the cylinder is always non-decreasing in n. However, when the bonds have time-coordinate 1 with probability p and 0 with probability 1 – p (0 < P < 1), then, for p sufficiently small, we get a counterexample.
General percolation and random graphs
Authors: McDiarmid Colin
Published: April 2006

I introduce some useful general results concerning clutter percolation and families of binary random variables arranged in independent subfamilies, and give several applications to the study of random graphs and digraphs.
Statistique des processus de Poisson stationnaires de sous-varietes lineaires affines
Authors: Fellous A. , Granara J.
Published: April 2006

This paper reviews the principal properties of stationary Poisson processes of k-dimensional linear affine subvarieties in a d-dimensional vector space, and outlines basic results for the estimation of parameters from a single incomplete realization of the process.
A second-order heavy traffic approximation for the queueGI/G/1
Authors: Köllerström Julian
Published: April 2006

A second-order heavy traffic approximation for the stationary waiting-time d.f.GforGI/G/1 queues is derived, the first-order term of which is Kingman's (1961), (1962a), (1965) exponential approximation. On the way to this result there are others of independent interest, such as a convolution equation relating this waiting time d.f.Gwith the d.f. of a related ladder height, an integral equation forGand some stochastic bounds forG. The main result requires a particular type of functional convergence that may also be of interest.
Persistence times of populations subject to random disasters
Authors: Haigh John
Published: April 2006

The model of a population which grows logistically, except for random disasters that reduce the population by a given amount at the instants of a Poisson process, which was examined by Hanson and Tuckwell (1978), is generalised in two ways:
Inbreeding in genetic algebras
Authors: Holgate Philip
Published: April 2006

The treatment of selfing in genetic algebras [3] can be extended by the definition of an analogous operator on the copular algebra which corresponds to brother–sister mating, one on the duplicate of the copular algebra corresponding to mating between double first cousins, etc.
Estimation of the correlation between relatives
Authors: Smith Cedric A. B.
Published: April 2006

Suppose, to take a simple example, we have a set of sibships. Each individual (sib) has a certain measured character x. We wish to estimate the interclass (or sib–sib) correlation for this character. Our model is as follows. In sibship s the observed values x are supposed drawn from a distribution with (true) mean ms and variance vs. The sibships are drawn from a population of sibships, so that the ms have a mean M and variance B, and the vs have mean A. (We also suppose that the number ns of sibs in sibship s is independent of ms and vs.) The interclass correlation is then easily seen to be

ρ = B/(A + B).

The obvious way to proceed is to estimate A and B first, and then find ρ. Take a sample of k sibships. In sibship s the observed mean is and the deviance (sum of squares of deviations) δs. Let n = σ ns = the total number of individuals in the sample, and δ = σ δs = the total deviance within sibships. Proceed iteratively as follows:
The geometry of random genetic drift VI. A random selection diffusion model
Authors: Antonelli Peter L. , Chapin Jared , Voorhees B. H.
Published: April 2006

The ray solution of Felsenstein'sn-allele random selection diffusion process is given for small values of the selection parameters. This solution holds away from, but not near, the boundary of frequency space. The solution is possible only because the coefficients of the associated Jacobi field equations agree uniformly with those for the case of zero selection up to fourth powers in the selection parameters, whilst the covariance of the diffusion has only quadratic dependence.
Polymerisation processes with intrapolymer bonding. III. Several types of unit
Authors: Whittle P.
Published: April 2006

The solution for the multitype version of the polymerisation process treated in earlier papers is examined, and found to present several new features, because of the necessarily complex integration path, and because of the greater number and variety of potential singularities. Cases of ‘Poisson interaction’ are particularly amenable to treatment, however, and we examine two particular cases of interest in Sections 5 and 6. One is a version of the Ising model, which shows phase transition and its character very simply. The other is a version of the farmer–trader model considered previously, which now manifests a number of features, including a discontinuous transition between agrarian and mercantile economies. A stratified Poisson-interaction model is examined in Section 7. The roles of the various singularities are clarified, and a behaviour analogous to that of the single-type case confirmed.
Les fonctions sphériques d'un couple de Gelfand symétrique et les chaînes de Markov
Authors: Letac Gérard
Published: April 2006

After an elementary description of Gelfand pairs, spherical functions and Plancherel measure, some explicit computations on the related Markov chains are performed. Random walks on polyhedra belong to this class of Markov chains; two more examples of chains on graphs are worked out, and the necessary and sufficient condition of transcience of random walks on p-adic numbers with spherical symmetry is given as an application of the techniques of the paper.
Two suggestions of how to define a multistate coherent system
Authors: Natvig B.
Published: April 2006

One inherent weakness of traditional reliability theory is that the system and the components are always described just as functioning or failed. However, recent papers by Barlow and Wu (1978) and El-Neweihi et al. (1978) have made significant contributions to start building up a theory for a multistate system of multistate components. Here the states represent successive levels of performance ranging from a perfect functioning level down to a complete failure level. In the present paper we will give two suggestions of how to define a multistate coherent system. The first one is more general than the one introduced in the latter paper, the results of which are, however, extendable. (This is also true for a somewhat more general model than ours, treated in independent work by Griffith (1980).) Furthermore, some new definitions and results are given (which trivially extend to the latter model). Our second model is similarly more general than the one introduced in Barlow and Wu (1978), the results of which are again extendable. In fact we believe that most of the theory for the traditional binary coherent system can be extended to our second suggestion of a multistate coherent system.
On the extinction time distribution of a branching process in varying environments
Authors: Fujimagari Tetsuo
Published: April 2006

The extinction time distributions of a class of branching processes in varying environments are considered. We obtain (i) sufficient conditions for the extinction probability q = 1 or q < 1; (ii) asymptotic formulae for the tail probability of the extinction time if q = 1; and (iii) upper bounds for 1 – q if q < 1. To derive these results, we give upper and lower bounds for the tail probability of the extinction time. For the proofs, we use a method that compares probability generating functions with fractional linear generating functions.
Saddle point approximation for the distribution of the sum of independent random variables
Authors: Lugannani Robert , Rice Stephen
Published: April 2006

In the present paper a uniform asymptotic series is derived for the probability distribution of the sum of a large number of independent random variables. In contrast to the usual Edgeworth-type series, the uniform series gives good accuracy throughout its entire domain. Our derivation uses the fact that the major components of the distribution are determined by a saddle point and a singularity at the origin. The analogous series for the probability density, due to Daniels, depends only on the saddle point. Two illustrative examples are presented that show excellent agreement with the exact distributions.
Measuring interference in the chiasma renewal formation process
Authors: Karlin Samuel , Liberman Uri
Published: April 2006

Multilocus recombination structure is introduced using recombination and linkage values. The notion of chiasma interference is discussed and the coincidence measure of interference is defined. The global nature of positive, negative or non-interference is characterized using linkage values. In the case of the renewal chiasma formation process with interdistance distributionFit is shown that the nature of interference depends on the ‘ageing' properties of F such as IFR, NBU, NBUE and DMRL.
Zone of influence models for competition in plantations
Authors: Gates D. J. , Westcott M.
Published: April 2006

A plantation is modelled in terms of discs of random size centred on the points of a two-dimensional lattice, together with rules for partitioning the union of these discs. The resulting sets are functions of random sequences, and some properties of these functions are deduced. Results are given which relate various properties of measurements on a plantation to measurements of individual plants and of plants in isolation. These include limit theorems for complete plantations and for samples. They also lead to a rigorous demonstration of certain well-known empirical relations between typical measurements and survival frequencies and to some new relations which are amenable to test. The model is a formulation of computer simulations which have had success in describing competition in plantations, but are too costly for routine forest management.
The geometry of random drift IV. Random time substitutions and stationary densities
Authors: Antonelli Peter L.
Published: April 2006

In the present paper a spatially homogeneous distance measure of Edwards and Cavalli-Sforza type is derived for multiple allele random genetic drift with a (possibly vanishing) symmetric mutation field, using the technique of random time substitution. Since the mutation field is of gradient type in gene frequency space, the transformed process is proved to be Brownian motion relative to a new Riemannian geometry and a new time measure. The new geometry is conformally related to spherical geometry of the original process but is not of constant curvature, generally. A formula relating the stationary density of the old and new process is derived and the edge length formula for the new geometry on the n-simplex frequency space is given and analysed.
The envelope of an oscillatory process and its upcrossings
Authors: Hasofer A. M. , Petocz P.
Published: April 2006

LetX(t) be a real-valued stochastic process with sample functions which are almost surely continuously differentiable. We say thatX(t) has an upcrossing at leveluat timetifX(t) =u,X′(t) > 0.
Distribution of arrival times in cosmic ray showers
Authors: Green H. S.
Published: April 2006

The theoretical analyses of the extensive air showers developing from the cosmic radiation has its origins in the work of Carlson and Oppenheimer (1937) and Bhabha and Heitler (1937), at a time when it was thought that such showers were initiated by electrons. The realization that protons and other nuclei were the primary particles led to a reformulation of the theory by Heitler and Janossy (1949), Messel and Green (1952) and others, in which the production of energetic pions and the three-dimensional development of air showers were accounted for. But as the soft (electromagnetic) component of the cosmic radiation is the most prominent feature of air showers at sea level, there has been a sustained interest in the theory of this component. Most of the more recent work, such as that by Butcher and Messel (1960) and Thielheim and Zöllner (1972) has relied on computer simulation; but this method has disadvantages in terms of accuracy and presentation of results, especially where a simultaneous analysis of the development of air showers in terms of several physical variables is required. This is so for instance when the time of arrival is one of the variables. Moyal (1956) played an important part in the analytical formulation of a stochastic theory of cosmic ray showers, with time as an explicit variable, and it is essentially this approach which will be adopted in the following. The actual distribution of arrival times is cosmic ray showers, for which results are obtained, is of current experimental interest (McDonald, Clay and Prescott (1977)).
Rates of convergence for time series regression
Authors: Hannan E. J.
Published: April 2006

Consider, initially, a time series regression model of the simplest kind, namely

Assume that x(t) is second-order stationary with zero mean and absolutely continuous spectrum with density f(ω) so that

The y(t) are taken to be part of a sequence generated entirely independently of x(t) and will be treated as constants. Let βN be the least squares estimate of β and

Call the numerator and denominator of b(N), respectively, c(N), d(N). We shall use K for a positive finite constant, not always the same one. We have the following result, which is Menchoff's inequality [3].
Time-reversible diffusions
Authors: Kent John
Published: April 2006

Symmetric diffusions on manifolds with boundary are studied. Symmetric diffusions are nicer to work with than non-symmetric diffusions because (1) it is easier to tell if an equilibrium density exists, and (2) it is easier to find the equilibrium density when it does exist. If an equilibrium density exists, a symmetric diffusion istime reversible.On the line essentiallyalldiffusions are symmetric.

Using symmetric diffusions, it is shown that a large family of densities can be realized as equilibrium densities of time-reversible diffusions. Some examples are given.
The diffusion approximation for tandem queues in heavy traffic
Authors: Harrison J. Michael
Published: April 2006

Consider a pair of single server queues arranged in series. (This is the simplest example of a queuing network.) In an earlier paper [2], a limit theorem was proved to justify a heavy traffic approximation for the (two-dimensional) equilibrium waiting-time distribution. Specifically the waiting-time distribution was shown to be approximated by the limit distribution F of a certain vector stochastic process Z. The process Z was defined as an explicit, but relatively complicated, transformation of vector Brownian motion, and the general problem of determining F was left unsolved.

It is shown in this paper that Z is a diffusion process (continuous strong Markov process) whose state space S is the non-negative quadrant. On the interior of S, the process behaves as an ordinary vector Brownian motion, and it reflects instantaneously at each boundary surface (axis). At one axis, the reflection is normal, but at the other axis it has a tangential component as well. The generator of Z is calculated.

It is shown that the limit distribution F is the solution of a first-passage problem for a certain dual diffusion process Z∗. The generator of Z∗ is calculated, and the analytical theory of Markov processes is used to derive a partial differential equation (with boundary conditions) for the density f of F. Necessary and sufficient conditions are found for f to be separable (for the limit distribution to have independent components). This extends slightly the class of explicit solutions found previously in [2]. Another special case is solved explicitly, showing that the density is not in general separable.
Uniform rates of convergence in extreme-value theory
Authors: Smith Richard L.
Published: April 2006

Rates of convergence are derived for the convergence in distribution of renormalised sample maxima to the appropriate extreme-value distribution. Related questions which are discussed include the estimation of the principal error term and the optimality of the renormalising constants. Throughout the paper a close parallel is drawn with the theory of slow variation with remainder. This theory is used in proving most of the results. Some applications are discussed, including some models of importance in reliability.
A stochastic model of neural response
Authors: Walsh John B.
Published: April 2006

We propose a model of a passive nerve cylinder undergoing random stimulus along its length. It is shown that this model is approximated by the solution of a stochastic partial differential equation. Numerous properties of the sample paths are derived, such as their modulus of continuity, quadratic and quartic variation, and it is shown that the solution exhibits the phenomenon of flicker noise. The first-passage problem is studied, and it is shown to be connected with a first-hitting time for an infinite-dimensional diffusion.
Testing the hypothesis that a point is Poisson
Authors: Davies Robert B.
Published: April 2006

The testing of the hypothesis that a point process is Poisson against a one-dimensional alternative is considered. The locally optimal test statistic is expressed as an infinite series of uncorrelated terms. These terms are shown to be asymptotically equivalent to terms based on the various orders of cumulant spectra. The efficiency of tests based on partial sums of these terms is found.
Trace inequalities for mixtures of Markov chains
Authors: Singer Burton , Spilerman Seymour
Published: April 2006

In a wide variety of multi-wave panel studies in economics and sociology, comparisons between the observed transition matrices and predictions of them based on time-homogeneous Markov chains have revealed a special kind of discrepancy: the trace of the observed matrices tends to be larger than the trace of the predicted matrices. A common explanation for this discrepancy has been via mixtures of Markov chains.

Specializing to mixtures of Markov semi-groups of the form

we exhibit classes of stochastic matrices M, probability measures µ and time intervals Δ such that

for k = 2, 3 and 4. These examples contradict the substantial literature which suggests — implicitly — that the above inequality should be reversed for general mixtures of Markov semi-groups.
Markovian contact processes
Authors: Mollison Denis
Published: April 2006

Markovian contact processes(mcp's) include some of the commoner models for the spatial spread of population processes, such as the ‘simple’ and ‘general’ epidemics, percolation processes, and birth, death and migration processes. They are here set in a framework which relates them to each other, and to a particular basic model, thecontact birth process(cbp); indeed they are defined as modifications of the cbp. The immediate advantage of this is that bounds for the velocity of the cbp, which can be obtained from the linear equation describing its expected numbers (provided the contact distribution has exponentially bounded tail) apply also to general mcp's. Existence of moments (and for some processes their time-derivatives) ofSt(θ), the distance to the furthest individual in an arbitrary direction θ, can also be deduced whenever the corresponding moment of the contact distribution exists.

An important subclass consists ofpopulation-monotonemcp's, for which the distributions ofStcan be shown to besubconvolutive, so that the work of Hammersley (1974) can be applied to obtain convergence theorems forSt/t; and these can be extended to convergence of the convex hull of the set of inhabited points. These results are particularly valuable because they apply to some non-linear processes, e.g. the simple epidemic. Some special results on the cbp (Section 6) emphasize the differences between linear and non-linear spatial stochastic processes.

Although the paper is written throughout in terms of continuous-time processes, the results on subconvolutive distributions are actually more easily applied in the discrete-time case. (Conversely, the approach used in the accompanying paper by Biggins (pp. 62–84), which is written in terms of discrete-time processes, can be extended to deal also with continuous-time processes.)
Martingale central limit theorems and asymptotic estimation theory for multitype branching processes
Authors: Asmussen Søren , Keiding Niels
Published: April 2006

This paper considers the problem of estimating the growth rate ρ of ap-type Galton–Watson process {Zn}. To this end, a general approach of possible independent interest to central limit theorems for discrete-time branching processes is developed. The idea is to adapt martingale central limit theory to martingale difference triangular arrays indexed by the set of all individuals ever alive. Iterated logarithm laws are derived by similar methods. Asymptotic distribution results and the a.s. asymptotic behaviour are derived for a maximum likelihood estimator based upon all parent–offspring combinations in a given numberNof generations, and for the estimatorwhich depends on the total generation sizesonly.
A time series approach to the study of the simple subcritical Galton–Watson process with immigration
Authors: Venkataraman K. N.
Published: April 2006

The principal aim of this paper is to exhibit applications of techniques of time series analysis for establishing limit distribution theorems of statistical relevance on a subcritical Galton–Watson processXwith immigration. In this approach the results obtained by Heyde and Seneta, Quine, and Klimko and Nelson are re-established in a more concise form on adopting new methods of proof, which seek to unify these results. In addition, Quenouille-type limit theorems onXare proved leading to the construction of Quenouille-type goodness-of-fit tests forX.It appears that Billingsley's central limit theorem for martingales is appropriate for proving the basic result, Theorem 1.1. This is done on converting the entire problem as a martingale problem through a use of Lemma 2 of Venkataraman (1968).
Steinhaus's geometric location problem for random samples in the plane
Authors: Hochbaum Dorit , Steele J. Michael
Published: April 2006

Let

whereXi, 1 ≦i≦n, are i.i.d. and uniformly distributed in [0, 1]2. It is proved thatMn∽cn1–p/2a.s. for 1 ≦p<2. This result is motivated by recent developments in the theory of algorithms and the theory of subadditive processes as well as by a well-known problem of H. Steinhaus.