bluuebunny/crossref_metadata_2025 huggingface dataset:
On large sample sequential analysis with applications to survivorship data
Authors: Breslow Norman
Published: July 2016

Although his work on the application of invariance concepts to the sequential testing of composite hypotheses is better known, Cox (1963) has also outlined a large sample approach to the same problem. His method is based on Bartlett's (1946) recognition that the sequence of maximum likelihood estimates (MLE) of the parameter of interest, calculated from an increasing number of observations, resembles asymptotically a random walk of normally distributed variables. However, the large sample theory needed to justify this approach rigorously is left largely implicit. At the end of his paper, Cox suggests that these methods may be extended to yield a sequential comparison of survival curves (Armitage (1959)), a suggestion which has been reiterated as a research problem in the monograph of Wetherill (1966).
Some results on a traffic model of Rényi
Authors: Brown Mark
Published: July 2016

In [5] Renyi considers the following traffic model: Vehicles enter a highway at times 〈Ti
, i = 1, 2, … 〉, forming a homogeneous Poisson process of intensity λ. The vehicle entering at time Ti
will choose a velocity Vi
and will travel at that constant velocity. The random variables 〈Vi
, i = 1, 2, …〉 are independently and identically distributed (i.i.d.) and independent of 〈Ti
〉 with c.d.f. F satisfying

All vehicles travel in the same direction.
Random size trees on the one and two dimensional integer lattice
Authors: Verhagen A.M.V.
Published: July 2016

A stochastic process in which any tree in a forest planted on an integer lattice eliminates its four neighbours when it exceeds their heights, is studied for the case when all heights are independent samples from a continuous distribution. The proportion of the trees of the forest eliminated in this manner is determined for both the one and the two dimensional integer lattices.
Used item replacement policy
Authors: Bhat B.R.
Published: July 2016

The block replacement policy, wherein items are replaced at regular intervals of time and on failure, is rather wasteful because sometimes almost new items are also removed. As an alternative a policy of replacement by new items at regular intervals of time and by used items on failure, is suggested. The consequences of this policy, called used item replacement policy, are studied for Erlangian and sub-exponential life-time distributions. The latter distribution which is the difference of two negative exponential distributions, does not seem to have received much attention in the literature so far.
Some non-stationary point processes with stationary forward recurrence time distribution
Authors: Rudemo Mats
Published: July 2016

Examples are given of point processes that are non-stationary but have stationary forward recurrence time distributions. They are obtained by modification of stationary Poisson and renewal processes.
Calculating extinction probabilities for the birth and death chain in a random environment
Authors: Torrez William C.
Published: July 2016

In a previous investigation (Torrez (1978)) conditions were given for extinction and instability of a stochastic process (Zn
) evolving in a random environment controlled by an irreducible Markov chain (Yn
) with state space 𝒴 The process (Yn, Zn
) is Markovian with state space 𝒴 × {0,1, ···, N} where 𝒴 = {1,· ··,m} and the marginal process (Zn
) is a birth and death chain on {0,1,· ··,N}, with 0 and N made absorbing, when conditioned on a fixed sequence of environmental states of (Yn
). This paper provides bivariate finite difference methods for calculating (i) P(Zn
→ 0) when this probability is not one; and (ii) the expected duration of the process Zn. For (i), the cases when the transition probabilities of the (Yn
)-conditioned process (Zn
) are non-homogeneous and homogeneous are considered separately. Examples are given to illustrate these methods.
Continuity of limit random variables in the branching random walk
Authors: Biggins J. D. , Grey D. R.
Published: July 2016

There are some martingales associated with the branching random walk which are natural generalizations of the classical martingale occurring in the Galton–Watson process. Some continuity properties of the distributions of their limits are discussed.
Filtered renewal processes with a two-sided impact function
Authors: Andersen Per Kragh
Published: July 2016

An expression for the characteristic function of a stationary filtered renewal process with a two-sided impact function is derived, and formulae for the moments are obtained.
A uniqueness problem for the envelope of an oscillatory process
Authors: Hasofer A. M.
Published: July 2016

In a previous paper, the author has described a method for obtaining envelope processes for oscillatory stochastic processes. These are processes which can be represented as the output of a time-varying linear filter whose input is a stationary process.

It is shown in this paper that the proposed definition of the envelope process may not be unique, but may depend on the particular representation of the oscillatory process chosen.

It is then shown that for a class of oscillatory processes which is of particular interest, the class of transient processes, there is a class of natural representations which all lead to a unique envelope process.
Characterizations of distributions through some identities
Authors: Prakasa Rao B. L. S.
Published: July 2016

Some characterizations of continuous and discrete exponential families of distributions are given. These characterizations are used to obtain some limit theorems.
The distribution of the convex hull of a Gaussian sample
Authors: Eddy William F.
Published: July 2016

The distribution of the convex hull of a random sample ofd-dimensional variables is described by embedding the collection of convex sets into the space of continuous functions on the unit sphere. Weak convergence of the normalized convex hull of a circular Gaussian sample to a process with extreme-value marginal distributions is demonstrated. The proof shows that an underlying sequence of point processes converges to a Poisson point process and then applies the continuous mapping theorem. Several properties of the limit process are determined.
On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1
Authors: De Meyer A. , Teugels J. L.
Published: July 2016

For the distribution function of the busy period in the M/G/l queueing system with traffic intensity less than one it is shown that the tail varies regularly at infinity iff the tail of the service time varies regularly at infinity.
Likelihood and nearest-neighbor distance properties of multidimensional Poisson cluster processes
Authors: Baudin Michel
Published: July 2016

The probability generating functional representation of a multidimensional Poisson cluster process is utilized to derive a formula for its likelihood function, but the prohibitive complexity of this formula precludes its practical application to statistical inference. In the case of isotropic processes, it is however feasible to compute functions such as the probability Q(r) of finding no point in a disc of radius r and the probability Q(r | 0) of nearest-neighbor distances greater than r, as well as the expected number C(r | 0) of points at a distance less than r from a given point. Explicit formulas and asymptotic developments are derived for these functions in the n-dimensional case. These can effectively be used as tools for statistical analysis.
On the limit of the Markov binomial distribution
Authors: Wang Y. H.
Published: July 2016

Let X
1
X
2, · ·· be a Markov Bernoulli sequence with initial probabilities p of success and q = 1 – p of failure, and probabilities 1 – (1 – π) p, (1 – π) p in the first row and (1 – π) (1 – p), (1 – π) p + πin the second row of the transition matrix. If we define Sn
= Σ
i=1

n

Xi
, then the limit distribution P{Sn = k} is obtained when n →∞, np →λ.
Correlations and regressions in ‘family’ data
Authors: Smith Cedric A. B.
Published: July 2016

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.
Substitution times for mutant nucleotides
Authors: Watterson G. A.
Published: July 2016

It is shown that, for one model of genetic evolution, the times at which mutations at nucleotide sites become fixed in a population tend to cluster together rather than behave as a Poisson process.
The human lifespan
Authors: Benjamin Bernard
Published: July 2016

This paper examines the distribution of deaths in life tables for the population of England and Wales as a means of exploring the way in which the length of life is steadily advancing. To assess the possibility for future extension, some extreme assumptions are made about the reduction in mortality rates for certain causes. A distinction is made between ‘anticipated' deaths and ‘senescent' deaths, the latter group falling within bounds either side of a terminal peak in the distribution of deaths by age. For males, the extreme assumptions yield a peak at age 86 with some 85 per cent of deaths within the senescent area. For females, the peak is at about age 92 with 90 per cent of deaths in the senescent area.
Comparison of two treatments when there may be an initial effect
Authors: Scott Elizabeth L.
Published: July 2016

Consider situations where the treatment may cause an initial effect and may also cause a long-range effect. We want to evaluate the treatment, or to compare two treatments, when the effect of treatment may result from the two distinct mechanisms,M1andM2. We may wish to evaluateM1andM2separately, but we may also want to evaluate their combined effectM3. Examples are given and the general results are applied to the special case arising in weather modification studies and elsewhere: the possible effects are multiplicative and the distribution of non-zero variables is gamma with at most the scale parameter affected by treatment. An example demonstrates that the two components may be too weak to be judged significant while their sum is large and significant. The locally optimumC(α) test is used.

There is a brief discussion of the power function of the tests. The asymptotic power agrees well, in general, by the results of the Monte Carlo simulation for the testZ3of the combined effect. If the zero values are discarded and thenZ2employed, there is large bias in the power. The bias is more pronounced if the Wilcoxon, Mann–Whitney test is employed. Notice that the two effects under study may be acting in the same direction or they may be in opposition.
Approximations to hard-core models and their application to statistical analysis
Authors: Westcott Mark
Published: July 2016

This paper derives upper and lower bounds to the distribution functions of nearest-neighbour and minimum nearest-neighbour distances between N points generated by a hard-core model on the surface of a sphere. The use of these bounds in statistical inference is discussed.
Time series modelling — an integrated approach
Authors: McNeil Don
Published: July 2016

Some inadequacies of both the traditional (exponential smoothing) and Box-Jenkins approaches to time series forecasting of economic data are investigated. An approach is suggested which integrates these two methodologies. It is based on smoothing the data using straight line segments instead of differencing to obtain stationarity, and forecasting using an autoregressive-moving-average model for the residuals from the most recent linear segment. The efficiency of this approach is calculated theoretically using a series comprising integrated white noise.
Convergence to equilibrium of the two locus additive viability model
Authors: Karlin Samuel , Feldman Marcus W.
Published: July 2016

Consider a deterministic diploid population and a character determined by two loci with allelesA, aat the first andB, bat the second. When the effects of the alleles at each of the two loci on the relative fitnesses of the genotypes are additive, the selection matrix may be written
On the law of the iterated logarithm for inter-record times
Authors: Strawderman William E. , Holmes Paul T.
Published: July 2016

Let X
1, X2, X

3
, ··· 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 Ehrenfest multiurn model
Authors: Severo Norman C.
Published: July 2016

Denote by SN,

r

the set of r-tuples
n
having non-negative integer components summing to N, and by
e

i
the r-tuple having a one in the ith position and zeros elsewhere. For
m
and
n
belonging to SN r, we let P(
n

, t|
m

, s) represent the transition probability of moving from state
m
at time s to state
n
at time t. Bartlett's conservative process [1] assumes that for
n

–
ei
+
e

j
and
n
∈ SN,

r

and Δt > 0

where the λ
ij
(t) are non-negative continuous functions of t.
A note on the queueing systems GI/D/1 and D/G/1
Authors: Pakes A. G.
Published: July 2016

In this note we adopt the notation and terminology of Kingman (1966) without further comment. For the general single server queue one has

For the queueing systems GI/D/1 and D/G/1 we shall show that it is possible to make use of the special form of the service time and inter-arrival time distributions, respectively, to evaluate the right hand side of (1). A similar evaluation applies to the limiting distribution

when it exists. The results obtained could also be obtained from those of Finch (1969) and Henderson and Finch (1970) by using suitable limiting arguments.
A note on the queueing system Ek
/G/1
Authors: Henderson J. , Finch P. D.
Published: July 2016

In this note we present results for the queueing system Ek
/G/1 which are analogous to those obtained in Finch (1969) for the queueing system GI/Ek
/1. As in that paper we adopt the notation and terminology of Kingman (1966) without further comment.
A simple proof of a random stable limit theorem
Authors: Kimbleton Stephen R.
Published: July 2016

Random stable limit theorems have been obtained by several authors, e.g., [3], [4]. The purpose of this note is to give a rather elementary proof of the basic version of this theorem. Our proof may be viewed as the natural extension to stable laws of the method used by Rényi [2] in obtaining a random central limit theorem. Indeed, the only “outside” theorems used are Kolmogorov's inequality (which Rényi also uses) and a general theorem on necessary and sufficient conditions for convergence of a triangular array. It will also be observed that in the present theorem, the consideration of random variables in the domain of attraction of a stable law of index α = 1, introduces no additional difficulties.
Log-concavity and log-convexity in passage time densities of diffusion and birth-death processes
Authors: Keilson Julian
Published: July 2016

Diffusion and birth-death processes have basic theoretical and practical importance for statistics. Insight into the structure of transition distributions and passage time distributions for such processes has been given in recent years by Feller. Karlin, Kemperman, D. G. Kendall, Reuter and many others. An elementary account of this work and partial bibliography has been given elsewhere ([9], [10)]. Certain key passage time densities and sojourn time densities for such processes have a simple property of log-concavity or log-convexity and associated unimodality. Such properties provide information on the character of distributions unavailable from the spectral representations, Laplace transforms and series of convolutions at hand. These properties may also have value for purposes of estimation and optimization.
The progeny of a branching process
Authors: Doney R. A.
Published: July 2016

1. Let {Z(t), t ≧ 0} be an age-dependent branching process with offspring generating function and life-time distribution function G(t). Denote by N(t) the progeny of the process, that is the total number of objects which have been born in [0, t], counting the ancestor. (See Section 2 for definitions.) Then in the Galton-Watson process (i.e., when G(t) = 0 for t ≦ 1, G(t) = 1 for t > 1) we have the simple relation Nn
= Z
0 + Z
1 + ··· + Zn
, so that the asymptotic behaviour of Nn
as n → ∞ follows from a knowledge of the asymptotic behaviour of Zn
. In particular, if 1 < m = h'(1) < ∞ and Zn
(ω)/E(Zn
) → Z(ω) > 0 then also Nn
(ω)/E(Nn
) → Z(ω) > 0; since E(Zn
)/E(Nn
) → 1 – m
–1 this means that
A quantum-mechanical central limit theorem
Authors: Cushen C. D. , Hudson R. L.
Published: July 2016

The concepts of distribution operator, stochastic independence, convergence in distribution and normal distribution are formulated for pairs of canonically conjugate quantum-mechanical momentum and position operators. It is shown that if the sequence (pn
, qn
), n = 1, 2, ··· is stochastically independent and identically distributed with finite covariance and zero mean then the sequence of pairs of canonical observables

converges in distribution to a normal limit distribution.
On a characterization of the Poisson process
Authors: Srivastava R. C.
Published: July 2016

In a recent paper Rényi (1964) has obtained the following characterization of the Poisson process.
The principle of the diffusion of arbitrary constants
Authors: Barbour Andrew D.
Published: July 2016

Equations are derived describing a central limit type large population approximation for continuous time Markov lattice processes in one or more dimensions, such as are commonly encountered in biological models. A method of solving the equations using only the deterministic solution of the process is explained, and it is extended by the use of a martingale argument to provide more detailed information about the process.
Limiting behaviour of the distributions of the maxima of partial sums of certain random walks
Authors: Emery D. J.
Published: July 2016

It is shown that, under certain conditions, satisfied by stable distributions, symmetric distributions, distributions with zero mean and finite second moment and other distributions, the distribution function of the maxima of successive partial sums of identically distributed random variables has an asymptotic property. This property implies the regular variation of the tail of the distribution of the hitting times of the associated random walk, and hence that these hitting times belong to the domain of attraction of a stable law.
A multi-type critical age-dependent branching process with immigration
Authors: Weiner Howard J.
Published: July 2016

In a multi-type critical age-dependent branching process with immigration, the numbers of cell types alive at time t, each divided by t, as t becomes large, tends to a one-dimensional gamma distribution law. The method of proof employs generating functions and compution of asymptotic moments. Connections with earlier results and extensions are indicated.
Spectral analysis for a random process on the circle
Authors: Roy Roch
Published: July 2016

A random process on the circle is a family of random variables X(P,t) indexed by the position P on the unit circle and by the time t (t = 0, + 1, ···). For a homogeneous and stationary process, using its Fourier series expansion, we deduce a spectral representation of the covariance function. The purpose of the paper is to develop a spectral analysis when X(P,t) is observed at all the points on the circle at t = 0, 1, ···, T – 1. The asymptotic distribution of the family of finite Fourier transforms, of the family of periodograms and of the family of spectral densities estimates are obtained from results available for vector-valued time series. Also, a sample covariance function for X (P,t) is defined and its asymptotic distribution derived.
On a characterization of Poisson distributions
Authors: Aczél J.
Published: July 2016

The conjecture pronounced at the end of the paper of Srivastava and Srivastava (1970) is proved in this paper. It gives the following characterization of (bivariate) Poisson distributions. Suppose that items of two types have been observed certain numbers of times, but these original observations have been reduced due to a destructive process which is the product of two binomial distributions and that the probabilities of these reduced numbers are the same whether damaged or undamaged. Then the original random variables had a bivariate Poisson distribution with zero mutual dependence coefficient.
On time-dependent linear transformations of non-stationary stochastic processes
Authors: Tong H.
Published: July 2016

We have introduced a definition of a slowly changing time-dependent linear transformation on a class of non-stationary stochastic processes and have studied the “spectral” relationship between the input and output processes. In addition, using this definition, we have extended one important property of coherency and the concept of residual variance bound pertinent to the theory of stationary stochastic processes to the non-stationary case.
A finite dam with exponential release
Authors: Yeo G. F.
Published: July 2016

This paper considers a finite dam with independently and identically distributed (i.i.d.) inputs occurring in a Poisson process; the special cases where the inputs are (i) deterministic and (ii) negative exponentially distributed are considered in detail. The instantaneous release trate is proportional to the content, i.e., there is an exponential fall in conten except when inputs occur. This model may arise in several other situations such as a geiger counter or integrated shot noise. The distribution of the number of inputs, and of the time, to first overflowing is obtained in terms of generating functions; in Case (i) the solution is obtained through recurrence relations involving iterated integrals which can be evaluated numerically, and in Case (ii) using a series solution of a second order differential equation. Numerical results, in particular for the first two moments, are obtained for various values of the parameters of the model, and compared with a large number of simulations. Some remarks are also made about the infinite dam.
Another definition of genealogical distance
Authors: Mallows C. L.
Published: July 2016

Mycielski and Ulam have introduced three metrics on the set of all members of a species that reproduces by pairwise mating; their discussion concerns only a very orderly society in which the whole population is divided into disjoint generations, with random mating, universal monogamy and exactly two offspring of each mating. Here we discuss another metric, derived from certain probabilistic considerations, that is free of all such restrictions. It does not derive from Mendelian laws.
On the transient state probabilities for a queueing model where potential customers are discouraged by queue length
Authors: Natvig Bent
Published: July 2016

Earlier work by Hadidi and Conolly and contemporary work by the author point to the great operational advantages of state-dependent queueing models. Let pin
(t) be the state probabilities and p∗
in
the corresponding L.T.'s relative to the single server birth-and-death queueing process with parameters λn
= λ/(n + 1), n ≥ 0, μn = μ, n ≥ 1. We have obtained an exact formulation of p
∗

i0
, p
∗

in
(n ≥ 1) being determined recursively. An exact expression for p
10(t) is given in the case of low traffic intensities, and this has been approximated efficiently. Numerical evaluations show that the steady-state is reached very rapidly.
The distribution on the shortest distance between random cuts on opposite strands of DNA
Authors: Litwin Samuel
Published: July 2016

Experiments involving random enzymatic or radioactive multiple cutting of single strands in double stranded DNA will occasionally cut it at pairs of positions on opposite strands that are within about 12 base pairs. In this case the DNA may unravel and come apart. The distribution of closest opposing breaks as a function of λ, the incidence of breaks per unit length, is obtained and graphs of the probability of critical distances as functions ofλare plotted. It is shown that for very largeλthe distance between closest opposite breaks is approximately exponential with parameter ½λ2.
Bounds on the delay distribution in GI/G/1 queues
Authors: Ross Sheldon M.
Published: July 2016

Bounds are obtained for the limiting distribution of the delay in queue for a GI/G/1 system via Martingale theory. These bounds are somewhat stronger than similar bounds recently obtained by Kingman. Simplifications of the bounds are obtained in the special cases where the service distribution is either IFR, DFR, NBU or NWU.
Strongly consistent estimation in a controlled Markov renewal model
Authors: Kolonko Michael
Published: July 2016

The optimal control of dynamic models which are not completely known to the controller often requires some kind of estimation of the unknown parameters. We present conditions under which a minimum contrast estimator will be strongly consistent independently of the control used. This kind of estimator is appropriate for the adaptive or ‘estimation and control' approach in dynamic programming under uncertainty. We consider a countable-state Markov renewal model and we impose bounding and recurrence conditions of the so-called Liapunov type.
The secretary problem with an unknown number of candidates
Authors: Abdel-hamid A. R. , Bather J. A. , Trustrum G. B.
Published: July 2016

When the number of candidates is unknown, the problem of selecting the best during a sequence of interviews has many reasonable solutions. A simple condition for admissibility is established and it is shown that the class of Bayes solutions obtained by treating the number of candidates as a random variable is by no means complete. On the other hand, there is a single improper prior distribution for which the extended Bayes solutions constitute the whole family of admissible procedures.
The structure of elementary pure birth processes
Authors: Pfeifer Dietmar
Published: July 2016

A complete characterization of elementary pure birth processes is given by means of record counting processes from independent (non-identically) distributed random variables.
The stationary moments of Poisson-driven non-linear dynamical systems
Authors: Smith Charles E. , Cobb Loren
Published: July 2016

Moment recursion relations have previously been derived for the stationary probability density functions of continuous-time stochastic systems with Wiener (white noise) input. These results are extended in this paper to the case of Poisson (shot noise) input. The non-linear dynamical systems are expressed, in general, as stochastic differential equations, with an independent increment input. The transition probability density function evolves according to the appropriate Kolmogorov equation. Moments of the stationary density are obtained from the Fourier transform of the stationary density. The moment relations can be used to estimate the parameters of linear and non-linear stochastic systems from empirical moments, given either Wiener or Poisson input.
First-passage time of Markov processes to moving barriers
Authors: Tuckwell Henry C. , Wan Frederic Y. M.
Published: July 2016

The first-passage time of a Markov process to a moving barrier is considered as a first-exit time for a vector whose components include the process and the barrier. Thus when the barrier is itself a solution of a differential equation, the theory of first-exit times for multidimensional processes may be used to obtain differential equations for the moments and density of the first-passage time of the process to the barrier. The procedure is first illustrated for first-passage-time problems where the solutions are known. The mean first-passage time of an Ornstein–Uhlenbeck process to an exponentially decaying barrier is then found by numerical solution of a partial differential equation. Extensions of the method to problems involving Markov processes with discontinuous sample paths and to cases where the process is confined between two moving barriers are also discussed.
Looking backwards in time in the Moran model in population genetics
Authors: Clifford Peter , Sudbury Aidan
Published: July 2016

The Moran model in population genetics may also be regarded as an invasion process (or voter model) on the completen-graph. Tracing lines of ancestry is equivalent to studying the dual process. Using this technique certain results concerning lines of ancestry can be derived very rapidly.
A discrete-time storage process with a general release rule
Authors: Glynn John E.
Published: July 2016

A discrete-time storage system with a general release rule and stationary nonnegative inflows is examined. A simple condition is found for the existence of a stationary storage and outflow for a general possibly non-monotone release function. It is also shown that in the Markov case (i.e. independent inflows) these distributions are unique under certain conditions. It is demonstrated that under these conditions the stationary behaviour in the Markov case varies continuously with parametric changes in the release rule. This result is used to prove convergence of a finite state space approximation for the Markov storage system.
On the limiting behaviour of Downton's carrier epidemic in the case of a general infection mechanism
Authors: Booth James G.
Published: July 2016

A generalization of Downton's (1968) carrier-borne epidemic process is considered in which the assumption of homogeneous mixing is replaced by a general infection mechanism. Using a method and notation similar to that of Gani and Purdue (1984), a recursive algorithm for the joint distribution of the total size and the total observed size of the epidemic is obtained.
Collision probabilities for convex sets
Authors: Weil Wolfgang
Published: July 2016

Let K, L ⊂ En
be non-empty, closed, convex sets, K bounded, and suppose boundary sets αof K and ß of L are painted. If K undergoes a random motion such that K and L touch, the probability for a paint-to-paint contact is expressed by curvature measures of K and L. This generalizes and simplifies previous work of Molter (1986) on infinite cylinders L touching a convex body K.
Laplace transform inversion and passage-time distributions in Markov processes
Authors: Harrison Peter G.
Published: July 2016

Products of the Laplace transforms of exponential distributions with different parameters are inverted to give a mixture of Erlang densities, i.e. an expression for the convolution of exponentials. The formula for these inversions is expressed both as an explicit sum and in terms of a recurrence relation which is better suited to numerical computation. The recurrence for the inversion of certain weighted sums of these transforms is then solved by converting it into a linear first-order partial differential equation. The result may be used to find the density function of passage times between states in a Markov process and it is applied to derive an expression for cycle time distribution in tree-structured Markovian queueing networks.
Procedures for the evaluation of strategies for resource allocation in a stochastic environment
Authors: Glazebrook K. D.
Published: July 2016

Existing procedures for strategy evaluation based on earlier work by the author have led (inter alia) to procedures for the assessment of heuristics for stochastic scheduling and to the development of an approach to sensitivity analysis. Recent theoretical work has now led to the development of more effective evaluation procedures. The key results are reported here.
The limit distribution of the number of rare mutants
Authors: Finkelstein Mark , Tucker Howard G. , Veeh Jerry Alan
Published: July 2016

We study the number of mutants in a mutation process in which reverse mutation is allowed and in which both normal and mutant organisms reproduce at the same rate. Under a mild side condition on the rate of forward mutation we find necessary and sufficient conditions for the number of mutants to converge in distribution. We find the probability generating function of the limit distribution, when it exists. We present an example which shows that the mild side condition cannot be relaxed.
Certain state-dependent processes for dichotomised parasite populations
Authors: Kemp A. W. , Newton J.
Published: July 2016

The paper re-examines Quinn and MacGillivray's (1986) stationary birth-death process for a population of fixed size N consisting of two types of parasite, active and passive, and sets up a more elaborate model for the dichotomy between parasites on hosts with and without open wounds resulting from previous parasite attacks. The probability generating functions for the stationary count distributions are obtained, allowing limiting forms of the distributions to be investigated.
Percolation of coalescing random walks
Authors: Nguyen Bao Gia
Published: July 2016

We study the shape of the binary tree containing 0 that is created from percolation of coalescing random walks. The key result is a duality lemma describing the shape of the tree. Furthermore, we show that

and

where A(R
0), M(R
0), L(R
0) are respectively the area, the number of external nodes and the length of the longest path of the tree R
0.
On the stationary distribution of some extremal Markovian sequences
Authors: Alpuim M. T. , Athayde E.
Published: July 2016

This paper is concerned with the Markovian sequence Xn
= Zn
max{Xn–

1, Yn
},n ≧ 1, where X
0 is any random variable, {Zn
} and {Yn
} are independent sequences of i.i.d. random variables both independent of X
0. We consider the problem of characterizing the class of stationary distributions arising in such a model and give criteria for a d.f. to belong to it. We develop further results when the Zn
's are random variables concentrated on the interval [0, 1], namely having a beta distribution.
A competitive best-choice problem with Poisson arrivals
Authors: Enns E. G. , Ferenstein E. Z.
Published: July 2016

Two competitors observe a Poisson stream of offers. The offers are independent and identically distributed random variables from some continuous distribution. Each of the competitors wishes to accept one offer in the interval [0,T] and each aims to select an offer larger than that of his competitor. Offers are observed sequentially and decisions to accept or reject must be made when the offers arrive. Optimal strategies and winning probabilities are obtained for the competitors under a priorized decision scheme. The time of first offer acceptance is also analyzed. In all cases the asymptotic results are obtained.
Stochastically maximizing the number of successes in a sequential assignment problem
Authors: Righter Rhonda
Published: July 2016

In the classical sequential assignment problem as introduced by Derman et al. (1972) there are n workers who are to be assigned a finite number of sequentially arriving jobs. If a worker of value p is assigned a job of value x the return is px, where we interpret the return as the probability that the given worker correctly completes the given job. The job value is a random value that is observed upon arrival, and jobs must be assigned or rejected when they arrive. Each worker can only do one job. Derman et al. showed that when the objective is to maximize the expected return, i.e., the expected number of correctly completed jobs, the optimal policy is a simple threshold policy, which does not depend on the worker values. Their result was extended by Albright (1974) to allow job arrivals according to a Poisson process and a single random deadline for job completion (which is equivalent to discounting). Righter (1987) further extended the result to permit workers to have independent random deadlines for job completions. Here we show that when there are independent deadlines a simple threshold policy that is independent of the worker values stochastically maximizes the number of correctly completed jobs, and therefore maximizes the expected number of correctly completed jobs. We also show that there is no policy that stochastically maximizes the number of correctly completed jobs when there is a single deadline. However, when there is single deadline and the objective is to maximize the probability that n jobs are done correctly by n workers, then the optimal policy is determined by a single threshold that is independent of n and of the worker values.
Queueing systems with restricted workload: an explicit solution
Authors: Van Dijk Nico M.
Published: July 2016

Queueing systems with non-exponential service distributions and a capacity restriction on the total residual service requirement (workload) are studied. Closed-form expressions of geometric form in the LCFS case and recursive representations in the FCFS case are obtained for the steady state workload distribution. The proof is of interest in itself and directly leads to extensions.
On the speed of convergence for the queue length process of the GI/G/K system in heavy traffic
Authors: Jin Xing , Wang Rongxin
Published: July 2016

This paper continues the study (initiated in [2]) of the GI/G/K system in heavy traffic. We give a Berry–Esseen type theorem for the rate of convergence of the queue length.
Asymptotic results for a problem of DNA breakage
Authors: Cowan Richard , Culpin David , Gates David
Published: July 2016

Double-stranded DNA molecules can be damaged by enzymic action or radiation, in a manner which creates randomly-located single-stranded breaks (nicks). The accumulation of these leads eventually to the double-stranded breakage of the molecule, because two opposite-strand nicks within a critical distance of each other establish conditions for breakage. We study the random variable N, defined as the number of nicks needed for double-stranded breakage to occur. We develop an asymptotic theory which is needed for practical computations.
A geometric rate of convergence to the equilibrium for Boltzmann processes with multiple particle interactions
Authors: Chauvin Brigitte , Giroux Gaston
Published: July 2016

We construct Boltzmann processes using the formalism of random trees. We are then able to extend previous results about convergence toward the equilibrium law to interactions involving random numbers of particles. We even show a geometric rate of convergence for an extended class of processes, especially for those having a scaling invariant interaction mechanism.
Ornstein–Uhlenbeck process with quadratic killing
Authors: Wenocur Michael L.
Published: July 2016

An Ornstein-Uhlenbeck process subject to a quadratic killing rate is analyzed. The distribution for the process killing time is derived, generalizing the analogous result for Brownian motion. The derivation involves the use of Hermite polynomials in a spectral expansion.
A comparison theorem for conditioned Markov processes
Authors: Roberts G. O.
Published: July 2016

Intuitively, the effect of conditioning a one-dimensional process to remain below a certain (possibly time-dependent) boundary is to ‘push' the process downwards. This paper investigates the effect of such conditioning, and finds the class of processes for which our intuition is accurate. It is found that ordinary stochastic inequalities are in general unsuitable for making statements about such conditioned processes, and that a stronger type of inequality is more appropriate.

The investigation is motivated by applications in estimation of boundary hitting time distributions.
Approximating the stationary distribution of an infinite stochastic matrix
Authors: Heyman Daniel P.
Published: July 2016

We are given a Markov chain with states 0, 1, 2, ···. We want to get a numerical approximation of the steady-state balance equations. To do this, we truncate the chain, keeping the first n states, make the resulting matrix stochastic in some convenient way, and solve the finite system. The purpose of this paper is to provide some sufficient conditions that imply that as n tends to infinity, the stationary distributions of the truncated chains converge to the stationary distribution of the given chain. Our approach is completely probabilistic, and our conditions are given in probabilistic terms. We illustrate how to verify these conditions with five examples.
On the convex hull of n random points on a circle
Authors: Carnal H. , Hüsler J.
Published: July 2016

We consider the convex hull of a sample of n randomly placed points in a unit circle. Obviously not all of the n points are needed to construct the convex hull. We show that asymptotically only the points belonging to a certain ring are used for the convex hull, where the size of the ring decreases at a certain rate such that the error of not completely correct construction of the convex hull by the particular subset of points tends to 0.
On three classical problems for Markov chains with continuous time parameters
Authors: Chen Mu-Fa
Published: July 2016

For a given transition rate, i.e., aQ-matrixQ= (qij) on a countable state space, the uniqueness of theQ-semigroupP(t) = (Pij(t)),the recurrence and the positive recurrence of the corresponding Markov chain are three fundamental and classical problems, treated in many textbooks. As an addition, this paper introduces some practical results motivated from the study of a type of interacting particle systems, reaction diffusion processes. The main results are theorems (1.11), (1.17) and (1.18). Their proofs are quite straightforward.
Some new results on queueing networks with batch movement
Authors: Henderson W. , Taylor P. G.
Published: July 2016

Product-form equilibrium distributions in networks of queues in which customers move singly have been known since 1957, when Jackson derived some surprising independence results. A product-form equilibrium distribution has also recently been shown to be valid for certain queueing networks with batch arrivals, batch services and even correlated routing.

This paper derives the joint equilibrium distribution of states immediately before and after a batch of customers is released into the network. The results are valid for either discrete- or continuous-time queueing networks: previously obtained results can be seen as marginal distributions within a more general framework. A generalisation of the classical ‘arrival theorem' for continuous-time networks is given, which compares the equilibrium distribution as seen by arrivals to the time-averaged equilibrium distribution.
On a discrete-time non-zero-sum Dynkin problem with monotonicity
Authors: Ohtsubo Yoshio
Published: July 2016

We consider a monotone case of the non-zero-sum stopping game with discrete time parameter which is called the Dynkin problem. Marner (1987) has investigated a stopping game with general monotone reward structures, but his monotonicity is too strong to apply our problem. We establish that there exists an explicit equilibrium point in our monotone case. We also give a simple example applicable to a duopolistic exit game.
The NBUC and NWUC classes of life distributions
Authors: Cao Jinhua , Wang Yuedong
Published: July 2016

A new class of life distributions, namely new better than used in convex ordering (NBUC), and its dual, new worse than used in convex ordering (NWUC), are introduced. Their relations to other classes of life distributions, closure properties under three reliability operations, and heritage properties under shock model and Laplace-Stieltjes transform are discussed.
Bell's inequality, probability modelling and quantum correlation experiments
Authors: Finch P. D.
Published: July 2016

Problems associated with setting up a probability model which generates the quantum theoretical probabilities for the two spin 1/2 particle system are examined. Arguments which claim to show that such a model cannot be constructed within classical probability theory under the assumption of local singlet states are also considered. It is shown that the model then in question is not a probability model in the sense that term is used elsewhere in science. An alternative model is proposed and its bearing on the Einstein-Bohr debate is briefly discussed.
The passage from random walk to diffusion in quantum probability
Authors: Parthasarathy K. R.
Published: July 2016

The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.
Coincident probabilities and applications in combinatorics
Authors: Karlin Samuel
Published: July 2016

For a strong Markov process on the line with continuous paths the Karlin–McGregor determinant formula of coincidence probabilities for multiple particle systems is extended to allow the individual component processes to start at variable times and run for variable durations. The extended formula is applied to a variety of combinatorial problems including counts of non-crossing paths in the plane with variable start and end points, dominance orderings, numbers of dominated majorization orderings, and time-inhomogeneous random walks.
Some applications of weak convergence to statistics
Authors: Loynes R. M.
Published: July 2016

Results showing the weak convergence of certain stochastic processes are used to derive both known and new (asymptotic) properties of signs of residuals from regression; other weak convergence results are derived, and used to determine the behaviour of runs of residuals.
On the limit of a supercritical branching process
Authors: Bingham N. H.
Published: July 2016

LetWbe the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the ‘Schröder' and ‘Böttcher' cases; both appear in work of Harris and Dubuc. The Schröder case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the Böttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.
Ignatov's theorem: a new and short proof
Authors: Engelen Ron , Tommassen Paul , Vervaat Wim
Published: July 2016

By Ignatov's theorem the sets of values in an i.i.d. sequence that are the kth largest at their appearance (k = 1, 2, ···) are supports of i.i.d. Poisson processes. The present paper contains an elementary and short proof for the case where the underlying distribution function F is discrete, and then extends the result to general F.
Homogeneous row-continuous bivariate markov chains with boundaries
Authors: Keilson J. , Zachmann M.
Published: July 2016

The matrix-geometric results of M. Neuts are extended to ergodic row-continuous bivariate Markov processes [J(t), N(t)] on state space B = {(j, n)} for which: (a) there is a boundary level N for N(t) associated with finite buffer capacity; (b) transition rates to adjacent rows and columns are independent of row level n in the interior of B. Such processes are of interest in the modelling of queue-length for voice-data transmission in communication systems.

One finds that the ergodic distribution consists of two decaying components of matrix-geometric form, the second induced by the finite buffer capacity. The results are obtained via Green's function methods and compensation. Passage-time distributions for the two boundary problems are also made available algorithmically.