bluuebunny/crossref_metadata_2025 huggingface dataset:
Random measures and their application to motion in an incompressible fluid
Authors: Port Sidney C. , Stone Charles J.
Published: July 2016

A generalization of a theorem of Geman and Horowitz concerning random measures is obtained. Their theorem in turn is used to extend a well-known result of Lumley, which asserts that for homogeneous turbulence in an incompressible fluid the probability distributions of the Eulerian and Lagrangian velocities coincide with each other at all times.
The stationary distribution of the infinitely-many neutral alleles diffusion model
Authors: Watterson G. A.
Published: July 2016

An expression is found for the stationary density of the allele frequencies, in the infinitely-many alleles model. It is assumed that all alleles are neutral, that there is a constant mutation rate, and that the population is sufficiently large for the diffusion model to apply. Bounds on some moments are calculated from the density, and some applications are made to the problem of which allele is oldest in a population. A postulate made by Ewens (1972), concerning the distribution of allele numbers in a finite random sample from the neutral diffusion population, is shown to be correct.
A proof of optimality for age replacement policies
Authors: Berg Menachem
Published: July 2016

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

In previous papers concerned with this procedure the authors searched for the optimal critical age to minimize the objective function under discussion. The purpose of this paper is to prove that the ARP itself is the optimal decision rule amongst all reasonable policies.
On a result of Rubin and Vere-Jones concerning subcritical branching processes
Authors: Hoppe Fred M.
Published: July 2016

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.
An exponential moving-average sequence and point process (EMA1)
Authors: Lawrance A. J. , Lewis P. A. W.
Published: July 2016

A construction is given for a stationary sequence of random variables {Xi
} which have exponential marginal distributions and are random linear combinations of order one of an i.i.d. exponential sequence {ε
i
}. The joint and trivariate exponential distributions of Xi

−1, Xi
and Xi

+ 1 are studied, as well as the intensity function, point spectrum and variance time curve for the point process which has the {Xi
} sequence for successive times between events. Initial conditions to make the point process count stationary are given, and extensions to higher-order moving averages and Gamma point processes are discussed.
Optimality of the shortest line discipline
Authors: Winston Wayne
Published: July 2016

We consider a queuing system consisting of a finite number of identical exponential servers. Each server has its own queue, and upon arrival each customer must be assigned to some server's queue. Under the assumption that no jockeying between queues is permitted, it is shown that the intuitively satisfying rule of assigning each arrival to the shortest line maximizes, with respect to stochastic order, the discounted number of customers to complete their service in any time t.
Mean duration time for a general epidemic process
Authors: Gladstien Billard
Published: July 2016

A general epidemic process is said to be completed whenever the number of susceptibles or the number of infectives reduces to zero. The mean duration time for this process to be completed is derived.
Single-shelf library-type Markov chains with infinitely many books
Authors: Nelson Peter R.
Published: July 2016

In a single-shelf library having infinitely many books B
1
, B
2
, …, the probability of selecting each book is assumed known. Books are removed one at a time and replaced in position k prior to the next removal. Books are moved either to the right or the left as is necessary to vacate position k. Those arrangements of books where after some finite position all the books are in natural order (book i occupies position i) are considered as states in an infinite Markov chain. When k > 1, we show that the chain can never be positive recurrent. When k = 1, we find the limits of ratios of one-step transition probabilities; and when k = 1 and the chain is transient, we find the Martin exit boundary.
Time and customer processes in queues with stationary inputs
Authors: Miyazawa Masakiyo
Published: July 2016

Two types of processes occurring in queues with stationary inputs are considered. They are called ‘time processes’ and ‘customer processes’. Sufficient conditions for the convergence of sample averages and the existence of limiting distributions for each type of processes are given. The results generalize those of Loynes (1962). The results are applied to three queueing processes and the Little's formula L = λW is obtained under rather general conditions.
Relative extinction times for independent branching processes
Authors: Slud Eric
Published: July 2016

The conditional probability that a Galton-Watson process with initial population j dies out before an independent Galton-Watson process initially i, is computed asymptotically as i or j gets large, given that both processes eventually die out. The results are interpreted qualitatively to discuss parameters of fitness for biological populations, especially with respect to the spread of mutant genes under evolutionary models.
Clumping effects in models of isolation by distance
Authors: Sudbury Aidan
Published: July 2016

Individuals situated on the infinite line breed and produce offspring that independently migrate a distance that is a random variable depending on the number in each family. It is shown that under very general conditions the individuals in the nth generation will tend to form clumps.
Ergodicity of diffusion and temporal uniformity of diffusion approximation
Authors: Norman M. Frank
Published: July 2016

Let {XN
(t), t ≧ 0}, N = 1, 2, … be a sequence of continuous-parameter Markov processes, and let TN
(t)f(x) = Ex
[f(XN
(t))]. Suppose that limN→∞

T

N
(t)f(x)= T(t)f(x), and that convergence is uniform over x and over t ∈ [0, K] for all K < ∞. When is convergence uniform over t ∈ [0, ∞)? Questions of this type are considered under the auxiliary condition that T(t)f(x) converges uniformly over x as t → ∞. A criterion for such ergodicity is given for semigroups T(t) associated with one-dimensional diffusions. The theory is illustrated by applications to genetic models.
A model for blue-green algae and gorillas
Authors: Morgan Byron J. T. , Leventhal B.
Published: July 2016

The linear birth-and-death process is elaborated to allow the elements of the process to live as members of linear clusters which have the possibility of breaking up. For the supercritical case, expressions, based on an approximation, are derived for the mean numbers of clusters of the various sizes as time → ∞. These expressions compare very well with exact solutions obtained by the method of Runge-Kutta. Exact solutions for the mean values for all time are given for when the death rate is zero.
On central limit and iterated logarithm supplements to the martingale convergence theorem
Authors: Heyde C. C.
Published: July 2016

Let {Sn
, n ≧ 1} be a zero, mean square integrable martingale for which so that Sn
→ S
∞ a.s., say, by the martingale convergence theorem. The paper is principally concerned with obtaining central limit and iterated logarithm results for Bn
(Sn
– S∞
) where the multipliers Bn
↑ ∞ a.s. An example on the Pólya urn scheme is given to illustrate the results.
On the autocorrelation structure for seasonal moving average models and its implications for the Cramér-Wold factorization
Authors: Godolphin E. J.
Published: July 2016

An account is given of the autocorrelation theory of seasonal moving average models, using a vector representation defined by the author. It is shown that the autocorrelation structure of a seasonal moving average is reducible to that of a certain non-seasonal moving average, independently of the value of the seasonal periodicity if sufficiently large. It is pointed out that existing methods for the Cramér–Wold factorization discussed in Godolphin (1976) can easily be adapted to contend with the seasonal case.
Estimation for multitype branching processes
Authors: Quine M. P. , Durham P.
Published: July 2016

It is shown that certain estimators of the matrix of offspring means and the vector of stationary means in a subcritical multitype branching process with immigration are strongly consistent and obey the central limit theorem under fairly natural conditions. The results give some evidence of the robustness of the analogous autoregressive time series model in the case of large samples.
Evaluation of the first-passage time probability to a square root boundary for the Wiener process
Authors: Sato Shunsuke
Published: July 2016

This paper gives an asymptotic evaluation of the probability that the Wiener path first crosses a square root boundary. The result is applied to estimate the moments of the first-passage time distribution of the Ornstein–Uhlenbeck process to a constant boundary.
A strong law of Erdös-Rényi type for cumulative processes in renewal theory
Authors: Steinebach Josef
Published: July 2016

Let {Nt
}
t >0 be a renewal counting process (cf. Parzen (1962), p. 160) with underlying failure times let be a sequence of non-negative random variables and {Zt
}
t >0 an associated cumulative process, i.e. if Nt
= 1, 2, …, and Zt
= 0, if Nt
= 0. By convention set Z
0 = 0. Consider the maximum increment of the process {Zt
}
t >0
in [0, T] over a time K, 0 < K < T, divided by K. Under appropriate conditions it is shown that for a wide range of numbers a there exist constants C(a), uniquely determined by a and the distributions of the X
i's and Yj
's, such that D(T, C log T) converges to a with probability 1. This result provides a renewal theoretic variant of Erdös and Rényi's (1970) ‘new law of large numbers’.
La formule de Steiner pour les érosions
Authors: Matheron G.
Published: July 2016

If A and K are compact convex sets in ℝn, a Steiner-type formula is valid for the erosion of A by K if and only if A is open with respect to K. The if part is proved in the general case. The only if part is conjectured, and proved only in the case n = 2.
Distances on circles, toruses and spheres
Authors: Silverman B. W.
Published: July 2016

Families of heavily dissociated random variables are defined and discussed. These include families of the form g(Yi
, Yj
) for some suitable function g of two arguments and independent uniformly distributed random variables Y
1, Y
2, ··· on the circle, the torus or the sphere. The weak convergence of the empirical distribution process is discussed. The particular case of distances between pairs of observations on the circle is considered in greater detail.
Random paths through a convex region
Authors: Enns E. G. , Ehlers P. F.
Published: July 2016

The distribution of the length of random secants through a convex region is formulated in terms of the intersection volume of the convex region with its translated self. This method allows a more straightforward approach to calculating secant-length distributions for various measures of randomness. The results are applied to calculating the transit-time distribution of particles traversing a convex region. Several examples are given.
Exponential spectra as a tool for the study of server-systems with several classes of customers
Authors: Keilson J.
Published: July 2016

For a single-server system having several Poisson streams of customers with exponentially distributed service times, busy period densities, waiting time densities, and idle state probabilities are completely monotone. The exponential spectra for such densities are of importance for understanding the transient behavior of such systems. Algorithms are given for the computation of such spectra. Applications to heavy traffic situations and priority systems are also discussed.
On the persistence of random genetic drift
Authors: Antonelli Peter L. , Morgan Kenneth
Published: July 2016

It is demonstrated that standard Brownian motion in the tangent plane at the centroid of frequency space does not well approximate the discrete Wright—Fisher process for more than 2N generations where N is the effective population size. This result is obtained using Wright's concept of negligible mutation rate for the study of systematic evolutionary effects together with Ludwig's notion of the persistence of a dynamical system. This work may be viewed as a mathematical elaboration of a portion of Wright's shifting balance theory of evolution.
Extinction probabilities in predator–prey models
Authors: Hitchcock S. E.
Published: July 2016

Two stochastic models are developed for the predator-prey process. In each case it is shown that ultimate extinction of one of the two populations is certain to occur in finite time. For each model an exact expression is derived for the probability that the predators eventually become extinct when the prey birth rate is 0. These probabilities are used to derive power series approximations to extinction probabilities when the prey birth rate is not 0. On comparison with values obtained by numerical analysis the approximations are shown to be very satisfactory when initial population sizes and prey birth rate are all small. An approximation to the mean number of changes before extinction occurs is also obtained for one of the models.
On transient regenerative processes
Authors: Murphree Emily , Smith Walter L.
Published: July 2016

A transient cumulative process based upon a sequence of possibly infinite lifetimes is defined, and examples of such a process are described. Given a mild condition on the improper lifetime distribution and given that all lifetimes observed by time t are finite, the expected value of this transient process at t is related to the expected value of a cumulative process based upon proper lifetimes. This relationship is exploited to show that the conditional behavior of the transient process is analogous to that of a proper process and, in particular, the transient process is asymptotically normal.
A useful generalization of renewal theory: counting processes governed by non-negative Markovian increments
Authors: Kijima Masaaki , Sumita Ushio
Published: July 2016

Let N(t) be a counting process associated with a sequence of non-negative random variables (X
j)1
∞ where the distribution of Xn

+1 depends only on the value of the partial sum Sn
= Σj=1
n
Xj
. In this paper, we study the structure of the function H(t) = E[N(t)], extending the ordinary renewal theory. It is shown under certain conditions that h(t) = (d/dt)H(t) exists and is a unique solution of an extended renewal equation. Furthermore, sufficient conditions are given under which h(t) is constant, monotone decreasing and monotone increasing. Asymptotic behavior of h(t) and H(t) as t → ∞ is also discussed. Several examples are given to illustrate the theoretical results and to demonstrate potential use of the study in applications.
Expected waiting times in a multiclass batch arrival retrial queue
Authors: Kulkarni V. G.
Published: July 2016

Expressions are derived for the expected waiting times for the customers of two types who arrive in batches (in a compound Poisson fashion) at a single-server queueing station with no waiting room. Those who cannot get served immediately keep returning to the system after random exponential amounts of time until they get served. The result is shown to agree with similar results for three special cases studied in the literature.
On the single-server queue with the preemptive-resume last-come–first-served queue discipline
Authors: Fakinos D.
Published: July 2016

The paper considers the GI/G/1 queueing system under the assumption of a last-come–first-served queue discipline, where each customer begins service immediately upon his arrival. At the next arrival, the previous service is interrupted but no loss of service is involved. It has been shown that when the system is considered exclusively at arrival epochs or exclusively at departure epochs, then the equilibrium distribution of the queue-size is geometric, while the remaining durations of the corresponding services are independent random variables each one distributed as the idle period in the dual (inverse) queue. In this paper alternative simpler proofs of the above results are given.
The departure process for the M/M/1 queue
Authors: Hubbard John R. , Pegden Claude Dennis , Rosenshine Matthew
Published: July 2016

The problem of determining the probability of j departures during the time interval (0, t) from an M/M/l queue empty at t = 0 is considered. A closed-form solution is obtained. It is shown that this solution is unique and invariant under interchanging the arrival rate and service rate. Finally, sample computational representations of the solution are developed and results of a simple computation are provided.
Finiteness of moments of partial busy periods for M/G/C queues
Authors: Ghahramani Saeed
Published: July 2016

Conditions for finiteness of moments of the following quantities have been found: the duration of a busy period of an Μ /G/∞ system; the duration of a partial busy period of an M/G/C loss system, and the duration of a partial busy period of an M/G/C queue.
The allocation of offensive and defensive resources in a territorial game
Authors: Cowan Richard
Published: July 2016

How should a team allocate its players to achieve the optimal balance between offence and defence? By posing a simple territorial game between two teams, insight into this question is gained. The saddlepoint solution to the game is derived.
Sur certaines fonctionnelles exponentielles du mouvement brownien réel
Authors: Yor Marc
Published: July 2016

Dufresne [1] recently showed that the integral of the exponential of Brownian motion with negative drift is distributed as the reciprocal of a gamma variable. In this paper, it is shown that this result is another formulation of the distribution of last exit times for transient Bessel processes. A bivariate distribution of such integrals of exponentials is also obtained explicitly.
Concavity and reflected Lévy processes
Authors: Kella Offer
Published: July 2016

Simple necessary and sufficient conditions for a function to be concave in terms of its shifted Laplace transform are given. As an application of this result, we show that the expected local time at zero of a reflected Lévy process with no negative jumps, starting from the origin, is a concave function of the time variable. A special case is the expected cumulative idle time in an M/G/1 queue. An immediate corollary is the concavity of the expected value of the reflected Lévy process itself. A special case is the virtual waiting time in an M/G/1 queue.
A remark on the proof of Itô's formula for C2 functions of continuous semimartingales
Authors: Kendall Wilfrid S.
Published: July 2016

The Itô formula is the fundamental theorem of stochastic calculus. This short note presents a new proof of Itô's formula for the case of continuous semimartingales. The new proof is more geometric than previous approaches, and has the particular advantage of generalizing immediately to the multivariate case without extra notational complexity.
Limit laws for the maxima of chain-dependent sequences with positive extremal index
Authors: Turkman K. F. , Oliveira M. F.
Published: July 2016

We extend the results on the extremal properties of chain-dependent sequences considered in Turkman and Walker (1983) by assuming conditions similar to those given by Leadbetter and Nandagopalan (1987) which permit clustering of high values.
Strong stationary times and eigenvalues
Authors: Matthews Peter
Published: July 2016

This note gives a new strong stationary time (SST) for reversible finite Markov chains. A modification of the initial distribution is represented as a mixture of distributions which have eigenvector interpretations, and for which good simple SSTs exist. This provides some insight into the relationship between SSTs and eigenvalues. Connections to duality and the threshold phenomenon are discussed.
On large deviations of empirical measures in the τ-topology
Authors: De Acosta A.
Published: July 2016

We prove a generalization of Sanov's theorem in which the state space S is arbitrary and the set of probability measures on S is endowed with the τ -topology.
A queueing network analysis of dynamic reconfigurability in a hierarchical information network
Authors: Bhat U. Narayan , Nance Richard E.
Published: July 2016

Hierarchical information networks are important in applications where the information management must support an existing tree-structured organization. Embedded computer-communication systems in military applications, with a dominant hierarchical command structure, are the most prominent examples. Also typical of such applications is the variability in message demand (both sources and intensity) depending on the external conditions encountered by the encapsulating system (the system supported by the embedded computer-communication system). Using a queueing network model for a hierarchical information network, we compare the effect of limited dynamic reconfiguration on expected transmission delays. The limited reconfigurability takes the form of apex transition among a proper subset of the communication nodes designated as the apex candidate set. Each apex candidate can assume the ultimate position under designated conditions. This network architecture is called a dynamic hierarchy. The model includes N + 1 nodes (0, …, N) with 0 identifying the apex node. We assume that message processing at each node is described by an M/M/1 model (single server with Poisson arrivals and exponential service times). Further message transfers among the nodes are served by communication links which also behave as M/M/1 queues.

Two distinctive features characterize the queueing network model:

1. The assignment of a set of weights to the nodes dependent on the hierarchical level reflects the increasing importance of information as it is transferred to higher levels.

2. The dynamic hierarchy requires a communications protocol that partitions the analysis of network delay into three periods: regular operation, reconfiguration, and adjustment. Characterization of the performance of the dynamic hierarchy entails the description of message transmission delay as a composite of the three periods.
Transient analysis of M/M/1 queues in discrete time by general server vacations
Authors: Böhm W. , Mohanty S. G.
Published: July 2016

In this contribution we consider an M/M/1 queueing model with general server vacations. Transient and steady state analysis are carried out in discrete time by combinatorial methods. Using weak convergence of discrete-parameter Markov chains we also obtain formulas for the corresponding continuous-time queueing model. As a special case we discuss briefly a queueing system with a T-policy operating.
Markov-modulated single-server queueing systems
Authors: Prabhu N. U. , Tang L. C
Published: July 2016

We consider single-server queueing systems that are modulated by a discrete-time Markov chain on a countable state space. The underlying stochastic process is a Markov random walk (MRW) whose increments can be expressed as differences between service times and interarrival times. We derive the joint distributions of the waiting and idle times in the presence of the modulating Markov chain. Our approach is based on properties of the ladder sets associated with this MRW and its time-reversed counterpart. The special case of a Markov-modulated M/M/1 queueing system is then analysed and results analogous to the classical case are obtained.
Aspects of Lagrangian probability distributions
Authors: Sibuya Masaaki , Miyawaki Norihiko , Sumita Ushio
Published: July 2016

Lagrangian distributions are reviewed from the viewpoint of the Galton-Watson process. They are related to the busy period in queuing systems and to the first visit in random walks.

A property of the distributions is remarked for the application to vacant vehicles in a new transit system. Combinatorial identities of multinomial and binomial coefficients and related recurrences are shown by a probabilistic method. Based on the identities and recurrences, random forests generated by the Poisson and geometric Galton-Watson processes are characterized.
Approximating last-exit probabilities of a random walk, by application to conditional queue length moments within busy periods of M/GI/1 queues
Authors: Daley D. J. , Servi L. D.
Published: July 2016

Certain last-exit and first-passage probabilities for random walks are approximated via a heuristic method suggested by a ladder variable argument. They yield satisfactory approximations of the first- and second-order moments of the queue length within a busy period of an M/D/1 queue. The approximation is applied to the wider class of random walks that arise in studying M/GI/1 queues. For gamma-distributed service times the queue length distribution is independent of the arrival rate. For other distributions where the arrival rate affects the queue length distribution, we have to use conjugate distributions in order to exploit a local central limit property. The limit underlying the approximation has the nature of a Brownian excursion.

The source of the problem lies in recent queueing inference work; the connection with Takács' interests comes from both queueing theory and the ballot theorem.
On the solution of Kac-type partial differential equations
Authors: Arató Mátyás
Published: July 2016

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].
An explicit solution to a particular Markov chain of M/G/1 type
Authors: Neuts Marcel F.
Published: July 2016

A Markov chain of M/G/1 type, which arose in a problem of communications engineering, is analyzed by a combination of matrix analysis and appropriate series expansions. The highly explicit results obtainable for this model owe much to analytic methods introduced by Professor Takács. Special features of the numerical implementation of the proposed solution are also discussed. The simplifications proper to this model can also be used in some related, more complex models to be discussed elsewhere.
Estimation of cause-effect relationship under noise
Authors: Prékopa András
Published: July 2016

Events that occur consecutively or simultaneously cause some other event as effect. The latter can be observed with noise, and the problem is to estimate the weights of the causes in the realization of the effect.
A simple increasing process
Authors: Syski Ryszard
Published: July 2016

An example of a simple integrable increasing process, possessing special properties relative to different filtrations, is examined from the point of view of probabilistic potential theory.
Some comparison results for multitype epidemic models
Authors: Clancy Damian
Published: July 2016

We consider a model for the spread of an epidemic through a population divided into m groups, in which infectives move from group to group and infect only within their current group. For both deterministic and stochastic versions of this model, the effect on the total size of the epidemic of varying the speed with which infectives move between groups is considered. We also compare the distribution of the total size of this model with that of a suitably matched model in which infectives cannot move between groups, but are able to infect outside their own group.
A strong law and a central limit theorem for controlled Galton-Watson processes
Authors: Viaud Daniel Pierre Loti
Published: July 2016

Through the study of a simple embedded martingale we obtain an extension of the Kesten–Stigum theorem and prove a central limit theorem for controlled Galton-Watson processes.
Brownian motions on shape and size-and-shape spaces
Authors: Le Huiling
Published: July 2016

The diffusions on the shape and size-and-shape spaces induced by brownian motions on the pre-size-and-shape spaces have been investigated in several papers (cf.). We here address the dual problem: the character of the diffusions on the pre-shape and pre-size-and-shape spaces which induce brownian motions on the shape and size-and-shape spaces. In particular we show that the shape and size-and-shape spaces for k labelled points in ℝ
m
are stochastically complete if k > m and obtain the heat kernels of certain diffusions which induce brownian motions on the size-and-shape spaces.
Crossings of max-stable processes
Authors: Davis Richard A. , Resnick Sidney I.
Published: July 2016

The expected number of upcrossings for a max-stable process is computed and compared with known results for stable processes. Asymptotically the formulas are of the same order.
Shock models by underlying counting process
Authors: Pellerey Franco
Published: July 2016

Suppose that a device is subjected to shocks governed by a counting process N = {N(t), t ≧0}. The probability that the device survives beyond time t is then H̄(t)=Σk=0
∞Q̄ℙ[N(t)=k], where Q̄k
is the probability of surviving k shocks. It is known that H is NBU if the interarrivals Uk
, ∊ ℕ+, are independent and NBU, and Q̄

k+j
≦ Q̄k
· Q̄j
holds whenever k, j ∊ ℕ. Similar results hold for the classes of the NBUE and HNBUE distributions. We show that some other ageing classes have similar properties.
Applications of the hazard rate ordering in reliability and order statistics
Authors: Boland Philip J. , El-Neweihi Emad , Proschan Frank
Published: July 2016

The hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their hazard rate functions. It is stronger than the usual stochastic order for random variables, yet is weaker than the likelihood ratio ordering. The hazard rate ordering is particularly useful in reliability theory and survival analysis, owing to the importance of the hazard rate function in these areas. In this paper earlier work on the hazard rate ordering is reviewed, and extensive new results related to coherent systems are derived. Initially we fix the form of a coherent structure and investigate the effect on the hazard rate function of the system when we switch the lifetimes of its components from the vector (T
1, · ··, Tn
) to the vector (T′
1, · ··, T′n
), where the hazard rate functions of the two vectors are assumed to be comparable in some sense. Although the hazard rate ordering is closed under the formation of series systems, we see that this is not the case for parallel systems even when the system is a two-component parallel system with exponentially distributed lifetimes. A positive result shows that for two-component parallel systems with proportional hazards (λ
1
r(t), λ
2
r(t))), the more diverse (λ
1, λ2
) is in the sense of majorization the stronger is the system in the hazard rate ordering. Unfortunately even this result does not extend to parallel systems with more than two components, demonstrating again the delicate nature of the hazard rate ordering.

The principal result of the paper concerns the hazard rate ordering for the lifetime of a k-out-of-n system. It is shown that if τ k|n
is the lifetime of a k-out-of-n system, then τ k|n
is greater than τ k+

1|n
in the hazard rate ordering for any k. This has an interesting interpretation in the language of order statistics. For independent (not necessarily identically distributed) lifetimes T
1, · ··, Tn
, we let Tk:n
represent the kth order statistic (in increasing order). Then it is shown that Tk +

1:n
is greater than Tk:n
in the hazard rate ordering for all k = 1, ···, n − 1. The result does not, however, extend to the stronger likelihood ratio order.
Asymptotic distributions of extremes of extremal Markov sequences
Authors: Adke S. R. , Chandran C.
Published: July 2016

Let {ξ
n
, n ≧1} be a sequence of independent real random variables, F denote the common distribution function of identically distributed random variables ξ
n
, n ≧1 and let ξ
1 have an arbitrary distribution. Define Xn+

1 = k max(Xn, ξ n

+1), Yn

+ 1 = max(Yn, ξ n

+1) – c, Un

+1 = l min(Un, ξ n

+1), Vn+

1 = min(Vn, ξ n

+1) + c, n ≧ 1, 0 < k < 1, l > 1, 0 < c < ∞, and X
1 = Υ
1
= U
1 = V
1 = ξ
1. We establish conditions under which the limit law of max(X
1, · ··, Xn
) coincides with that of max(ξ
2, · ··, ξ n+

1) when both are appropriately normed. A similar exercise is carried out for the extreme statistics max(Y
1, · ··, Yn
), min(U
1,· ··, Un
) and min(V
1, · ··, Vn
).
A generalization of ‘expectation equals reciprocal of intensity' to non-stationary exponential distributions
Authors: Feinberg Eugene A.
Published: July 2016

An observer watches one of a set of Poisson streams. He may switch from one stream to another instantaneously. If an arrival occurs in a stream while the observer is watching another stream, he does not see the arrival. The experiment terminates when the observer sees an arrival. We derive a formula which states essentially that the expected total time that the observer watches a stream is equal to the probability that he sees the arrival in this stream divided by the intensity of the stream. This formula is valid independently of the observation policy. We also discuss applications of this formula.
Weak limit for the stationary distribution of the infinite-alleles model
Authors: Avni Haim
Published: July 2016

The limit behavior of the stationary distribution of the infinite-alleles model is reduced to a single Laplace transform formula. Some known results, such as Ewens' sampling formula, the distribution of the order-statistics and the frequency spectrum density are shown to follow from this relation. All the results are obtained within the framework of the configuration process, without recourse to finite alleles models.

In view of some recent results by Kingman (1977), the results apply wherever the Ewens' sampling formula is valid.
A limit theorem for patch sizes in a selectively-neutral migration model
Authors: Sawyer Stanley
Published: July 2016

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

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

The proofs are by using multiple kinship coefficients for a stepping stone population.
On Cohen's stochastic generalization of the strong ergodic theorem of demography
Authors: Lange Kenneth
Published: July 2016

Cohen has generalized the classical strong ergodic theorem of demography to a stochastic setting. In this setting population projection matrices are chosen according to some homogeneous Markov chain. If this Markov chain converges to the same long-run distribution regardless of its starting point, then one can define an induced Markov chain on the product space of projection matrices and age structure vectors that also has a long-run distribution independent of its starting point. The present paper gives more natural conditions under which Cohen's result holds.
Limiting results for arrays of binary random variables on rectangular lattices under sparseness conditions
Authors: Saunders Roy , Kryscio Richard J. , Funk Gerald M.
Published: July 2016

In this article we give limiting results for arrays {Xij
(m, n) (i, j)
Dmn
} of binary random variables distributed as particular types of Markov random fields over m x n rectangular lattices Dmn. Under some sparseness conditions which restrict the number of X
ij (m, n)'s which are equal to one we show that the random variables (l = 1, ···, r)

converge to independent Poisson random variables for 0 < d1 < d2
< · ·· < dr
when m→∞ nd∞. The particular types of Markov random fields considered here provide clustering (or repulsion) alternatives to randomness and involve several parameters. The limiting results are used to consider statistical inference for these parameters. Finally, a simulation study is presented which examines the adequacy of the Poisson approximation and the inference techniques when the lattice dimensions are only moderately large.
Random exchanges of information
Authors: Boyd David W. , Steele J. Michael
Published: July 2016

Suppose that n persons each know a different piece of information, and that whenever a pair of persons talk on the telephone each tells the other all the information that he knows at the time. If calls are made at random, we show that the expected number of calls necessary until everyone knows all n pieces of information is asymptotically 1.5 n log n. This sharpens an earlier result of J. W. Moon.
An optimal betting strategy for repeated games
Authors: Gottlieb Gary
Published: July 2016

We consider the problem of finding a betting strategy for an infinite sequence of wagers where the optimality criterion is the minimization of the expected exit time of wealth from an interval. We add the side constraint that the right boundary is hit first with at least some specified probability. The optimal strategy is derived for a diffusion approximation.
The limiting behaviour of the maximal spacing generated by an i.i.d. sequence of Gaussian random variables
Authors: Deheuvels Paul
Published: July 2016

LetMnbe the maximal spacing generated in the sample's range byΧ1,· ··,Χn, independent and identically distributed Gaussian N(0, 1) random variables. We obtain the limiting distribution ofand show that

according to whether ε> 0 orε< 0.
Bernoulli, multinomial and Markov chain thinning of some point processes and some results about the superposition of dependent renewal processes
Authors: Chandramohan J. , Liang Lung-Kuang
Published: July 2016

We show that Bernoulli thinning of arbitrarily delayed renewal processes produces uncorrelated thinned processes if and only if the renewal process is Poisson. Multinomial thinning of point processes is studied. We show that if an arbitrarily delayed renewal process or a doubly stochastic Poisson process is subjected to multinomial thinning, the existence of a single pair of uncorrelated thinned processes is sufficient to ensure that the renewal process is Poisson and the double stochastic Poisson process is at most a non-homogeneous Poisson process. We also show that a two-state Markov chain thinning of an arbitrarily delayed renewal process produces, under certain conditions, uncorrelated thinned processes if and only if the renewal process is Poisson and the Markov chain is a Bernoulli process. Finally, we identify conditions under which dependent point processes superpose to form a renewal process.
An approximate solution of the integral equation of renewal theory
Authors: Deligönül Z. Șeyda
Published: July 2016

In this study, an approximation to the solution of the renewal integral equation is constructed. Performance of the new method is evaluated and it is shown that the approximation provides an upper bound for the renewal function when the hazard function is a non-increasing function of time.
A note on a continuous-time Markov manpower model
Authors: Davies G. S.
Published: July 2016

We examine a Markov manpower system in continuous time where demotion rates are 0 and promotion rates are time-dependent. The transient and limiting behaviours of the model are discussed, and illustrated with examples.
A note on the asymptotic properties of correlated random walks
Authors: Roerdink J. B. T. M.
Published: July 2016

We describe a simple relation between the asymptotic behavior of the variance and of the expected number of distinct sites visited during a correlated random walk. The relation is valid for multistate random walks with finite variance in dimensions 1 and 2. A similar relation, valid in all dimensions, exists between the asymptotic behavior of the variance and of the probability of return to the origin.
The residual process for non-linear regression
Authors: MacNeill Ian B. , Jandhyala V. K.
Published: July 2016

Limit processes for sequences of partial sums of non-linear regression residuals are obtained under assumptions on regressor functions imposed by Jennrich (1969). The limit process and covariance kernel are calculated explicitly for functions of exponential type.
Boundaries with negative jumps for the Brownian motion
Authors: Park C.
Published: July 2016

First-exit-time problems for Brownian motion have been studied extensively because of their theoretical importance as well as their practical applications. Except for a very few special cases such as straight-line barriers, the distribution (or density) of the first-exit time cannot be expressed in a closed form. In general the distribution and the density appear as solutions of Volterra integral equations. To solve such an equation analytically, some regularity conditions are needed for the barrier function including differentiability. This paper gives various integral equations involving the distribution and the density of the first-exit time for any sectionally continuous barrier, and then shows how to solve those integral equations numerically.
On stopping rules in proofreading
Authors: Chow Chung-Wen , Schechner Zvi
Published: July 2016

Stopping rules in the context of proofreading are studied. Previous results by Yang et al. [6] are corrected and a computational procedure to obtain an optimal rule is presented.
Some probability distributions for neutron transport in a half-space
Authors: Mullikin T. W.
Published: July 2016

We will consider a problem which arises in the theories of neutron transport and radiative (photon) transfer. We will consider, in a particular case, the random motion of a particle in a scattering medium and determine asymptotics for the probability distribution of the random variable for the number of collisions before exit from the medium.
Exponential ergodicity in Markov renewal processes
Authors: Teugels Jozef L.
Published: July 2016

In [3], Kendall proved a solidarity theorem for irreducible denumerable discrete time Markov chains. Vere-Jones refined Kendall's theorem by obtaining uniform estimates [14], while Kingman proved analogous results for an irreducible continuous time Markov chain [4], [5].

We derive similar solidarity theorems for an irreducible Markov renewal process. The transient case is discussed in Section 3, and Section 4 deals with the positive recurrent case. Recently Cheong also proved solidarity theorems for Semi-Markov processes [1]. His theorems use the Markovian structure, while our emphasis is on the renewal aspects of Markov renewal processes.

An application to the M/G/1 queue is included in the last section.
On a multi-type critical age-dependent branching process
Authors: Weiner H. J.
Published: July 2016

We will consider a branching process with m > 1 distinguishable particle types as follows. At time 0, one newly born cell of type i is born (i = 1, 2, ···, m). Cell type i lives a random lifetime with continuous distribution function Gi
(t), Gi
(0+) = 0. At the end of its life, cell i is replaced by j
1 new cells of type 1, j
2 new cells of type 2, ···, jm
new cells of type m with probability , and we define the generating functions

for i = 1,···,m, where and . Each new daughter cell proceeds independently of the state of the system, with each cell type j governed by Gj(t) and hj(s).
A stochastic model for two interacting populations
Authors: Becker Niels G.
Published: July 2016

To explain the growth of interacting populations, non-linear models need to be proposed and it is this non-linearity which proves to be most awkward in attempts at solving the resulting differential equations. A model with a particular non-linear component, initially proposed by Weiss (1965) for the spread of a carrier-borne epidemic, was solved completely by different methods by Dietz (1966) and Downton (1967). Immigration parameters were added to the model of Weiss and the resulting model was made the subject of a paper by Dietz and Downton (1968). It is the aim here to further generalize the model by introducing birth and death parameters so that the result is a linear birth and death process with immigration for each population plus the non-linear interaction component.
Migratory populations in branching processes
Authors: Crump Kenny S.
Published: July 2016

A general age-dependent branching process is studied in which the population is distributed in N colonies. Migration from one colony to another is possible and no particular geometrical configuration is assumed for the colonies. Asymptotic expressions for the first two moments are given in several different cases.
A two-unit standby redundant system with repair and preventive maintenance
Authors: Osaki Shunji , Asakura Tatsuyuki
Published: July 2016

A two-unit standby redundant system with repair and preventive maintenance is discussed. Assuming that the failure, repair, inspection, and preventive maintenance (preventive repair) time distributions of each unit for the system are all arbitrary, the Laplace-Stieltjes transform (L.S.T.) of the distribution of time to the first system down is derived and the mean time obtained from it. It is further shown that under suitable assumptions, the preventive maintenance policy for the system is effective in the sense that it increases the mean time to the first system down.
Linear programming and continuous markovian decision problems
Authors: Mine Hisashi , Tabata Yoshio
Published: July 2016

This paper is concerned with a continuous time parameter Markovian sequential decision process, and presents a method which transforms a given continuous parameter problem into a discrete one. It is proved that the optimal stationary policy for the resulting discrete time parameter Markovian decision process is also the optimal stationary policy for the original continuous one, and vice versa. The resulting discrete parameter problem may be more easily solved than the continuous one by applying the linear programming method. A simple numerical example is presented.