bluuebunny/crossref_metadata_2025 huggingface dataset:
Rate of convergence to equilibrium of marked Hawkes processes
Authors: Brémaud P. , Nappo G. , Torrisi G. L.
Published: July 2016

In this article we obtain rates of convergence to equilibrium of marked Hawkes processes in two situations. Firstly, the stationary process is the empty process, in which case we speak of the rate of extinction. Secondly, the stationary process is the unique stationary and nontrivial marked Hawkes process, in which case we speak of the rate of installation. The first situation models small epidemics, whereas the results in the second case are useful in deriving stopping rules for simulation algorithms of Hawkes processes with random marks.
Total duration of negative surplus for the compound Poisson process that is perturbed by diffusion
Authors: Zhang Chunsheng , Wu Rong
Published: July 2016

In this paper, we consider the compound Poisson process that is perturbed by diffusion (CPD). We derive formulae for the Laplace transform, expectation and variance of total duration of negative surplus for the CPD and also present some examples of the CPD with an exponential individual claim amount distribution and a mixture exponential individual claim amount distribution.
Doubly stochastic Hilbertian processes
Authors: Guillas Serge
Published: July 2016

In this paper, we consider a Hilbert-space-valued autoregressive stochastic sequence (X

n
) with several regimes. We suppose that the underlying process (I

n
) which drives the evolution of (X

n
) is stationary. Under some dependence assumptions on (I

n
), we prove the existence of a unique stationary solution, and with a symmetric compact autocorrelation operator, we can state a law of large numbers with rates and the consistency of the covariance estimator. An overall hypothesis states that the regimes where the autocorrelation operator's norm is greater than 1 should be rarely visited.
Transient analytical solution of M/D/1/N queues
Authors: Garcia Jean-Marie , Brun Olivier , Gauchard David
Published: July 2016

An analytical expression of the time-dependent probability distribution of M/D/1/N queues initialised in an arbitrary deterministic state is derived. We also obtain a simple analytical expression of the differential equation governing the transient average traffic which only involves probabilities of boundary states. As a by-product, a closed form solution of the departure rate from the system is also determined.
Conditional job-observer property for multitype closed queueing networks
Authors: Daduna Hans , Szekli Ryszard
Published: July 2016

For functionals of multitype closed queueing networks, a conditional job-observer property is shown which provides more insight into the classical job-observer property. Applications and examples are given, including the classical job-observer property for the number of customers in a network, a representation of cycle time distributions and a basic formula for sojourn times.
Sooner and later waiting time problems for patterns in Markov dependent trials
Authors: Han Qing , Hirano Katuomi
Published: July 2016

In this paper, we investigate sooner and later waiting time problems for patternsS0andS1in multistate Markov dependent trials. The probability functions and the probability generating functions of the sooner and later waiting time random variables are studied. Further, the probability generating functions of the distributions of distances between successive occurrences ofS0and between successive occurrences ofS0andS1and of the waiting time until therth occurrence ofS0are also given.
On the distance between the distributions of random sums
Authors: Roos Bero , Pfeifer Dietmar
Published: July 2016

In this paper, we consider the total variation distance between the distributions of two random sums S

M
and S

N
with different random summation indices M and N. We derive upper bounds, some of which are sharp. Further, bounds with so-called magic factors are possible. Better results are possible when M and N are stochastically or stop-loss ordered. It turns out that the solution of this approximation problem strongly depends on how many of the first moments of M and N coincide. As approximations, we therefore choose suitable finite signed measures, which coincide with the distribution of the approximating random sum S

N
if M and N have the same first moments.
On the geometric ergodicity of hybrid samplers
Authors: Fort G. , Moulines E. , Roberts G. O. , Rosenthal J. S.
Published: July 2016

In this paper, we consider the random-scan symmetric random walk Metropolis algorithm (RSM) on ℝ

d
. This algorithm performs a Metropolis step on just one coordinate at a time (as opposed to the full-dimensional symmetric random walk Metropolis algorithm, which proposes a transition on all coordinates at once). We present various sufficient conditions implying V-uniform ergodicity of the RSM when the target density decreases either subexponentially or exponentially in the tails.
Remarks on the absolute maximum of a Lévy process
Authors: Bratiychuk Mykola
Published: July 2016

Asymptotic behaviour of the distribution of the absolute maximum of a process with independent increments is studied depending on the properties of the Lévy measure of the process. Some applications to the risk process are also considered.
Random sequential packing in Euclidean spaces of dimensions three and four and a conjecture of Palásti
Authors: Blaisdell B. Edwin , Solomon Herbert
Published: July 2016

A conjecture of Palásti [11] that the limiting packing density β d
in a space of dimension d equals β d
where ß is the limiting packing density in one dimension continues to be studied, but with inconsistent results. Some recent correspondence in this journal [7], [8], [13], [14], [15], [16], [18], [19], [20] as well as some other papers indicate a lively interest in the subject. In a prior study [3], we demonstrated that the conjectured value in two dimensions was smaller than the actual density. Here we demonstrate that this is also so in three and four dimensions and that the discrepancy increases with dimension.
A variation of Weiss's carrier-borne epidemic model
Authors: Routleff C.
Published: July 2016

A carrier-borne epidemic model, in which the carrier removal process is influenced by the number of detected infected individuals, is considered. The distribution of the total size of the epidemic is found and expressions for the expected ultimate size of the epidemic and the expected area under the trajectory of carriers are given.
On queueing systems by retrials
Authors: Kulkarni Vidyadhar G.
Published: July 2016

A general result for queueing systems with retrials is presented. This result relates the expected total number of retrials conducted by an arbitrary customer to the expected total number of retrials that take place during an arbitrary service time. This result is used in the analysis of a special system where two types of customer arrive in an independent Poisson fashion at a single-server service station with no waiting room. The service times of the two types of customer have independent general distributions with finite second moments. When the incoming customer finds the server busy he immediately leaves and tries his luck again after an exponential amount of time. The retrial rates are different for different types of customers. Expressions are derived for the expected number of retrial customers of each type.
The Hurst effect under trends
Authors: Bhattacharya R. N. , Gupta Vijay K. , Waymire Ed
Published: July 2016

Necessary and sufficient conditions for the so-called Hurst effect are given in the case of a weakly dependent stationary sequence of random variables perturbed by a trend. As a consequence of this general result it is shown that the Hurst effect is present in the case of weakly dependent random variables with a small monotonic trend of the formf(n) =c(m+n)ß, where m is an arbitrary non-negative parameter andcis not 0. For – ½ <ß< 0 the Hurst exponent is shown to be precisely given by 1 +ß.Forß≦ – ½ and forß= 0 the Hurst exponent is 0.5, while forß > 0it is 1. This simple mathematical model, motivated by empirical evidence in various geophysical records, demonstrates the presence of the Hurst effect in a direction not explored before.
Conditional response times in the M/G/1 processor-sharing system
Authors: Asare B. K. , Foster F. G.
Published: July 2016

The expected response time of a job that requires processing time t and meets n jobs on arrival in the M/G/1 processor-sharing system is derived.
A property of longtailed distributions
Authors: Embrechts Paul , Omey Edward
Published: July 2016

We investigate sufficient conditions so that is subexponential. Here F is a distribution function on [0, ∞[, with finite mean. Some applications to risk theory and rates of convergence in renewal theory are given.
A correlated random walk model for two-dimensional diffusion
Authors: Henderson Robin , Renshaw Eric , Ford David
Published: July 2016

A two-dimensional lattice random walk is studied in which, at each stage, the direction of the next step is correlated to that of the previous step. An exact expression is obtained from the characteristic function for the dispersion matrix of the n-step transition probabilities, and the limiting diffusion equation is derived.
Mixed percolation on the square lattice
Authors: Wierman John C.
Published: July 2016

In a planar percolation model, faces of the underlying graph, as well as the sites and bonds, may be viewed as random elements. With this viewpoint, Whitney duality allows construction of a planar dual percolation model for each planar percolation model, which applies to mixed models with sites, bonds, and faces open or closed at random. Using self-duality for percolation models on the square lattice, information is obtained about the percolative region in the mixed model.
An operator-analytic approach to the Jackson network
Authors: Massey William A.
Published: July 2016

Operator methods are used in this paper to systematically analyze the behavior of the Jackson network. Here, we consider rarely treated issues such as the transient behavior, and arbitrary subnetworks of the total system. By deriving the equations that govern an arbitrary subnetwork, we can see how the mean and variance for the queue length of one node as well as the covariance for two nodes vary in time.

We can estimate the transient behavior by deriving a stochastic upper bound for the joint distribution of the network in terms of a judicious choice of independent M/M/1 queue-length processes. The bound we derive is one that cannot be derived by a sample-path ordering of the two processes. Moreover, we can stochastically bound from below the process for the total number of customers in the network by an M/M/1 system also. These results allow us to approximate the network by the known transient distribution of the M/M/1 queue. The bounds are tight asymptotically for large-time behavior when every node exceeds heavy-traffic conditions.
Characterization of a class of second-order density functions
Authors: Mehr C. B.
Published: July 2016

Distributions of some random variables have been characterized by independence of certain functions of these random variables. For example, let X and Y be two independent and identically distributed random variables having the gamma distribution. Laha showed that U = X + Y and V = X | Y are also independent random variables. Lukacs showed that U and V are independently distributed if, and only if, X and Y have the gamma distribution. Ferguson characterized the exponential distribution in terms of the independence of X – Y and min (X, Y). The best-known of these characterizations is that first proved by Kac which states that if random variables X and Y are independent, then X + Y and X – Y are independent if, and only if, X and Y are jointly Gaussian with the same variance. In this paper, Kac's hypotheses have been somewhat modified. In so doing, we obtain a larger class of distributions which we shall call class λ1. A subclass λ0 of λ1 enjoys many nice properties of the Gaussian distribution, in particular, in non-linear filtering.
On quasi-stationary distributions in absorbing continuous-time finite Markov chains
Authors: Darroch J. N. , Seneta E.
Published: July 2016

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.
The Factorisation of the rectangular distribution
Authors: Lewis T.
Published: July 2016

Questions of the decomposability of distribution functions into real-valued components of bounded variation were discussed by P. Lévy (1964) in relation to the nature of the components, whether non-decreasing (distribution functions in particular) or absolutely continuous (a.c.) or both. Hanson (1965), in a review of Lévy's paper, raised the question of whether or not a rectangular distribution could be decomposed into two a.c. distributions. In fact, D. G. Kendall had conjectured earlier (Kendall (1960)) that no such decomposition is possible. The object of this paper is to state and prove the truth of Kendall's conjecture. “Decomposition” or “factorisation” will be understood throughout the paper to mean decomposition into distributions. Decompositions of the rectangular distribution into one a.c. and one discrete factor are well known (see, e.g., Lukacs (1960) pp. 128–9), and decompositions in which both factors are singular continuous (s.c.) have been discovered by Kendall and by P. M. Lee; it is shown here that no other combinations of factor-type can exist. References to other work on related decomposability properties are given in the papers by Lévy and Kendall cited above.
Finding the edge of a Poisson forest
Authors: Ripley B. D. , Rasson J.-P.
Published: July 2016

The points of a homogeneous Poisson process within a compact convex set are observed. We consider how to reconstruct this domain from the observations.
Velocities of propagation for stepping-stone models of population growth
Authors: Renshaw Eric
Published: July 2016

A population is composed of an infinite number of colonies situated at the integer points of a single co-ordinate axis. Each colony develops according to a simple birth and death process and migration is allowed between nearest neighbours only. Exact expressions are derived for both the asymptotic velocity of propagation and the shape of the wave profile as the population spreads along the axis, and it is shown that they are identical with the saddle point approximations developed by Daniels.
A limit theorem for point processes by applications
Authors: Puri Prem S.
Published: July 2016

Let 0 ≦ T
1 ≦ T
2 ≦ ·· · represent the epochs in time of occurrences of events of a point process N(t) with N(t) = sup{k : Tk
≦ t}, t ≧ 0. Besides certain mild conditions on the process N(t) (see Conditions (A1)– (A3) in the text) we assume that for every k ≧ 1, as t →∞, the vector (t – TN

(t), t – TN

(t)–1, · ··, t – TN

(t)–k+1) converges in law to a k-dimensional distribution which coincides with that of a random vector ξ k
= (ξ
1, · ··, ξ k
) necessarily satisfying P(0 ≦ ξ
1 ≦ ξ
2 ≦ ·· ·≦ ξ
k) = 1. Let R(t) be an arbitrary function defined for t ≧ 0, satisfying 0 ≦ R(t) ≦ 1, ∀0 ≦ t <∞, and certain mild conditions (see Conditions (B1)– (B4) in the text). Then among other results, it is shown that

The paper also deals with conditions under which the limit (∗) will be positive. The results are applied to several point processes and to the situations where the role of R(t) is taken over by an appropriate transform such as a probability generating function, where conditions are given under which the limit (∗) itself will be a transform of an honest distribution. Finally the results are applied to the study of certain characteristics of the GI/G/∞ queue apparently not studied before.
Sequential random packing in the plane
Authors: Weiner Howard J.
Published: July 2016

The Palasti conjecture on the asymptotic mean proportion of coverage is verified for the sequential random packing of rectangular cars with sides parallel to rectangular boundaries in the models of Rényi and Solomon. The extension to n dimensions is given. An extension to a random car size model is indicated.
Dryness of discrete dams: comments on a paper by Tin and Phatarfod
Authors: Balagopal K.
Published: July 2016

The utilisation factor for a discrete dam, defined as the stationary probability of non-emptiness of the dam just before release, is obtained for a class of models that includes the Odoom–Lloyd model, the Anis–Lloyd model, the model of Herbert, etc. The method used points to a different approach to measuring dam utilisation via the actual utilisation, which appears to be more fruitful, and this is discussed in the last section.
The Effect of an age replacement to a standby redundant system
Authors: Mine Hisashi , Asakura Tatsuyuki
Published: July 2016

A multiple unit standby redundant system with preventive maintenance is presented. The Laplace transform of the probability density function of the time to the first emptiness for the system and its mean time are derived. As a special case, a two-unit standby redundant system with preventive maintenance is investigated in detail.
Integral representations of transition probabilities and serial covariances of certain markov chains
Authors: Daley D. J.
Published: July 2016

This note is concerned with integral representations of some transition probabilities of countable state space Markov chains embedded in birth and death processes, of the serial covariances of functions defined on such chains when stationary, and finally the properties of the spectral measure of stationary processes with monotonically decreasing or completely monotonic serial covariances.
Linear-Intercept distributions do not characterize plane sets
Authors: Mallows C. L. , C. Clark J. M.
Published: July 2016

All sets mentioned in this paper will be assumed to be appropriately measurable without further comment, and events of probability zero will be neglected. We start with a fixed (large, convex) subset R of the plane, termed the “retina”.
The convex hull of a uniform sample from the interior of a simple d-polytope
Authors: Van Wel Barthold F.
Published: July 2016

An asymptotic expression is given for the expected number of vertices of the convex hull of a uniform sample from the interior of a d-dimensional simple polytope. This extends a result derived by Rényi and Sulanke for sample points in the plane.
Stopping rules for proofreading
Authors: Ferguson T. S. , Hardwick J. P.
Published: July 2016

A manuscript with an unknown random numberMof misprints is subjected to a series of proofreadings in an effort to detect and correct the misprints. On thenthproofreading, each remaining misprint is detected independently with probabilitypn– 1. Each proofreading costs an amountCP> 0, and if one stops afternproofreadings, each misprint overlooked costs an amountcn> 0. Two models are treated based on the distribution ofM.In the Poisson model, the optimal stopping rule is seen to be a fixed sample size rule. In the binomial model, the myopic rule is optimal in many important cases. A generalization is made to problems in which individual misprints may have distinct probabilities of detection and distinct overlook costs.
Waiting time and workload in queues with periodic Poisson input
Authors: Lemoine Austin J.
Published: July 2016

This paper develops moment formulas for asymptotic workload and waiting time in a single-server queue with periodic Poisson input and general service distribution. These formulas involve the corresponding moments of waiting-time (workload) for the M/G/1 system with the same average arrival rate and service distribution. In certain cases, all the terms in the formulas can be computed exactly, including moments of workload at each ‘time of day.' The approach makes use of an asymptotic version of the Takács [12] integro-differential equation, together with representation results of Harrison and Lemoine [3] and Lemoine [6].
On the Extinction of the S–I–S stochastic logistic epidemic
Authors: Kryscio Richard J. , Lefèvre Claude
Published: July 2016

We obtain an approximation to the mean time to extinction and to the quasi-stationary distribution for the standard S–I–S epidemic model introduced by Weiss and Dishon (1971). These results are a combination and extension of the results of Norden (1982) for the stochastic logistic model, Oppenheim et al. (1977) for a model on chemical reactions, Cavender (1978) for the birth-and-death processes and Bartholomew (1976) for social diffusion processes.
Sojourn times in finite Markov processes
Authors: Rubino Gerardo , Sericola Bruno
Published: July 2016

Sojourn times of Markov processes in subsets of the finite state space are considered. We give a closed form of the distribution of the nth sojourn time in a given subset of states. The asymptotic behaviour of this distribution when time goes to infinity is analyzed, in the discrete time and the continuous-time cases. We consider the usually pseudo-aggregated Markov process canonically constructed from the previous one by collapsing the states of each subset of a given partition. The relation between limits of moments of the sojourn time distributions in the original Markov process and the moments of the corresponding holding times of the pseudo-aggregated one is also studied.
On a duality between a non-Markoyian storage/production process and a Markovian dam process with state-dependent input and output
Authors: Kaspi Haya , Perry David
Published: July 2016

We consider a storage/production process where the production rate is state dependent, the demand arrival is a renewal process, and the sizes of the demands are i.i.d exponentially distributed random variables. The resulting content process is non-Markovian but regenerative. We construct a dual Markovian dam process with drift, jump rate and jump sizes that are state dependent and use it to compute the limiting one-dimensional distribution of the content process.
On the characterization of point processes with the order statistic property without the moment condition
Authors: Puri Prem S.
Published: July 2016

The paper characterizes point processes with the order statistic property without the unnecessary condition of finiteness of the first moment of the process, a condition imposed by previous researchers. It shows that the class of these processes is composed only of mixed Poisson processes up to a time-scale transformation and of the mixed sample processes. It also introduces a multivariate analog of the order statistic property and characterizes completely the class of multivariate point processes with this property.
On the Distribution and covariance structure of the present value of a random income stream
Authors: Keilson J.
Published: July 2016

The present value is studied when I(t) is a stationary random income stream. The stationary distribution of V(t) for a family of simple streams modeled by stationary finite Markov chains is given explicitly. The process V(t) is shown to be observable in a special sense when I(t) is time-reversible.
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: July 2016

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.
Optimal control of arrivals to multiserver queues in a random environment
Authors: Helm Werner E. , Waldmann Karl-Heinz
Published: July 2016

We study the problem of optimal customer admission to multiserver queues. These queues are assumed to live in an extraneous environment which changes in a semi-Markovian way. Arrivals, service mechanism and random reward/cost structure may all depend on these surroundings. Included as special cases are SM/M/c queues, in particular G/M/c queues, in a random environment. By a direct inductive approach we establish optimality of a generalized control-limit rule depending on the actual environment. Particular emphasis is laid on different applications that show the versatility of the proposed setup.
A note on the volume of small random simplexes
Authors: Gates John
Published: July 2016

This note shows how the initial density of the volume of a random simplex in a body, K, depends upon the fourth moment of sectional area and so rejects the conjecture that this initial density is independent of K.
A limit theorem for population-size-dependent branching processes
Authors: Klebaner F. C.
Published: July 2016

An analogue of the Kesten–Stigum theorem, and sufficient conditions for the geometric rate of growth in the rth mean and almost surely, are obtained for population-size-dependent branching processes.
Counting figures in planar random configurations
Authors: Kellerer A. M.
Published: July 2016

Random configurations are considered that are generated by a Poisson process of figures in the plane, and a recent result is used to derive formulae for the estimation of the number of figures, and their mean area and perimeter. The formulae require merely the determination of the area, the perimeter, and the Euler–Poincaré characteristic of the random configurations in a fixed field of view. There are no similar formulae for the standard deviations of the estimates; their magnitudes in typical cases are therefore assessed by Monte Carlo simulations.
The first-passage density of a continuous gaussian process to a general boundary
Authors: Durbin J.
Published: July 2016

Under mild conditions an explicit expression is obtained for the first-passage density of sample paths of a continuous Gaussian process to a general boundary. Since this expression will usually be hard to compute, an approximation is given which is computationally simple and which is exact in the limit as the boundary becomes increasingly remote. The integral of this approximating density is itself approximated by a simple formula and this also is exact in the limit. A new integral equation is derived for the first-passage density of a continuous Gaussian Markov process. This is used to obtain further approximations.
Moments for stationary and quasi-stationary distributions of markov chains
Authors: Seneta E. , Tweedie R. L.
Published: July 2016

A necessary and sufficient set of conditions is given for the finiteness of a general moment of the R -invariant measure of an R -recurrent substochastic matrix. The conditions are conceptually related to Foster's theorem. The result extends that of [8], and is illustratively applied to the single and multitype subcritical Galton–Watson process to find conditions for Yaglom-type conditional limit distributions to have finite moments.
Presentation of phase-type distributions as proper mixtures
Authors: Assaf David , Langberg Naftali A.
Published: July 2016

It is shown that any phase-type distribution can be represented as a proper mixture of two distinct phase-type distributions. Using different terms, it is shown that the class of phase-type distributions does not include any extreme ones. A similar result holds for the subclass of upper-triangular phase-type distributions.
Exponential trends of first-passage-time densities for a class of diffusion processes with steady-state distribution
Authors: Nobile A. G. , Ricciardi L. M. , Sacerdote L.
Published: July 2016

The asymptotic behavior of the first-passage-time p.d.f. through a constant boundary is studied when the boundary approaches the endpoints of the diffusion interval. We show that for a class of diffusion processes possessing a steady-state distribution this p.d.f. is approximately exponential, the mean being the average first-passage time to the boundary. The proof is based on suitable recursive expressions for the moments of the first-passage time.
On a modified counter with prolonging dead time
Authors: Dvurečenskij A. , Ososkov G. A.
Published: July 2016

Emitted particles arrive at the counter with prolonging dead time so that the interarrival times and the lengths of impulses in any dead time are independent but not necessarily identically distributed random variables, and whenever the counter is idle then the following evolution starts from the beginning. For this class of counters we derive the probability laws of the numbers of particles arriving at the counters during their dead times, and the Laplace transform of the cycle, respectively.
Inequalities for random flats meeting a convex body
Authors: Schneider Rolf
Published: July 2016

We choose a uniform random point in a given convex bodyKinn-dimensional Euclidean space and through that point the secant ofKwith random direction chosen independently and isotropically. Given the volume ofK, the expectation of the length of the resulting random secant ofKwas conjectured by Enns and Ehlers [5] to be maximal ifKis a ball. We prove this, and we also treat higher-dimensional sections defined in an analogous way. Next, we consider a finite number of independent isotropic uniform random flats meetingK, and we prove that certain geometric probabilities connected with these again become maximal whenKis a ball.
Some results on population-size-dependent Galton-Watson processes
Authors: Höpfner Reinhard
Published: July 2016

Some classes of population-size-dependent Galton-Watson processes are considered where extinction occurs with probability 1. Results on the asymptotic behaviour of the probability of survival up to time t, mean population size and conditioned limit distributions are found to hold. They correspond to those obtained in the study of Galton-Watson processes with immigration stopped at 0.
Rates of convergence to the stationary distribution for k-dimensional diffusion processes
Authors: Davies P. L.
Published: July 2016

Using a coupling technique, rates of convergence in total variation of certain k -dimensional diffusion processes to the stationary distribution are obtained. The results place assumptions only on the coefficients of the elliptic differential operator governing the diffusion.
A planar random motion governed by the two-dimensional telegraph equation
Authors: Orsingher Enzo
Published: July 2016

In this paper a planar random motion governed by the two-dimensional telegraph equation is presented. It is proved that the particle performing motion is at any time t within a circle centred at the starting point and with radius . The explicit density of the particle position is obtained. Results concerning the trend of motion are also given.
The Wright–Fisher model with varying selection
Authors: Weber N. C.
Published: July 2016

The Wright–Fisher model with varying population size is examined in the case where the selective advantage varies from generation to generation. Models are considered where the selective advantage is not always in favour of a particular genotype. Sufficient conditions in terms of the selection coefficients and the population growth are given to ensure ultimate homozygosity.
Learning algorithms for Markov decision processes
Authors: Kurano Masami
Published: July 2016

This study is concerned with finite Markov decision processes whose dynamics and reward structure are unknown but the state is observable exactly.

We establish a learning algorithm which yields an optimal policy and construct an adaptive policy which is optimal under the average expected reward criterion.
Scale renormalization and random solutions of the Burgers equation
Authors: Rosenblatt M.
Published: July 2016

Solutions of the Burgers equation with a stationary (spatially) stochastic initial condition are considered. A class of limit laws for the solution which correspond to a scale renormalization is considered.
On the probability densities of an Ornstein–Uhlenbeck process with a reflecting boundary
Authors: Ricciardi L. M. , Sacerdote L.
Published: July 2016

We show that the transition p.d.f. of the Ornstein–Uhlenbeck process with a reflection condition at an assigned state S is related by integral-type equations to the free transition p.d.f., to the transition p.d.f. in the presence of an absorption condition at S, to the first-passage-time p.d.f. to S and to the probability current. Such equations, which are also useful for computational purposes, yield as an immediate consequence all known closed-form results for Wiener and Ornstein–Uhlenbeck processes.
Non-random shuffling for multiple decks
Authors: Gottlieb Gary
Published: July 2016

Non-randomness that arises in the shuffling of multiple numbers of decks of playing cards is modeled. An efficient way to exploit the non-randomness is derived, and its effect on the win rate for a simple game of chance is derived.
An upper bound on the performance of queues with returning customers
Authors: Greenberg Betsy S. , Wolff Ronald W.
Published: July 2016

Multiple channel queues with Poisson arrivals, exponential service distributions, and finite capacity are studied. A customer who finds the system at capacity either leaves the system for ever or may return to try again after an exponentially distributed time. Steady state probabilities are approximated by assuming that the returning customers see time averages. The approximation is shown to result in an upper bound on system performance.
Sampling random polygons
Authors: George Edward I.
Published: July 2016

Every realization of a Poisson line process is a set of lines which subdivides the plane into a population of non-overlapping convex polygons. To explore the unknown statistical features of this population, an alternative stochastic construction of random polygons is developed. This construction, which is based on an alternating sequence of random angles and side lengths, provides a fast simulation method for obtaining a random sample from the polygon population. For the isotropic case, this construction is used to obtain a random sample of 2500000 polygons, providing the most precise estimates to date of some of the unknown distributional characteristics.
Large deviations of tail estimators based on the Pareto approximation
Authors: Smith Richard L. , Weissman Ishay
Published: July 2016

We consider the relative error of a tail function when this is approximated by y–α
using an estimator of Hill's for α. The results combine recent work of Davis and Resnick on tail estimation with Anderson's work on large deviations in extreme-value theory. Treating separately the domains of attraction of Φα and Λ, we obtain general conditions for the relative error to tend to 0 as u →∞, y → ∞ simultaneously. The results serve as warning against the automatic extrapolation of estimates based on extreme-value approximations.
Monotone and associated Markov chains, with applications to reliability theory
Authors: Lindqvist Bo Henry
Published: July 2016

We study monotone and associated Markov chains on finite partially ordered state spaces. Both discrete and continuous time, and both time-homogeneous and time-inhomogeneous chains are considered. The results are applied to binary and multistate reliability theory.
The single-server queue with service depending on queue size and with the preemptive-resume last-come–first-served queue discipline
Authors: Fakinos D.
Published: July 2016

This paper studies theGI/G/1 queueing system assuming that customers have service times depending on the queue size and also that they are served in accordance with the preemptive-resume last-come–first-served queue discipline. Expressions are given for the limiting distribution of the queue size and the remaining durations of the corresponding services, when the system is considered at arrival epochs, at departure epochs and continuously in time. Also these results are applied to some particular cases of the above queueing system.
Queues with semi-Markovian arrivals
Authors: Çinlar Erhan
Published: July 2016

A queueing system with a single server is considered. There are a finite number of types of customers, and the types of successive arrivals form a Markov chain. Further, the nth interarrival time has a distribution function which may depend on the types of the nth and the n–1th arrivals. The queue size, waiting time, and busy period processes are investigated. Both transient and limiting results are given.
A note on telephone traffic with losses
Authors: Erlander Sven
Published: July 2016

Consider the following service system.
A generalisation of erlang's formulas in queueing theory
Authors: Mejzler D.
Published: July 2016

We follow the terminology of Khintchine's monograph [4].

Let us consider a random stream of calls entering a service system which consists of n lines. The service durations are assumed to be identically distributed random variables which are independent both of each other and of the course of the incoming stream.
General models for r-molecular reactions
Authors: Tallis G.M. , Leslie R.T.
Published: July 2016

In the present paper we consider the r-molecular reversible reaction rA⇌B from several viewpoints. The deterministic theory for integral reaction orders is considered first and is subsequently extended to cover the case of fractional order reactions. Stochastic models are then proposed, the analyses being carried through by spectral methods and, in the case of first order reactions, the first passage time problem is also examined. Finally, we use a diffusion theory approach to the problem to obtain results which are valid for a large number of molecules.
Diffusion approximations in collective risk theory
Authors: Iglehart L. Donald
Published: July 2016

Collective risk theory is concerned with the random fluctations of the total assets, the risk reserve, of an insurance company. Consider a company which only writes ordinary insurance policies such as accident, disability, fire, health, and whole life. The policyholders pay premiums regularly and at certain random times make claims to the company. A policyholder's premium, the gross risk premium, is a positive amount composed of two components. The net risk premium is the component calculated to cover the payments of claims on the average, while the security risk premium, or safety loading, is the component which protects the company from large deviations of claims from the average and also allows an accumulation of capital. When a claim occurs the company pays the policyholder a positive amount called the positive risk sum.
The minimum of a stationary Markov process superimposed on a U-shaped trend
Authors: Daniels H.E.
Published: July 2016

1. This paper was motivated by some questions of Barnett and Lewis (1967) concerning extreme winter temperatures. The temperature during the winter can be hopefully regarded as generated by a stationary Gaussian process superimposed on a locally U-shaped trend. One is interested in statistical properties of the minimum of sample paths from such a process, and of their excursions below a given level. Equivalently one can consider paths from a stationary process crossing a curved boundary of the same form. Problems of this type are discussed by Cramer and Leadbetter (1967), extensively in the trend-free case and in less detail when a trend is present, following the method initiated by Rice (1945). While results on moments are easy to obtain, explicit results for the actual probability distributions are not usually available. However, in the important case when the level of values of interest is far below the mean, the asymptotic independence of up-crossing times makes it possible to derive simple approximate distributions. (See Cramer and Leadbetter (1967) page 256, Keilson (1966).)
Random paths through convex bodies
Authors: Coleman Rodney
Published: July 2016

Several mechanisms under which the randomness of straight line paths through convex bodies can arise are described. Some general results are given relating four of these mechanisms, and the corresponding distributions of the lengths of the straight line paths are found for the circle, the rectangle, and the cube.
On a continuous/discrete time queueing system with arrivals in batches of variable size and correlated departures
Authors: Sharma S. D.
Published: July 2016

This paper studies the behaviour of a first-come-first-served queueing network with arrivals in batches of variable size and a certain service time distribution. The arrivals and departures of customers can take place only at the transition time marks and the intertransition time obeys a general distribution. The Laplace transforms of the probability generating functions for the queue length are obtained in the two cases; (i) when departures are correlated; (ii) when departures are uncorrelated; and the steady state results are derived therefrom. It has also been shown that the steady state continuous time solution is the same as the steady state discrete time solution.
On the distribution of the inter-record times in an increasing population
Authors: Yang Mark C. K.
Published: July 2016

It is shown in this note that if the population increases geometrically, then the asymptotic distribution for the inter-record times is also geometric. The records in Olympic games are used as an example. Also, it is noted that the rapid breaking of Olympic records is not due mainly to the increase in population.
The behavior near the origin of the supremum functional in a process with stationary independent increments
Authors: Resnick Sidney I. , Rubinovitch Michael
Published: July 2016

Let {X(t),t ≧0} be a process with stationary independent increments which is stochastically continuous with right-continuous paths and normalized so that X(0)=0. Let Z
1(t) = X(t), Z
2(t) = sup0≦s≦t

X(s) and Z
3 (t) = largest positive jump of X in (0, t] if there is one; = 0 otherwise. Then for i = 1,2,3 and x > 0: lim
t↓0
t
—1
P[Zi
(t) > x] = M
+(x) at all points of continuity of M
+, the Lévy measure of X.
The dependence of betting strategies on the probability of winning
Authors: Subelman Eduardo J.
Published: July 2016

We consider the optimal wagers to be made by a gambler facing cointossing games who desires to maximize the expected value of the utility of his final fortune in a fixed number n of plays. In the case of fixed probability of a win, the optimal bet is shown to be increasing in the probability. In the case of unknown probability of a win, the wager is shown to be monotone in the prior distribution under the monotone likelihood ratio ordering of these distributions.
Estimation of the criticality parameter of a superitical branching process with random environments
Authors: Nanthi K.
Published: July 2016

For a supercritical branching process x = {xn
; n ≧ 0, x
0 = 1} with random environments, define when xn > 0; and = 1 when xn
= 0. When x is assumed to satisfy the standard regularity assumptions, under the non-extinction hypothesis, is a strongly consistent and asymptotically unbiased estimator for the criticality parameter π and is asymptotically normal. A strongly consistent estimator,

is also proposed for the associated variance, σ
2.
A martingale approach to central limit theorems for exchangeable random variables
Authors: Weber N. C.
Published: July 2016

In this paper it will be shown that by an appropriate choice of σ-fields, martingale methods can be used to obtain simple proofs of many of the central limit theorems known for triangular arrays of exchangeable random variables.
Limit theorems for Markov processes via a variant of the Trotter-Kato theorem
Authors: Rosenkrantz Walter A. , Dorea C. C. Y.
Published: July 2016

A variant of the Trotter–Kato theorem due to Kurtz (1969) is used to give new and simpler proofs of functional central limit theorems for Markov processes. Applications include theorems of Bellman and Harris (1951), Stone (1961), Karlin and McGregor (1965), Gihman and Skorokhod (1972) and Rosenkrantz (1975). In addition our methods yield a novel counterexample to the so-called ‘diffusion approximation'.
A positive recurrence criterion associated with multidimensional queueing processes
Authors: Rosberg Zvi
Published: July 2016

A criterion is given for positive recurrence of a multidimensional, aperiodic, irreducible Markov chain with a denumerable state space. This criterion extends to the multidimensional case Foster's one-dimensional criterion. The multidimensional criterion consists of several conditions, one for each coordinate of the process. The usefulness of this criterion is shown through a queueing network example.
Stability in a non-homogeneous Markov chain model in manpower systems
Authors: Vassiliou P.-C. G.
Published: July 2016

Necessary and sufficient conditions for stability, imposed firstly on the initial structure and the sequence of recruitment, and secondly on the initial structure and the sequence of expansion are provided in forms of two theorems. Also the limiting behaviour of the expected relative grade sizes is studied if we drop the conditions for stability imposed on the initial structure and keep the same sequence of expansion. Finally we examine the limiting behaviour of the expected grade sizes if we drop the assumption of a continuously expanding system.
The spread of infection in a heterogeneous population
Authors: Cane Violet , McNamee Rosanne
Published: July 2016

A simple interaction model is proposed for describing variable meeting-rates in a population exposed to a contagious disease. In comparison with the usual assumption of a uniform meeting-rate it predicts, on average, a more rapid spread of infection initially and a slower spread finally; it also predicts that, for a given removal rate, an epidemic could develop among a smaller number of susceptibles.
A finite algorithm for the rank regression problem
Authors: Osborne M. R.
Published: July 2016

A new approach to the minimization of polyhedral convex functions is applied to give a finite algorithm for the rank regression problem. Numerical results for the Daniel and Wood example are presented.
Some classes of conditional inference procedures
Authors: Williams E. J.
Published: July 2016

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

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

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

With endomorphic parameters, since the surrogates and the statistics independent of them are functions of θ, tests of independence of these sets of statistics provide a means of making inferences about θ.
Identities relating forward and backward treatments of optimisation
Authors: Whittle P.
Published: July 2016

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.
On the relation between types and tokens in literary text
Authors: Brainerd Barron
Published: July 2016

The ratio of the number Xn
of different words (types) in a text of length n (token) words to n has received considerable attention in the literature of statistical linguistics. The present note contains two stochastic models for Xn
based on an inhomogeneous discrete Markov process of the pure birth type where the transition probabilities take certain forms depending only upon n. These models are then tested against data obtained from the plays of William Shakespeare.