bluuebunny/crossref_metadata_2025 huggingface dataset:
NEW COMBINATIONS IN ZANTHOXYLUM L. (1753)
Authors: Waterman P. G.
Published: April 2006

The evidence for the combination of the rutaceous genera Zanthoxylum and Fagara is presented and the new combinations are proposed.
ON THE TYPIFICATION OF HEXAGONIA POLLINI PER FR.
Authors: Donk M. A.
Published: April 2006

The author does not agree with M. E. P. K. Fidalgo's replacement of Hexagonia mori Pollini per Fr. as the type of Hexagonia Pollini per Fr. by Hexagonia crinigera Fr. Her arguments in favour of this substitution are scrutinized below and found to be untenable. The correct name for Hexagonia sensu K. Fid. is Scenidium (Kl.) ex O.K. For Hexagonia sensu Bond. & S. (not validly published) the new name Apcxona Donk is proposed.
NOMENCLATURE OF THE SUBDIVISIONS OF THE LAMIACEAE
Authors: Sanders Roger W. , Cantino Philip D.
Published: April 2006

A list is provided of all the legitimate names available for application to subtribes, tribes, subfamilies, and segregate families of the Lamiaceae. The names Elsholtzieae, Prunelleae and Tetrachondraceae are formally validated. Also presented are earlier generically based names which are invalid or inoperative under Art. 35.2 of the Code. Correct names are provided for the familial subdivisions recognized in three widely accepted systematic treatments of the Lamiaceae. Action by the Sydney Congress that affects nomenclature of the subdivisions of families is discussed.
Suprageneric nomenclature in the Caryophyllaceae
Authors: Rabeier Richard K. , Bittrich Volker
Published: April 2006

Rabeler, R. K. & Bittrich, V.: Suprageneric nomenclature in the Caryophyllaceae. – Taxon 42: 857‐863.1993. – ISSN 0040‐0262.

All known suprageneric names within the Caryophyllaceae are listed.
Generating functions for a class of one-dimensional random walks
Authors: Percus Ora Engelberg , Percus Jerome K.
Published: April 2006

A weighted Markov chain technique is used to find the generating functions for several restricted one-dimensional random walks. Examples with simple restrictions concern the number of penetrations of a penetrable barrier. Examples with compound restrictions include the number of full crossings of the origin. A typical asymptotic evaluation is carried out.
Spectral inference over narrow bands
Authors: Hannan E. J. , Thomson P. J.
Published: April 2006

We consider a vector, discrete, time sequence x(n), n = 0, ± 1 ···, of p components Xj(n), j = 1, ··· p. For the most part we shall assume x(n) to be strictly stationary and with finite variances. Thus if μ is the mean vector we shall have
Storage with deterministic outputs and inputs subject to breakdown
Authors: Wu P. M.
Published: April 2006

In this paper, we shall consider a storage model in which a bin of finite capacity is fed by specified r(r > 0) or zero input depending on the content of the bin and the operation of the input machines (i.e., whether or not they are working on the bin). These machines are allowed to fill the bin until its level reaches maximum and they are then used to produce some secondary products not relevant to the storage problem. When the content of the bin decreases to a given minimum level, these machines are again set to feed the bin. The whole process is then repeated. It is often desirable in practice to keep the minimum level as low as possible, to attain stability in the process, or for maintenance purposes. When the machines are subject to breakdowns, it becomes necessary to consider the reliability of the operational system.
On random n-fold coincidences
Authors: Ivkovic Z. , Kocher M. , Stanojevic C. V.
Published: April 2006

The problem of random n-fold coincidences for Geiger counters was considered by Janossy [1] and Schrödinger [2]. In this note we consider a more general problem related to the following model.
On the law of the iterated logarithm for inter-record times
Authors: Strawderman William E. , Holmes Paul T.
Published: April 2006

Let X1, X2, X3, ··· be independent, identically distributed random variables on a probability space (Ω, F, P); and with a continuous distribution function. Let the sequence of indices {Vr} be defined as

Also define

The following theorem is due to Renyi [5].
A note on the cost of the simple epidemic
Authors: Jerwood D.
Published: April 2006

The concept of cost has not, until recently, been investigated in any detail within the framework of epidemic theory. In the case of minor illnesses a pertinent estimate of cost may be the man-hours lost in industry during an outbreak. For the general epidemic, in which the removals are immune recoveries (or deaths), this is conveniently represented by the area generated by the trajectory in time t of infectives I(t) until the epidemic ceases. A small element of area under the trajectory is I(t)δt and represents I(t) infectives withdrawn from the labour force for a small time δt; the total work-time lost is thus represented by where T is the duration of the epidemic.
The Galton-Watson process with infinite mean
Authors: Darling D. A.
Published: April 2006

Let Zn be the number of descendents in the nth generation of a simple Galton-Watson branching process, initiated by a single progenitor, Z0 = 1. If E(Z1) < ∞ the limiting distribution of Zn is known in some detail, and a comprehensive account is given in Seneta [1]. If E{Z1) = ∞ (the “explosive” case) the behavior of the distribution of Zn for large n seems not to be known. As shown by Seneta [1], there are no constants cn such that cnZn has a non-degenerate limiting distribution but it turns out that, under conditions given below, log(Zn + 1) has a limiting distribution.
A new family of life distributions
Authors: Birnbaum Z.W. , Saunders S.C.
Published: April 2006

A new two parameter family of life length distributions is presented which is derived from a model for fatigue. This derivation follows from considerations of renewal theory for the number of cycles needed to force a fatigue crack extension to exceed a critical value. Some closure properties of this family are given and some comparisons made with other families such as the lognormal which have been previously used in fatigue studies.
Simple random walks on semi-groups of nilpotency class two
Authors: Dowson D. C.
Published: April 2006

One of the earliest known distributions is that of the Binomial distribution which arises from a Bernoulli sequence defined on two symbols (or generators) a and b. The corresponding limit theorem is that of Demoivre and Laplace and states (in an obvious notation) that (r – np)/√npq converges to the standard Normal distribution N(0,1). If the generators do not commute the situation is a good deal more complicated and in order to say very much about the sequences generated we must be able to put them in some simple canonical form. One case in which this can certainly be done is when the two symbols generate a semi-group of nilpotency class two. This means that although ba ≠ ab, we do have ba = ab (b,a) where (b, a) is a symbol which commutes with both a and b. Each sequence can then be expressed in the form aαbβ(b,a)γ. In this paper we examine first the conditional distribution of γ given α and β for Bernoulli sequences in the symbols a and b and obtain central limit theorems when γ is appropriately normed. We then consider the more general problem of the m-generator semi-group of nilpotency class two and obtain the corresponding multi-dimensional central limit theorem in the case where the probability measure is discrete and is distributed over the generators.
Finite regular invariant measures for Feller processes
Authors: Beneš V. E.
Published: April 2006

In the study of dynamical systems perturbed by noise, it is important to know whether the stochastic process of interest has a stationary distribution. Four necessary and sufficient conditions are formulated for the existence of a finite invariant measure for a Feller process on a σ-compact metric (state) space. These conditions link together stability notions from several fields. The first uses a Lyapunov function reminiscent of Lagrange stability in differential equations; the second depends on Prokhorov's condition for sequential compactness of measures; the third is a recurrence condition on the ergodic averages of the transition operator; and the fourth is analogous to a condition of Ulam and Oxtoby for the nonstochastic case.
On road traffic with free overtaking
Authors: Thedéen Torbjörn
Published: April 2006

The cars are considered as points on an infinite road with no intersections. They can overtake each other without any delay and they travel at constant speeds. These are independent identically distributed random variables also independent of the initial positions of the cars. The main purpose of the paper is the study of the asymptotic distribution for the number of overtakings (and/or meetings) in increasing rectangles in the time-road plane. Under the assumption of (weighted) Poisson distributed cars along the time-axis we deduce the asymptotic distribution of the standardized number of overtakings (and/or meetings) for large rectangles in the time-road plane. Lastly we shall indicate an application of the results.
Negative binomial processes
Authors: Barndorff-Nielsen Ole , Yeo G. F.
Published: April 2006

This paper is concerned with negative binomial processes which are essentially mixed Poisson processes whose intensity parameter is given by the sum of squares of a finite number of independently and identically distributed Gaussian processes. A study is made of the distribution of the number of points of a k-dimensional negative binomial process in a compact subset of Rk, and in particular in the case where the underlying Gaussian processes are independent Ornstein-Uhlenbeck processes when more detailed results may be obtained.
A note on recent research in geometrical probability
Authors: Moran P. A. P.
Published: April 2006

The subject of geometrical probability has recently shown signs of renewed vigour which is partly due to the attractive nature of the many unsolved problems which it contains, and partly the ever widening field of applications. These applications are so diverse and so scattered in the literatures of different branches of science that when Professor M.G. Kendall and I published a monograph on the subject in 1963 some interesting papers were ignored. Others have since appeared and the purpose of the present paper is to survey, in a very brief manner, work not described in that book. In fact the present bibliography, which contains 94 items, is disjoint from the bibliography in our joint monograph.
Optimal search for a randomly moving object in a special case
Authors: Hellman Olavi
Published: April 2006

Search for an object whose motion is a symmetrical two-dimensional diffusion process is considered in the case where a given amount of search of constant density is distributed, during a given interval of time, over a fixed circular area. An expression for the probability of detection of the object during this interval of time is derived in the special case where the location of the object at time zero is given. The application of this expression would involve numerical work.
Stability of finite queue, tandem server systems
Authors: Hildebrand David K.
Published: April 2006

One of the troublesome problems in the extension of queueing theory to more complex systems of servers is the development of models suitable for the analysis of assembly lines and similar structures. Results in special cases are given in [2], [5], and [6]. A reasonable model would assume:
On survival probabilities and extinction times for some branching processes
Authors: Pollak Edward
Published: April 2006

We consider branching processes for which the first three moments of the distribution of offspring exist. Let f(t) andz be, respectively, the generating function of the distribution of offspring and the smallest positive root of the equation f(t) = t. Then if M = f'(z) and fn(t) is equal to the generating function of the distribution of nth generation descendants of a single individual, it is known that, quite generally, (z – fn(0)) / Mn tends toward a constant as n increases. A method is derived for obtaining upper and lower bounds for this constant, which gives an exact solution when there is a geometric distribution of offspring and good bounds when there are Poisson or negative binomial offspring distributions. With some further calculations, one can also obtain finite upper and lower bounds for the mean time to extinction of a line descended from an individual, given there is extinction. These bound apply even ifzis less than 1. Numerical values are given for the Poisson and negative binomial cases.
Feynman integral and phase space probability
Authors: Gilson J. G.
Published: April 2006

The relation between “forward” and “backward” definitions of quantum velocity is discussed. A connection between the Feynman Integral and Wigner's phase space distributions which was noted in an earlier paper is explored further and some new and useful formulae are derived.
On the autocorrelation and spectral functions of queues
Authors: Reynolds John F.
Published: April 2006

This paper considers the autocorrelation function of queue length and the corresponding spectral density (i.e., its Fourier transform). Some general expressions are obtained using generating functions and matrices, and applied to M/M/1 and M[x]/M/∞ queues.
A note on the characterization of a U-shaped probability density function
Authors: Ghosh P. , Shanbhag D. N.
Published: April 2006

In this note a characterization for a U-shaped probability density function is given using a well-known result for unimodal distributions due to Khinchine. As a corollary of this result, a characteristic property based on moments is found.
The random volume of interpenetrating spheres in space
Authors: Moran P. A. P.
Published: April 2006

The distribution of the volume occupied by random spheres in a cube is considered, both when the number of spheres is fixed and when their centres form a Poisson field. The mean and variance are obtained and in the latter case the distribution is proved to converge to normality. The probability of complete coverage is also obtained heuristically.
Branching processes with varying and random geometric offspring distributions
Authors: Keiding Niels , Nielsen John E.
Published: April 2006

The class of fractional linear generating functions is used to illustrate various aspects of the theory of branching processes in varying and random environments. In particular, it is shown that Church's theorem on convergence of the varying environments process admits of an elementary proof in this particular case. For random environments, examples are given on the asymptotic behavior of extinction probabilities in the supercritical case and conditional expectation given non-extinction in the subcritical case.
Characterization theorems using conditional expectations
Authors: Kotlarski Ignacy I. , Hinds William E.
Published: April 2006

Recent characterizations of probability measures by use of conditional expectations have included many probability measures defined on subsets of the real line. This paper contains results which allow similar characterizations by use of conditional expectations but are valid for probability measures defined on subsets of an arbitrary space. In the special case of a measure defined on subsets of R' the result is not always as sharp as known results, but these theorems can be applied to measures defined on subsets of Rn with n ≧ 1.
The maximum and time to absorption of a left-continuous random walk
Authors: Green P. J.
Published: April 2006

For a left-continuous random walk, absorbing at 0, the joint distribution of the maximum and time to absorption is derived. A description of the tails of the distributions and a conditional limit theorem are obtained for the cases where absorption is certain.
Probability of extinction of critical generation-dependent Galton–Watson processes
Authors: Fearn Dean H.
Published: April 2006

The limiting behavior of the probability of extinction of critical generation-dependent branching processes is determined. Lower bounds are determined for this probability. A means of classifying these branching processes is suggested.
The calculation of limit probabilities for denumerable Markov processes from infinitesimal properties
Authors: Tweedie Richard L.
Published: April 2006

The problem considered is that of estimating the limit probability distribution (equilibrium distribution) πof a denumerable continuous time Markov process using only the matrix Q of derivatives of transition functions at the origin. We utilise relationships between the limit vector πand invariant measures for the jump-chain of the process (whose transition matrix we write P∗), and apply truncation theorems from Tweedie (1971) to P∗. When Q is regular, we derive algorithms for estimating πfrom truncations of Q; these extend results in Tweedie (1971), Section 4, from q-bounded processes to arbitrary regular processes. Finally, we show that this method can be extended even to non-regular chains of a certain type.
A proof of optimality for age replacement policies
Authors: Berg Menachem
Published: April 2006

A well-known maintenance procedure for stochastically failing units is the age replacement policy (ARP). Under the ARP a unit is replaced on failure, or when it reaches a predetermined critical age.

In previous papers concerned with this procedure the authors searched for the optimal critical age to minimize the objective function under discussion. The purpose of this paper is to prove that the ARP itself is the optimal decision rule amongst all reasonable policies.
Weak limits of sample range
Authors: Haan Laurens De
Published: April 2006

Necessary and sufficient conditions are obtained for the weak convergence of the sample range of i.i.d. random variables as the number of observations tends to infinity.
Solving a singular diffusion equation occurring in population genetics
Authors: Jensen Louis
Published: April 2006

A technique for solving a partial differential equation is presented. The technique is based upon the known solution of a similar equation. The method is used to attempt to solve the equation describing the change in the frequency of an allele in the presence of selection and random drift in a finite population. Two cases can be solved within a reasonable degree of approximation: (a) the viabilities are additive and (b) heterozygotes are symmetrically overdominant to the homozygotes. The solutions in both cases are compared with the exact discrete solutions found by powering the re evant transition matrix.
A storage model in which the net growth-rate is a Markov chain
Authors: Brockwell P. J.
Published: April 2006

The distribution of the times to first emptiness and first overflow, together with the limiting distribution of content are determined for a dam of finite capacity. It is assumed that the rate of change of the level of the dam is a continuous-time Markov chain with finite state-space (suitably modified when the dam is full or empty).
Averages for polygons formed by random lines in Euclidean and hyperbolic planes
Authors: Santaló L. A. , Yañez I.
Published: April 2006

We consider a countable number of independent random uniform lines in the hyperbolic plane (in the sense of the theory of geometrical probability) which divide the plane into an infinite number of convex polygonal regions. The main purpose of the paper is to compute the mean number of sides, the mean perimeter, the mean area and the second order moments of these quantities of such polygonal regions. For the Euclidean plane the problem has been considered by several authors, mainly Miles [4]–[9] who has taken it as the starting point of a series of papers which are the basis of the so-called stochastic geometry.
Some theorems on the transient covariance of Markov chains
Authors: Reynolds John F.
Published: April 2006

Several authors have considered the covariance structure of continuous parameter Markov chains. Most of this work has dealt with particular process ses, notably Morse (1955) who analysed the simple M/M/1 queue and Bene-(1961) who considered a telephone trunking model. Furthermore, the results obtained apply only when the process has attained its limiting (stationary) distribution. A recent paper by Reynolds (1968) gave some general results for finite chains, still assuming stationarity. This note generalises the results obtained therein, and considers the covariance structure during the transient period prior to attaining the stationary distribution where this exists. In the case where no such distribution exists, the results are valid throughout the whole lifetime of the process.
On possible rates of growth of age-dependent branching processes with immigration
Authors: Grey D. R.
Published: April 2006

It is shown that if ϕ is a given function out of a large class satisfying a certain regularity condition, then a supercritical age-dependent branching process {Z(t)} exists with deterministic immigration and given life-length and family-size distributions such that Z(t)/(eat ϕ(t)) converges in probability to a non-zero constant, a being the appropriate Malthusian parameter.

As an easy corollary one discovers the asymptotic behaviour of some processes with random immigration.
Functional and random central limit theorems for the Robbins-Munro process
Authors: McLeish D. L.
Published: April 2006

A functional central limit theorem extending the central limit theorem of Chung (1954) for the Robbins–Munro procedure is proved. It is shown that the asymptotic normality is preserved under certain random stopping rules.
A single-server queue with limited virtual waiting time
Authors: Takács Lajos
Published: April 2006

The limiting distributions of the actual waiting time and the virtual waiting time are determined for a single-server queue with Poisson input and general service times in the case where there are two types of services and no customer can stay in the system longer than an interval of length m.
A birth and death model of neuron firing
Authors: Logan B. F. , Shepp L. A.
Published: April 2006

A simple birth-death model of particle fluctuations is studied where at each discrete time a birth and/or death may occur. We show that if the probability of a birth does not depend on the number of particles present and if births and deaths are independent, then the times between successive deaths are independent geometrically distributed random variables, which is false in the general case. Since the above properties of the times between successive neuron firings have been observed in nerve cells, the model proposed in [2] obtains added credence.
Random environment branching processes with equal environmental extinction probabilities
Authors: Raffety Donald C.
Published: April 2006

R-positivity theory for Markov chains is used to obtain results for random environment branching processes whose environment random variables are independent and identically distributed and whose environmental extinction probabilities are equal. For certain processes whose eventual extinction is almost sure, it is shown that the distribution of population size conditioned by non-extinction at time n tends to a left eigenvector of the transition matrix. Limiting values of other conditional probabilities are given in terms of this left eigenvector and it is shown that the probability of non-extinction at time n approaches zero geometrically as n approaches ∞. Analogous results are obtained for processes whose extinction is not almost sure.
Two mean values which characterize the Poisson process
Authors: Çinlar Erhan , Jagers Peter
Published: April 2006

The Poisson process enjoys two special properties: the mean forward recurrence time at time t does not depend on t, and the mean backward recurrence time at time t is the “mean” of the interval distribution truncated at t. Poisson process is the only renewal process with these properties.
The cost of a general stochastic epidemic
Authors: Gani J. , Jerwood D.
Published: April 2006

This paper is concerned with the cost Cis = aWis + bTis (a, b > 0) of a general stochastic epidemic starting with i infectives and s susceptibles; Tis denotes the duration of the epidemic, and Wis the area under the infective curve. The joint Laplace-Stieltjes transform of (Wis, Tis) is studied, and a recursive equation derived for it. The duration Tis and its mean Nis are considered in some detail, as are also Wis and its mean Mis. Using the results obtained, bounds are found for the mean cost of the epidemic.
Best asymptotic value of the error exponent with random coding on discrete memoryless channels
Authors: Kambo N. S. , Singh Samar
Published: April 2006

Generalizing a technique given by Wolfowitz, we calculate a formula for the asymptotic value of the error exponent with random coding on a discrete memoryless channel. We then evaluate this formula analytically for low rates and show that the exact value given by this formula agrees with the random coding exponent of Gallager and Fano. This proves that their exponent, which was only known to be a bound for low rates, gives the exact value and this brings out the inherent limitation of the random coding method.
Contributions to the theory of optimal resource allocation
Authors: Laska E. , Meisner M. , Siegel C.
Published: April 2006

A problem in optimal resource allocation is considered for n jobs with identically distributed service times admitting a monotone hazard function. If the hazard function is increasing, it is shown that the procedure of allocating the full resource individually to each job until its completion minimizes the expected completion time of the jth job. The procedure which at any instant of time equally allocated the resource among all of the remaining jobs is shown to minimize the expected total cumulative waiting time if the hazard is decreasing.
Diffusion limits of critical branching processes conditioned on extinction in the near future
Authors: Esty Warren W.
Published: April 2006

Limit laws and limiting diffusions are obtained for critical branching processes, either Galton-Watson or age-dependent, conditioned on extinction in the interval (T, cT], 1 < c, as T→∞, and also as T→∞ and c ↓ 1.
The extinction time of Markov branching processes
Authors: Wang Robert C.
Published: April 2006

This paper considers the expression of expectations and density function of the extinction time for a continuous-time Markov branching process.
A note on Spitzer's lemma and its application to the maximum of partial sums of dependent random variables
Authors: Solari M. E. , Dunnage J. E. A.
Published: April 2006

We give an expression for the expectation of max (0, S1, …, Sn) where Sk is the kth partial sum of a finite sequence of exchangeable random variables X1, …, Xn. When the Xk are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.
Multivariate extremal processes generated by independent non-identically distributed random variables
Authors: Weissman Ishay
Published: April 2006

Let be the kth largest among Xn1, …, Xn[nt], where Xni = (Xi – an)/bn, {Xi} is a sequence of independent random variables and bn > 0 and an are norming constants. Suppose that for each converges in distribution. Then all the finite-dimensional laws of converge. The limiting process is represented in terms of a non-homogeneous two-dimensional Poisson process.
Polynomial bounds for probability generating functions
Authors: Braun Henry
Published: April 2006

The problem of approximating an arbitrary probability generating function (p.g.f.) by a polynomial is considered. It is shown that if the coefficients rj are chosen so that LN(·) agrees with g(·) to k derivatives at s = 1 and to (N – k) derivatives at s = 0, then LN is in fact an upper or lower bound to g; the nature of the bound depends only on k and not on N. Application of the results to the problems of finding bounds for extinction probabilities, extinction time distributions and moments of branching process distributions are examined.
On some properties of the scan statistic on the circle and the line
Authors: Cressie Noel
Published: April 2006

The scan statistic is defined as the supremum of a particular continuous-time stochastic process, and is used as a test statistic for testing uniformity against a simple clustering type of alternative. Its distribution under the null hypothesis is investigated and weak convergence of the stochastic process to the appropriate Gaussian process is proved. An interesting link is forged between the circular scan statistic and Kuiper's statistic, which rids us of the trouble of estimating a nuisance parameter. Distributions under the alternative are then derived, and asymptotic power comparisons are made.
Three characterizations of population strategy stability
Authors: Hines W. G. S.
Published: April 2006

In addition to the concept of the evolutionarily stable strategy (ESS), developed specifically for considering intraspecific conflicts, concepts such as the Nash equilibrium from game theory and the attractor or sink from dynamical systems theory appear relevant to the problem of characterizing populations of stable composition. The three concepts mentioned are discussed for one simple standard population model. It is found that evolutionarily stable strategies of one type are necessarily Nash equilibrium strategies, although the converse is not true. The dynamical systems characterization is found to provide a model for populations susceptible to invasion by ‘co-operative' strategies, but capable of evolving back in average to the original equilibrium.
On the optimal assignment of servers and a repairman
Authors: Derman C. , Lieberman G. J. , Ross S. M.
Published: April 2006

We consider an N-server queue with arbitrary arrivals and service times which are random but with differing rates for different servers. Customers arriving when all servers are occupied do not enter the system. We show that the policy of always assigning an arrival to that free server whose service rate is largest (smallest) stochastically minimises (maximises) the number in the system. We then show that in a particular component-repair context with exponential repair times the policy of repairing failed components with the smallest failure rate stochastically maximises the number of working components.
Processor-sharing of two parallel lines
Authors: Konheim Alan G. , Meilijson Isaac , Melkman Abraham
Published: April 2006

An exponential server splits its service capacity between two independent Poisson streams of customers, unless one queue is empty, in which case the full service capacity is granted to the other queue. We present the stationary joint generating function of the two queue sizes explicitly in the form of an elliptic integral.
A limit theorem for patch sizes in a selectively-neutral migration model
Authors: Sawyer Stanley
Published: April 2006

Assume a population is distributed in an infinite lattice of colonies in a migration and random mating model in which all individuals are selectively equivalent. In one and two dimensions, the population tends in time to consolidate into larger and larger blocks, each composed of the descendants of a single initial individual. Let N(t) be the (random) size of a block intersecting a fixed colony at time t. Then E[N(t)] grows like √t in one dimension, t/log t in two, and t in three or more dimensions. On the other hand, each block by itself eventually becomes extinct. In two or more dimensions, we prove that N(t)/E[N(t)] has a limiting gamma distribution, and thus the mortality of blocks does not make the limiting distribution of N(t) singular. Results are proven for discrete time and sketched for continuous time.

If a mutation rate u > 0 is imposed, the ‘block structure' has an equilibrium distribution. If N(u) is the size of a block intersecting a fixed colony at equilibrium, then as u → 0 N(u)/E[N(u)] has a limiting exponential distribution in two or more dimensions. In biological systems u ≈ 10–6 is usually quite small.

The proofs are by using multiple kinship coefficients for a stepping stone population.
A note on coupling of birth and death processes
Authors: Lindvall Torgny
Published: April 2006

The purpose of this note is to show how well the coupling device is fitted for use in the study of birth and death process asymptotics: known achievements get new proofs, and new results on rate of ‘forgetfulness of initial state' and stochastic domination are established.
Scheduled maxima sequences
Authors: Deken Joseph G.
Published: April 2006

We define a vector-valued scheduled maxima sequence M by considering simultaneously the maxima of several i.i.d. sequences, with the number of observations considered from each sequence at any time determined by a random scheduling sequence J. It is shown that the max-min (vector) sequence derived from i.i.d. can be represented as a mixture of scheduled maxima sequences, giving results for this sequence and the range A functional limit theorem for the scheduled maxima sequence shows convergence to independent extremal processes. Embedding in a scheduled extremal process gives strong laws, central limit theorems, and laws of the iterated logarithm for the record time of the scheduled maxima sequence, and hence for the max-min sequence and the range.
Non-homogeneously paced random records and associated extremal processes
Authors: Gaver Donald P. , Jacobs Patricia A.
Published: April 2006

A study is made of the extremal process generated by i.i.d. random variables appearing at the events of a non-homogeneous Poisson process, 𝒫. If 𝒫 has an exponentially increasing rate function, then records eventually occur in a homogeneous Poisson process. The size of the latest record has a classical extreme value distribution.
One-sided excursions of brownian motion, and an application
Authors: Imhof J. P.
Published: April 2006

The value of a process on C[0,∞) when it starts its first one-sided excursion of width ≧ α is related to first hitting one of two boundaries. Explicit results for brownian motion are applied to a finite dam model.
Transient distribution of the number of segregating sites in a neutral infinite-sites model with no recombination
Authors: Griffiths R. C.
Published: April 2006

The transient distribution of the number of segregating sites in a sample from a large population of 2Ngenes is found. Segregating sites are split into those in common with the sites segregating in the initial population and those segregating due to new mutations. The waiting time distribution for a population to lose all of its initial sites is also studied. A neutral infinite-sites model with no recombination is used. This paper extends the work of Watterson (1975), from the stationary case; and of Li (1977) from the transient distribution in a sample of 2.
A non-random limit for the volume covered k times
Authors: Stam A. J.
Published: April 2006

Let n sets with volume ~ n-–1 be placed in Rm, independently and with the same distribution. As n →∞, the volume in V CRm, covered by exactly k of these sets under certain conditions converges to a non-random limit, which is the integral over V of a density that is of the type of a Poisson probability.
Adaptive control of a production-inventory system
Authors: Doshi Bharat T.
Published: April 2006

We study a production-inventory system in which input is deterministic, and its rate is the controlled parameter. The output is a compound Poisson process with exponential jump-size distribution. The parameters of the output process are unknown. The following adaptive policy is used: At each time t the unknown parameters are estimated by maximum likelihood method and the input rate which would be optimal if these were the true values of the parameter is used. The issues investigated are (1) the convergence of maximum likelihood estimates to the true value, and (2) the asymptotic properties of the cost under the adaptive policy.
The use of prior information in the estimation of the optimal interpolator parameter
Authors: Kabaila Paul V.
Published: April 2006

We consider the estimation of the parameter of the best interpolator from a restricted class of linear, time-invariant interpolators for the weakly stationary process {xt}. The estimation is supposed to be based on a part-realisation x1, · ··, xn and on certain prior information regarding the spectral density of the process. Two methods of estimation are proposed — one which uses x1, · ··, xn and the prior information and another which uses only x1, · ··, xn. For the methods proposed it is reasonable to conjecture that the method which utilises the prior information as well as x1, · ··, xn, will be more efficient. Reasonable as this conjecture may seem, its proof is by no means obvious and is the main subject of the paper.
Scheduling spares with exponential lifetimes in a two-component parallel system
Authors: Pinedo Michael
Published: April 2006

Consider a two-component parallel system. There is a stockpile of n spares, spare i having lifetime Xi, exponentially distributed with rate µi. The system breaks down when neither component functions. Every time a spare expires it has to be replaced, to avoid system breakdown. We investigate the policies which maximize the expected lifetime of the system using these n spares.
An exponential Markovian stationary process
Authors: Tavares L. Valadares
Published: April 2006

A new markovian process {Xi : i = 0, 1, 2, ·· ·} following a negative exponential distribution and with the same autocorrelation function as the lag-1 autoregressive process is proposed and studied in this paper. The exact distribution of the maxima and of the minima of n consecutive Xi values are obtained and the exact expected upcrossing interval is given for any crossing level.
Optimality of the one step look-ahead stopping times
Authors: Abdel-Hameed M.
Published: April 2006

The optimality of the one step look-ahead stopping rule is shown to hold under conditions different from those discussed by Chow, Robbins and Seigmund [5]. These results are corollaries of the following theorem: Let {Xn, n = 0, 1, …}; X0 = x be a discrete-time homogeneous Markov process with state space (E, ℬ). For any ℬ-measurable function g and α in (0, 1], define Aαg(x) = αExg(X1) – g(x) to be the infinitesimal generator of g. If τ is any stopping time satisfying the conditions: Ex[αNg(XN)I(τ > N)]→0 as as N → ∞, then Applications of the results are considered.
Distance distributions associated with poisson processes of geometric figures
Authors: Berman Mark
Published: April 2006

Consider a process of identically-shaped (but not necessarily equal-sized) figures (e.g. points, clusters of points, lines, spheres) embedded at random in n-dimensional space. A simple technique is derived for finding the distribution of the distance from a fixed point, chosen independently of the process of figures, to the k th nearest figure. The technique also shows that the distribution is independent of the distribution of the orientations of the figures. It is noted that the distribution obtained above (for equal-sized figures) is identical to the distribution of the distance from a fixed figure to the k th nearest of a random process of points.
An estimate for the tails of the distribution of the supremum for a class of stationary multiparameter Gaussian processes
Authors: Cabaña E. M. , Wschebor M.
Published: April 2006

Using Slepian processes as a standard of comparison, estimates are given for the probability that a centered multiparameter stationary Gaussian process reaches a constant barrier u on a subset of the parameter domain.
On the variance of a trimmed Poisson distribution
Authors: Kemperman J. H. B.
Published: April 2006

Consider a queue for a special performance or sale. A person with arrival number i will decide to join the queue with probability pi. Here the pi denote fixed constants. Let X denote the number of arrivals, X″ the number of customers leaving the queue and X′ = X – X″ the number of customers who stay with the queue. For the case that pi+1 ≦ pi and X has a Poisson distribution, it is shown that Var (Xʺ) ≧ E(Xʺ) and Var (X′) ≦ E(X′). There are also results for the case where {pi} and X are rather arbitrary.
A representation for the limiting random variable of a branching process with infinite mean and some related problems
Authors: Cohn Harry , Pakes Anthony G.
Published: April 2006

It is known that for a Bienaymé– Galton–Watson process {Zn} whose mean m satisfies 1 < m < ∞, the limiting random variable in the strong limit theorem can be represented as a random sum of i.i.d. random variables and hence that convergence rate results follow from a random sum central limit theorem.

This paper develops an analogous theory for the case m = ∞ which replaces ‘sum' by ‘maximum'. In particular we obtain convergence rate results involving a limiting extreme value distribution. An associated estimation problem is considered.
Stochastic analysis of the raw silk reeling process
Authors: Miura M.
Published: April 2006

We consider the variation in size of raw silk in the fixed size raw silk reeling process, and the relations between occurrences of feeding ends and some properties of cocoons from the stochastic viewpoint. The results obtained by stochastic analysis appear reasonable for the reeling experiment using three groups of cocoons.
Asymptotic properties of subcritical Galton-Watson processes
Authors: Bagley J. H.
Published: April 2006

Results connecting Yaglom's limit and Z1 are obtained. For example, if α > 1, E[Zα1] and converge or diverge together.
Un modele probabiliste utile en biologie moleculaire
Authors: Joffe Anatole
Published: April 2006

Let X1 ··· Xn be i.i.d.r.v.'s uniformly distributed on [0, l]. Let X(1) ··· X(n) be the ordered r.v.'s 0≦ X(1) ≦ X(2)···≦ X(n) ≦ l. We obtain the n + 1 intervals (X(i), X(i+1))ni=0, with X(0) = 0, X(n+1)= l of length Li = X(i+1) – X(i). Let N(c) be the number of intervals of length Li < c. We have where

The study of Nc is of some interest in molecular biology and was studied in [3] where the author found a complicated expression for the probability for a randomly chosen segment to have a length less than or equal to c; he then obtains an approximation. From the experimental point of view, it seems more reasonable to work with the mean since one can obtain an unbiased estimator that way.
Correlations and regressions in ‘family’ data
Authors: Smith Cedric A. B.
Published: April 2006

We study the situation in which individuals occur in ‘families' or similar groups, individuals within a ‘family' being correlated with one another, as for example a biological population. In such a population, the number of individuals will usually vary from one family to another. We assume that a sample chosen from the population consists of whole families rather than unrelated individuals. A similar situation might occur in experimental design, if individuals occur in ‘blocks' (of varying sizes) within which they share a common environment. In the simplest case considered here two measurements are made on each individual, namely y, the character of interest, andx(not necessarily a random variable), some other character which is believed to influence y. We discuss how to estimate the regression ofyonx, and the within and between family variance components fory(and hence the intrafamily correlation) when the effect ofxis eliminated. Generalizations of this are briefly discussed.
Identities relating forward and backward treatments of optimisation
Authors: Whittle P.
Published: April 2006

A general form of Green's theorem is used to derive relations between expected costs for an optimisation problem and the distribution of state variable. A characterisation of the optimal stopping set and an alternative proof of the Howard improvement lemma emerge as non-trivial consequences.
Optimal Sequential selection of n random variables under a constraint
Authors: Chen R. W. , Nair V. N. , Odlyzko A. M. , Shepp L. A. , Vardi Y.
Published: April 2006

We observe a sequence {Xk}k≧1 of i.i.d. non-negative random variables one at a time. After each observation, we select or reject the observed variable. A variable that is rejected may not be recalled. We want to select N variables as soon as possible subject to the constraint that the sum of the N selected variables does not exceed some prescribed value C > 0. In this paper, we develop a sequential selection procedure that minimizes the expected number of observed variables, and we study some of its properties. We also consider the situation where N → ∞and C/N → α > 0. Some applications are briefly discussed.
On the moments of a self-correcting process
Authors: Vere-Jones D. , Ogata Y.
Published: April 2006

The existence of ordinary and exponential moments of a point process with conditional intensity of the formis deduced from a Markov chain representation fort – ρN(t). These results form an application of recent theorems of Tweedie (1983a, b) and are used to obtain laws of large numbers for a range of functionals of the process.
The steady-state behaviour of multistate monotone systems
Authors: Natvig Bent , Streller Arnfried
Published: April 2006

In this paper the steady-state behaviour of multistate monotone systems of multistate components is considered by applying the theory for stationary and synchronous processes with an embedded point process. After reviewing some general results on stationary availability, stationary interval availability and stationary mean interval performance probabilities, we concentrate on systems with independently working and separately maintained components. For this case an explicit formula is given for the mean time which the system in steady state sojourns in states not below a fixed critical level.
Asymptotic analysis of a queueing system by a two-dimensional state space
Authors: Blanc J. P. C.
Published: April 2006

A technique is developed for the analysis of the asymptotic behaviour in the long run of queueing systems with two waiting lines. The generating function of the time-dependent joint queue-length distribution is obtained with the aid of the theory of boundary value problems of the Riemann–Hilbert type and by introducing a conformal mapping of the unit disk onto a given domain. In the asymptotic analysis an extensive use is made of theorems on the boundary behaviour of such conformal mappings.
A characterization of the geometric distribution
Authors: Sreehari M.
Published: April 2006

Let X1, X2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2:n, …, Xn:n. We prove that if the random variable X2:n – X1:n is independent of the events [X1:n = m] and [X1:n = k], for fixed k > m > 1, then the Xi's are geometric. This is related to a characterization problem raised by Arnold (1980).
The renewal-process stationary-excess operator
Authors: Whitt Ward
Published: April 2006

This paper describes the operator mapping a renewal-interval distribution into its associated stationary-excess distribution. This operator is monotone for some kinds of stochastic order, but not for the usual stochastic order determined by the expected value of all non-decreasing functions. Conditions for a renewal-interval distribution to be larger or smaller than its associated stationary-excess distribution for several kinds of stochastic order are determined in terms of familiar notions of ageing. Convergence results are also obtained for successive iterates of the operator, which supplement Harkness and Shantaram (1969), (1972) and van Beek and Braat (1973).
A stress–strength system
Authors: Ebrahimi Nader
Published: April 2006

Suppose Y is the strength of a system subjected to a sequence of stresses X1, X2, X3, · ··. Let N(t) be the number of stresses occurring by time t and let T be the lifetime of the system. Then the survival distribution of is given by

where and . In this note we aim at determining from the above model various non-parametric classes of life distributions which are of interest in reliability theory.