bluuebunny/crossref_metadata_2025 huggingface dataset:
Stationarity properties of neural networks
Authors: Asmussen Søren , Turova Tatyana S.
Published: July 2016

A neural model with N interacting neurons is considered. A firing of neuron i delays the firing times of all other neurons by the same random variable θ(i), and in isolation the firings of the neuron occur according to a renewal process with generic interarrival time Y
(i). The stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes. Further, the distribution of the period of activity of a neuron is studied for the symmetric case where θ(i) and Y
(i) do not depend upon i. The tools are probabilistic and involve path decompositions, Palm theory and random walks.
Slow variation and uniqueness of solutions to the functional equation in the branching random walk
Authors: Kyprianou A. E.
Published: July 2016

In this short communication, some of the recent results of Liu (1998) and Biggins and Kyprianou (1997), concerning solutions to a certain functional equation associated with the branching random walk, are strengthened. Their importance is emphasized in the context of travelling wave solutions to a discrete version of the KPP equation and the connection with the behaviour of the rightmost particle in thenth generation.
Random walks on finite semigroups
Authors: Barnes George R. , Cerrito Patricia B. , Levi Inessa
Published: July 2016

The purpose of this paper is to study the asymptotic properties of Markov chains on semigroups. In particular, the structure of transition matrices representing random walks on finite semigroups is examined. It is shown that the transition matrices associated with certain semigroups are block diagonal with identical blocks. The form of the blocks is determined via the algebraic structure of the semigroup.
New characterizations of the no-aging property and the ℓ1-isotropic model
Authors: Chick Stephen E. , Mendel Max B.
Published: July 2016

The no-aging property and the ℓ1-isotropic model it implies have been introduced to overcome certain shortcomings of the exponential model. However, its definition is abstract and not very useful for practitioners. This paper presents several additional characterizations of the no-aging property. Included are (1) characterizations that appropriately generalize the memoryless property and the constant-failure-rate property of the exponential, (2) behavioral characterizations based on fair bets, and (3) geometric characterizations of the survival and density function and differential-geometric characterizations based on tensor methods.
On a distributed clustering process of Coffman, Courtois, Gilbert and Piret
Authors: van den Berg J. , Ermakov A.
Published: July 2016

Coffman, Courtois, Gilbert and Piret (1991) have introduced a flow process in graphs, where each vertex gets an initial random resource, and where at each time vertices with large resources tend to attract resources from neighbours. The initial resources are assumed to be i.i.d., with a continuous distribution.

We are mainly interested in the following question: does, with probability 1, the resource of each vertex change only finitely many times?

Coffman et al. concentrate mainly on the case where the graph corresponds with the integer points on the line, in which case it is easily seen that the answer to the above question is positive. For higher-dimensional lattices they make general remarks which suggest that the answer to the above question is still positive. However, no proof seems to be known.

We restrict to the case of the square lattice, and, by a percolation approach, we reduce the question above to the question whether a certain quantity, which can be obtained from a finite computation, is sufficiently small. This computation is, however, still too long to be executed in an acceptable time. We therefore apply Monte Carlo simulation for this finite problem, which gives overwhelming evidence that, for the square lattice, the answer to the main question is positive.
A note on losses in M/GI/1/n queues
Authors: Righter Rhonda
Published: July 2016

Let L

n
be the number of losses during a busy period of an M/GI/1/n queueing system. We develop a coupling between L

n
and L

n+1 and use the resulting relationship to provide a simple proof that when the mean service time equals the mean interarrival time, EL

n
= 1 for all n. We also show that L

n
is increasing in the convex sense when the mean service time equals the mean interarrival time, and it is increasing in the increasing convex sense when the mean service time is less than the mean interarrival time.
Optimal investment strategies in a CIR framework
Authors: Deelstra Griselda , Grasselli Martino , Koehl Pierre-François
Published: July 2016

We study an optimal investment problem in a continuous-time framework where the interest rates follow Cox-Ingersoll-Ross dynamics. Closed form formulae for the optimal investment strategy are obtained by assuming the completeness of financial markets and the CRRA utility function. In particular, we study the behaviour of the solution when time approaches the terminal date.
Lower bounds for point-to-point wandering exponents in Euclidean first-passage percolation
Authors: Howard C. Douglas
Published: July 2016

In first-passage percolation models, the passage time T(0,L) from the origin to a point L is expected to exhibit deviations of order |L|χ from its mean, while minimizing paths are expected to exhibit fluctuations of order |L|ξ away from the straight line segment . Here, for Euclidean models in dimension d, we establish the lower bounds ξ ≥ 1/(d+1) and χ ≥(1-(d-1)ξ)/2. Combining this latter bound with the known upper bound ξ ≤ 3/4 yields that χ ≥ 1/8 for d=2.
On a better burn-in procedure
Authors: Cha Ji Hwan
Published: July 2016

A new burn-in procedure for a repairable component is proposed. During a burn-in period, the failed component is only minimally repaired rather than being completely repaired. This procedure is shown to be economical and efficient when the minimal repair method is applicable during a burn-in process. The properties of the optimal burn-in time b
∗ and block replacement policy T
∗ are also given.
Stein factor bounds for random variables
Authors: Weinberg Graham V.
Published: July 2016

A probabilistic proof of a Stein factor bound is given, extending the Poisson case in Xia (1999).
On the nature of the binomial distribution
Authors: Vellaisamy P. , Punnen Abraham P.
Published: July 2016

We examine how the binomial distribution B(n,p) arises as the distribution S

n
= ∑
i=1

n

X

i
of an arbitrary sequence of Bernoulli variables. It is shown that B(n,p) arises in infinitely many ways as the distribution of dependent and non-identical Bernoulli variables, and arises uniquely as that of independent Bernoulli variables. A number of illustrative examples are given. The cases B(2,p) and B(3,p) are completely analyzed to bring out some of the intrinsic properties of the binomial distribution. The conditions under which S

n
follows B(n,p), given that S

n-1 is not necessarily a binomial variable, are investigated. Several natural characterizations of B(n,p), including one which relates the binomial distributions and the Poisson process, are also given. These results and characterizations lead to a better understanding of the nature of the binomial distribution and enhance the utility.
The stochastic equation Y
t+1 = A
t
Y
t + B
t with non-stationary coefficients
Authors: Horst Ulrich
Published: July 2016

In this paper, we consider the stochastic sequence {Y

t
}
t∊ℕ defined recursively by the linear relation Y

t+1 = A

t

Y

t
+ B

t
in a random environment which is described by the non-stationary process {(A

t
, B

t
)}
t∊ℕ. We formulate sufficient conditions on the environment which ensure that the finite-dimensional distributions of {Y

t
}
t∊ℕ converge weakly to the finite-dimensional distributions of a unique stationary process. If the driving sequence {(A

t
, B

t
)}
t∊ℕ becomes stationary in the long run, then we can establish a global convergence result. This extends results of Brandt (1986) and Borovkov (1998) from the stationary to the non-stationary case.
Empirical convergence rates for continuous-time Markov chains
Authors: Pritchard Geoffrey , Scott David J.
Published: July 2016

We consider the problem of estimating the rate of convergence to stationarity of a continuous-time, finite-state Markov chain. This is done via an estimator of the second-largest eigenvalue of the transition matrix, which in turn is based on conventional inference in a parametric model. We obtain a limiting distribution for the eigenvalue estimator. As an example we treat an M/M/c/c queue, and show that the method allows us to estimate the time to stationarity τ within a time comparable to τ.
Optimal stopping on trajectories and the ballot problem
Authors: Tamaki Mitsushi
Published: July 2016

An urn contains m minus balls and p plus balls, and we draw balls from this urn one at a time randomly without replacement until we wish to stop. Let P

n
and M

n
denote the respective numbers of plus balls and minus balls drawn by time n and define Z
0 = 0, Z

n
= P

n
- M

n
, 1 ≤ n ≤ m + p. The main problem of this paper is to stop with maximum probability on the maximum of the trajectory formed by . This problem is closely related to the celebrated ballot problem, so that we obtain some identities concerning the ballot problem and then derive the optimal stopping rule explicitly. Some related modifications are also studied.
Sequential games
Authors: Siegrist Kyle , Steele John
Published: July 2016

We give a general construction of sequential games among multiple players, as well as a construction of the composition of sequential games. We obtain new properties of the optimal class of win-by-k games, including closure under composition and independence between the winner of the game and the number of points played. We obtain new results on the asymptotic efficiency of the n-point, win-by-k games.
Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates
Authors: Guo Xianping , Hernández-Lerma Onésimo
Published: July 2016

This paper is a first study of two-person zero-sum games for denumerable continuous-time Markov chains determined by given transition rates, with an average payoff criterion. The transition rates are allowed to be unbounded, and the payoff rates may have neither upper nor lower bounds. In the spirit of the ‘drift and monotonicity’ conditions for continuous-time Markov processes, we give conditions on the controlled system's primitive data under which the existence of the value of the game and a pair of strong optimal stationary strategies is ensured by using the Shapley equations. Also, we present a ‘martingale characterization’ of a pair of strong optimal stationary strategies. Our results are illustrated with a birth-and-death game.
Stochastic ordering for continuous-time processes
Authors: Irle A.
Published: July 2016

We consider the following ordering for stochastic processes as introduced by Irle and Gani (2001). A process (Y

t
)
t
is said to be slower in level crossing than a process (Z

t
)
t
if it takes (Y

t
)
t
stochastically longer than (Z

t
)
t
to exceed any given level. In Irle and Gani (2001), this ordering was investigated for Markov chains in discrete time. Here these results are carried over to semi-Markov processes with particular attention to birth-and-death processes and also to Wiener processes.
Finite-size corrections to Poisson approximations of rare events in renewal processes
Authors: Spouge John L.
Published: July 2016

Consider a renewal process. The renewal events partition the process into i.i.d. renewal cycles. Assume that on each cycle, a rare event called 'success’ can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence may be relatively slow, because each success corresponds to a time interval, not a point. In 1996, Altschul and Gish proposed a finite-size correction to a particular approximation by a Poisson point process. Their correction is now used routinely (about once a second) when computers compare biological sequences, although it lacks a mathematical foundation. This paper generalizes their correction. For a single renewal process or several renewal processes operating in parallel, this paper gives an asymptotic expansion that contains in successive terms a Poisson point approximation, a generalization of the Altschul-Gish correction, and a correction term beyond that.
Optimal stopping of a risk process: model with interest rates
Authors: Muciek Bogdan Krzysztof
Published: July 2016

The following problem in risk theory is considered. An insurance company, endowed with an initial capital a ≥ 0, receives premiums and pays out claims that occur according to a renewal process {N(t), t ≥ 0}. The times between consecutive claims are i.i.d. The sequence of successive claims is a sequence of i.i.d. random variables. The capital of the company is invested at interest rate α ∊ [0,1], claims increase at rate β ∊ [0,1]. The aim is to find the stopping time that maximizes the capital of the company. A dynamic programming method is used to find the optimal stopping time and to specify the expected capital at that time.
Estimation of the variance for the multitype Galton–Watson process
Authors: Nanthi K.
Published: July 2016

This paper is concerned with the estimation of the variance for the multitype Galton-Watson process X = {X

n
= (X

n
(1),…, X

n
(p)); n ≧ 0}. Two estimators for the variance matrix are obtained and asymptotic results for the estimators are given. The first is a maximum likelihood estimator based upon knowledge of individual offspring sizes, the second estimator is based on parent-offspring type combination counts only. Estimators for the asymptotic variances of the Asmussen and Keiding estimator and Becker estimator are also proposed.
Regularity conditions for semi-Markov and Markov chains in continuous time
Authors: Gerrard Russell
Published: July 2016

The classical condition for regularity of a Markov chain is extended to include semi-Markov chains. In addition, for any given semi-Markov chain, we find Markov chains which exhibit identical regularity properties. This is done either (i) by transforming the state space or, alternatively, (ii) by imposing conditions on the holding-time distributions. Brief consideration is given to the problem of extending the results to processes other than semi-Markov chains.
On the roles of the Bessel and Poisson distributions in chemical kinetics
Authors: Hall Peter
Published: July 2016

We consider the Darvey, Ninham and Staff model for reversible chemical reactions, in the case where the ratio of the rate constants is either very large or very small. It is shown that the distribution of the number of molecules at equilibrium may sometimes be closely approximated by the Poisson distribution, and on other occasions, by a distribution with much smaller tails than the Poisson. The second type of approximating distribution is termed a Bessel distribution, and its properties are studied.
Imperfect repair
Authors: Brown Mark , Proschan Frank
Published: July 2016

A device is repaired at failure. With probability p, it is returned to the ‘good-as-new' state (perfect repair), with probability 1 – p, it is returned to the functioning state, but it is only as good as a device of age equal to its age at failure (imperfect repair). Repair takes negligible time. We obtain the distribution Fp
of the interval between successive good-as-new states in terms of the underlying life distribution F. We show that if F is in any of the life distribution classes IFR, DFR, IFRA, DFRA, NBU, NWU, DMRL, or IMRL, then Fp
is in the same class. Finally, we obtain a number of monotonicity properties for various parameters and random variables of the stochastic process. The results obtained are of interest in the context of stochastic processes in general, as well as being useful in the particular imperfect repair model studied.
On a random mapping (T, Pj
)
Authors: Jaworski Jerzy
Published: July 2016

A random mapping (T,Pj
) of a finite set V into itself is studied. We give a new proof of the fundamental lemma of [6]. Our method leads to the derivation of several results which cannot be deduced from [6]. In particular we determine the distribution of the number of components, cyclical points and ancestors of a given point.
On the past history of an allele now known to have frequency p
Authors: Sawyer Stanley
Published: July 2016

Consider a diploid population of N individuals undergoing random mating and mutation as in the infinite-allele Wright model. Choose a particular gene in generation Q for large Q, and let f
t (0 ≦ t ≦ Q) be the frequency of the allelic type of the predecessor of that particular gene (or the gene itself) in generation t. By considering a ‘diffusion approximation' {xt
} of {ft
}, we find the distribution of the age of an allelic type now known to have frequency p, and of its distribution of frequencies since the allele came into existence. A novelty here is that the process {xt
} is not a diffusion, but a process with jumps; it has x = 0 as an inaccessible entrance boundary but periodically jumps to it from the interior of [0, 1]. The formulas obtained are the same as those derived by Maruyama and Kimura, who used a totally different approach.
The age distribution of Markov processes
Authors: Levikson Benny
Published: July 2016

A limiting distribution for the age of a class of Markov processes is found if the present state of the process is known. We use this distribution to find the age of branching processes. Using the fact that the moments of the age of birth and death processes and of diffusion processes satisfy difference equations and differential equations respectively, we find simple formulas for these moments. For the Wright–Fisher genetic model we find the probability that a given allele is the oldest in the population if all the gene frequencies are known. The proofs of the main results are based on methods from renewal theory.
On Bayesian models in stochastic scheduling
Authors: Gittins J. C. , Glazebrook K. D.
Published: July 2016

The D.A.I. theorem of Gittins and Jones has proved a powerful tool in solving sequential statistical problems. A generalisation of this theorem is presented. This generalisation enables us to solve certain stochastic scheduling problems where the items or jobs to be scheduled have random times to completion, the random times having distributions dependent upon parameters to which prior distributions are allocated. Such problems are of interest in many areas where scheduling is important.
Limit theorems for periodic queues
Authors: Harrison J. Michael , Lemoine Austin J.
Published: July 2016

Consider a single-server queue with service times distributed as a general random variable S and with non-stationary Poisson input. It is assumed that the arrival rate function λ (·) is periodic with average value λ and that ρ = λE(S) < 1. Both weak and strong limit theorems are proved for the waiting-time process W = {W
1, W
2, · ··} and the server load (or virtual waiting-time process) Z = {Z(t), t ≧ 0}. The asymptotic distributions associated with Z and W are shown to be related in various ways. In particular, we extend to the case of periodic Poisson input a well-known formula (due to Takács) relating the limiting virtual and actual waiting-time distributions of a GI/G/1 queue.
An extension of the Rao-Rubin characterization of the Poisson distribution
Authors: Shanbhag D. N.
Published: July 2016

Using Bernstein's theorem concerning absolutely monotonic functions, Rao and Rubin (1964) have established a characterization of the Poisson distribution. The present note arrives at an extended version of this result using a technique existing in the renewal theory. A variant of a related result due to Srivastava and Srivastava (1970) and an extension of the result of Talwalker (1970) are also presented.
Multistate coherent systems
Authors: El-Neweihi E. , Proschan F. , Sethuraman J.
Published: July 2016

The vast majority of reliability analyses assume that components and system are in either of two states: functioning or failed. The present paper develops basic theory for the study of systems of components in which any of a finite number of states may occur, representing at one extreme perfect functioning and at the other extreme complete failure. We lay down axioms extending the standard notion of a coherent system to the new notion of a multistate coherent system. For such systems we obtain deterministic and probabilistic properties for system performance which are analogous to well-known results for coherent system reliability.
Mixing of cluster point processes
Authors: Laslett G. M.
Published: July 2016

This paper deals with the preservation of strong forms of mixing of point processes under the clustering operation. Strong, ϕ -, and *-mixing possess an increasing degree of uniformity of their asymptotic independence, and this turns out to be a significant factor in their preservation. In particular, it is indicated that ϕ-mixing may be maintained only under very stringent conditions (bounded clusters), whereas strong mixing is maintained under considerably milder conditions.
Short distances, flat triangles and Poisson limits
Authors: Silverman Bernard , Brown Tim
Published: July 2016

Motivated by problems in the analysis of spatial data, we prove some general Poisson limit theorems for the U-statistics of Hoeffding (1948). The theorems are applied to tests of clustering or collinearities in plane data; nearest neighbour distances are also considered.
Approximations in finite-capacity multi-server queues by Poisson arrivals
Authors: Nozaki Shirley A. , Ross Sheldon M.
Published: July 2016

An approximation for the average delay in queue of an entering customer is presented for the M/G/K queuing model with finite capacity. The approximation is obtained by means of an approximation relating a joint distribution of remaining service time to the equilibrium service distribution.
Some results for infinite server poisson queues
Authors: Brown Mark , Ross Sheldon M.
Published: July 2016

We consider a queueing model in which arrivals occur according to a non-homogeneous Poisson process in batches of varying size, and in which a customer is served immediately upon arrival by one of an infinite number of servers.
Random secants of a convex body
Authors: Kingman J. F. C.
Published: July 2016

Let two points be taken at random in an n-dimensional convex body K, and let σ be the line joining them. The distribution of σ is found and compared with other distributions for random secants of K. More generally, if r + 1 ≦ npoints are taken in K, the distribution of the r-dimensional affine subspace containing them is computed. The results find application to the n-dimensional case of a problem of Sylvester.
On Wald's equations in continuous time
Authors: Hall W. J.
Published: July 2016

Various formulas of Wald relating to randomly stopped sums have well known continuous-time analogs, holding in particular for Wiener processes. However, sufficiently general forms of most of these do not appear explicitly in the literature. Recent papers by Robbins and Samuel (1966) and by Brown (1969) provide general results on Wald's equations in discrete time and these are here extended (Theorems 2 and 3) to Wiener processes and other homogeneous additive processes, that is, continuous-time processes with stationary independent increments. We also give an inequality (Theorem 1) related to Wald's identity in continuous time, and we derive, as corollaries of Wald's equations, bounds on the variance of an arbitrary stopping time. The Wiener process versions of these results find application in a variety of stopping problems. Specifically, all are used in Hall ((1968), (1969)); see also Bechhofer, Kiefer, and Sobel (1968), Root (1969), and Shepp (1967).
Limit theorems for the single server queue with traffic intensity one
Authors: Prabhu N. U.
Published: July 2016

We consider a single server queueing system with inter-arrival times {un
, n ≧ 1}, and service times {υn
, n ≧ 1} and the queue-discipline, ‘first-come, first-served’. It is assumed that {un
} and {υn
} are two independent renewal processes, and 0 > E(un
) =a < ∞, 0 < E(υn
) = b < ∞. The traffic intensity is P ρ b/a(0 > ρ > ∞). This paper is concerned with the case ρ = 1, where it is known that the various queueing processes such as the queue-length Q(t) and waiting time W(t) diverge to + ∞ in distribution as t → ∞. Borovkov [1], [2] and Brody [3] have obtained limit distributions for Q(t) and W(t) with appropriate location and scale parameters in the cases P ≧ 1. Here we investigate random variables related to the busy and idle periods in the system. To explain our approach, we consider the random variables Xn
= υn
– un
(n ≧ 1). Let S
0 ≡ 0, Sn
= X
1 + X
2 + ··· + Xn
(n ≧ 1), and define the sequence {Nk
, k ≧ 0} as
The stability of infinite-server networks with random routing
Authors: Berezner S. A. , Malyshev V. A.
Published: July 2016

We consider networks with a very large or infinite number of nodes, linked by cable channels. The request which comes to a node is ordered to occupy a certain route of successive channels. The functioning of the system is regulated by the reserving of channels in order of the arrivals of the requests. Under some general conditions the existence of an ergodic region for such networks is proved. The practical value of the result lies in the fact that these conditions do not depend on the size of the graph.
Optimal pest control through catastrophes
Authors: Kyriakidis E.G. , Abakuks Andris
Published: July 2016

This paper is concerned with the problem of controlling a simple immigration–birth process, which represents a pest population, by the introduction of catastrophes which, when they occur, reduce the population size to zero. The optimality criterion is that of minimising the long-term average cost per unit time of the process. Firstly, an optimal policy is found within a restricted class of stationary policies, which introduce catastrophes if and only if the population size is greater than or equal to some critical value x. The optimality of this policy within the wider class of all stationary policies is then verified by applying the general results of Bather (1976).
A characterization of Gumbel's bivariate exponential and lindley and Singpurwalla's bivariate lomax distributions
Authors: Roy Dilip
Published: July 2016

A bivariate exponential distribution due to Gumbel and a bivariate Lomax distribution due to Lindley and Singpurwalla have been characterized through a bivariate extension of a univariate characterizing property of the exponential and the Lomax distribution; the characterization is based on mean residual lives and hazard rates.
Control of asymptotic variability in non-homogeneous Markov systems
Authors: Vassiliou P.-C. G. , Georgiou A. C. , Tsantas N.
Published: July 2016

In this paper we provide two basic results. First, we find the set of all the limiting vectors of expectations, variances and covariances in an NHMS which are possible provided that we control the limit vector of the sequence of vectors of input probabilities. Secondly, under certain conditions easily met in practice we find the distribution of the limiting vector of expectations, variances and covariances to be multinomial with probabilities the corresponding limiting expected populations in the various states of the NHMS.
On the rate of convergence of some functionals of a stochastic process
Authors: Rachev S. , Todorovic P.
Published: July 2016

This paper is concerned with the rate of convergence of certain functionals associated with a stochastic process arising in the modelling of soil erosion. Some limit theorems are derived for the total crop production Sn
over a number n of years, and the rate of convergence of Sn
to its limit S is discussed. Some stability assumptions are considered, and particular stable geometric infinitely divisible processes analyzed.
Optimal control of a finite dam using P
M
λΤ policies and penalty cost: total discounted and long run average cases
Authors: Abdel-hameed M. , Nakhi Y.
Published: July 2016

Zuckermann [10] considers the problem of optimal control of a finite dam using policies, assuming that the input process is Wiener with drift term μ ≧ 0. Lam Yeh and Lou Jiann Hua [7] treat the case where the input is a Wiener process with a reflecting boundary at zero, with drift term μ ≧ 0, using the long-run average cost and total discounted cost criteria. Attia [1] obtains results similar to those of Lam Yeh and Lou Jiann Hua for the long-run average case and extends them to include μ < 0. In this paper we look further into the results of Zuckerman [10], simplify some of the work of Attia [1], [2], offering corrections to some of his formulae and extend the results of Lam Yeh and Lou Jiann Hua [7].
Derivatives of the expected delay in the GI/G/1 Queue
Authors: Chen Hong , Yao David D.
Published: July 2016

Consider the GI/G/1 queue in which the interarrival/service times are parameterized. The derivatives (with respect to the parameter) of the expected delay of the nth job and its limit are derived and expressed in terms of the expectation of certain stochastic derivatives. Applications of these derivatives in stochastic optimization and random walk are illustrated.
A batch-ballot problem and applications
Authors: Mendelson Haim
Published: July 2016

Let (X1,X2, · ··,Xn) be a vector of discrete splittable and interchangeable random variables, and letSj= Σji = 1Xi.This paper is concerned with probabilities of the formP{Sj< jb – dforj= 1, 2, 3, ···,n|Sn= r}.We focus on applications of these probabilities, study their properties and suggest how they may be computed. The applications are from the areas of computer science, economics, operations management, finance and risk theory.
Weak and strong law results for a function of the spacings
Authors: McCormick William P.
Published: July 2016

Let be i.i.d. uniform on (0,1) random variables and define Si,n
= Ui

,n–1
Ui–

1,n–1, i = 1, · ··, n where the Ui

–n–1 are the order statistics from a sample of size n – 1 and U
0,n–1 =0 and Un,n–

1 = 1. The Si,n
are called the spacings divided by U
1,· ··,Un–

1. For a fixed integer l, set . Exact and weak limit results are obtained for the Ml,n. Further we show that with probability 1

This extends results of Cheng.
Inequalities with applications to percolation and reliability
Authors: Van Den Berg J. , Kesten H.
Published: July 2016

A probability measure μ on ℝn
+ is defined to be strongly new better than used (SNBU) if for all increasing subsets . For n = 1 this is equivalent to being new better than used (NBU distributions play an important role in reliability theory). We derive an inequality concerning products of NBU probability measures, which has as a consequence that if μ
1, μ
2, ···, μn
are NBU probability measures on ℝ+, then the product-measure μ = μ × μ
2 × ··· × μn
on ℝn
+ is SNBU. A discrete analog (i.e., with N instead of ℝ+) also holds.

Applications are given to reliability and percolation. The latter are based on a new inequality for Bernoulli sequences, going in the opposite direction to the FKG–Harris inequality. The main application (3.15) gives a lower bound for the tail of the cluster size distribution for bond-percolation at the critical probability. Further applications are simplified proofs of some known results in percolation. A more general inequality (which contains the above as well as the FKG-Harris inequality) is conjectured, and connections with an inequality of Hammersley [12] and others ([17], [19] and [7]) are indicated.
Compartmental models with transfer delays: a semi-Markov approach
Authors: Leitnaker Mary G. , Purdue Peter
Published: July 2016

Compartmental models for which transfer from one compartment to another takes a non-negligible time have been studied in the deterministic case. These models rely on the use of differential equations with delayed arguments. In this paper we show how the well-known structure of the semi-Markov process can be used to analyse stochastic compartmental models with transfer delays. Evaluation of the limiting behavior is much simpler in the stochastic model than in previous deterministic formulations. In addition, time-dependent behavior can be analysed using numerical quadrature methods.
Uniform conditional variability ordering of probability distributions
Authors: Whitt Ward
Published: July 2016

Variability orderings indicate that one probability distribution is more spread out or dispersed than another. Here variability orderings are considered that are preserved under conditioning on a common subset. One density f on the real line is said to be less than or equal to another, g, in uniform conditional variability order (UCVO) if the ratio f(x)/g(x) is unimodal with the model yielding a supremum, but f and g are not stochastically ordered. Since the unimodality is preserved under scalar multiplication, the associated conditional densities are ordered either by UCVO or by ordinary stochastic order. If f and g have equal means, then UCVO implies the standard variability ordering determined by the expectation of all convex functions. The UCVO property often can be easily checked by seeing if f(x)/g(x) is log-concave. This is illustrated in a comparison of open and closed queueing network models.
Approximations of the optimal stopping problem in partial observation
Authors: Mazziotto G.
Published: July 2016

The resolution of the optimal stopping problem for a partially observed Markov state process reduces to the computation of a function — the Snell envelope — defined on a measure space which is in general infinite-dimensional. To avoid these computational difficulties, we propose in this paper to approximate the optimal stopping time as the limit of times associated to similar problems for a sequence of processes converging towards the true state. We show on two examples that these approximating states can be chosen such that the Snell envelopes can be explicitly computed.
A simulation study of Sylvester's problem in three dimensions
Authors: Do Kim-Anh , Solomon Herbert
Published: July 2016

This note gives estimates of solutions to the Sylvester problem in three dimensions. These results suggest that the solutions in four and higher dimensions are very close to the solution for the n -dimensional sphere which is known.
On stochastic comparison of random vectors
Authors: Shanthikumar J. George
Published: July 2016

We provide sufficient conditions under which two random vectors could be stochastically compared using the standard construction. These conditions are weaker than those discussed by Arjas and Lehtonen (1978) and Veinott (1965). Using these conditions we present extensions of (i) a result of Block et al. (1984) concerning the stochastic monotonicity of independent and identically distributed random variables conditioned on their partial order statistics, and (ii) a theorem of Efron (1965) regarding an increasing property of Pólya frequency functions. Applications of these extensions are also pointed out.
Antistarshapedness, dispersiveness and mixtures
Authors: Lynch James
Published: July 2016

Classes of distributions are defined in terms of antistarshapedness or dispersiveness. Necessary and sufficient conditions are given for these classes to be closed under mixtures. These conditions characterize distributions with log-concave densities.
Limit theorems for general shock models with infinite mean intershock times
Authors: Anderson Kevin K.
Published: July 2016

A general shock model in which the time intervals between shocks have infinite expectation is considered. Limit theorems for the first time the magnitude of a shock exceeds a and the historical maximum magnitude are given.
On the maximum of random fields represented by stochastic integrals over circles
Authors: Orsingher Enzo
Published: July 2016

In this paper we obtain an upper bound for the maximum of random fields of the form , where CP
denotes circles of fixed radius and dW(P′) is a plane white noise field.

The results presented are obtained by means of successive steps involving Slepian's lemma for random fields, inequalities on Brownian fields and planar stochastic integrals.
A necessary condition on minimal cube numberings
Authors: Harper L. H.
Published: July 2016

Let G = (V, E) be a graph with N vertices and A a set of N real numbers. Then a one-to-one mapping ϕ: V → A is called a numbering of G. The elements of A will always be assumed ordered a
1 ≦ a
2 ≦ … ≦ a

N
. In [3] and [7] it was shown how to construct numberings of the n-cube, in fact all numberings, which minimize Σ
e ∈ E
Δ
e
where ∆
e
= |ϕ(v) – ϕ(w)| and e is the edge between vertices v and w. A variant problem of considerable interest is to do the same for Σ
e ∈ E
(Δ
e
)2. It is conjectured that when A = {1,2, …,2n} the natural numbering is the unique minimizer of Σ
e ∈ E
(Δ
e
)2 for every n, but this has so far resisted all efforts. Theorem 1 in the following is the result of attempts to get weaker results, namely to thin out the ranks of those numberings which could possibly minimize Σ
e ∈ E
(Δ
e
)2. The second problem posed and solved in this paper is a generalization of the results in [3], where the n-cube becomes the n-torus.
Asymptotic properties and equilibrium conditions for branching Poisson processes
Authors: Lewis P.A.W.
Published: July 2016

Some previously obtained asymptotic results for branching Poisson processes are extended and sharpened. It is shown that under rather general conditions the number of events in both the transient and the equilibrium processes, suitably normalized, have a unit normal distribution. Finally, unique initial conditions are derived for the equilibrium process.
Shock models with underlying birth process
Authors: A-Hameed M. S. , Proschan F.
Published: July 2016

This paper extends results of Esary, Marshall and Proschan (1973) and A-Hameed and Proschan (1973). We consider the life distribution of a device subject to a sequence of shocks occurring randomly in time according to a nonstationary pure birth process: given k shocks have occurred in [0, t], the probability of a shock occurring in (t, t + Δ] is λ kλ (t)Δ + o (Δ). We show that various fundamental classes of life distributions (such as those with increasing failure rate, or those with the ‘new better than used' property, etc.) are obtained under appropriate assumptions on λ k, λ (t), and on the probability of surviving a given number of shocks.
On quasi-stationary distributions for multi-type Galton-Watson processes
Authors: Buiculescu Mioara
Published: July 2016

In this paper we consider the long term behaviour till absorption of multitype Galton-Watson processes using the theory of quasi-stationary distributions.
Multistate reliability models
Authors: Griffith William S.
Published: July 2016

In this paper an axiomatic development of multistate systems is presented. Three types of coherence based on the strength of the relevancy axiom are studied. The strongest of these has been investigated previously by El-Neweihi, Proschan, and Sethuraman [3]. One of the weaker types of coherence permits wider applicability to real life situations without sacrificing any of the mathematical results obtained by El-Neweihi, Proschan, and Sethuraman. The concept of system performance is formalized through expected utility and the effect of component improvement on system performance is studied using a generalization of Birnbaum's reliability importance.
On the variance of the sample correlation between two independent lattice processes
Authors: Richardson Sylvia , Hemon Denis
Published: July 2016

Consider two stochastically independent, stationary Gaussian lattice processes with zero means, {X(u), u (Z
2} and {Y(u), u (Z
2}. An asymptotic expression for the variance of the sample correlation between {X(u)} and {Y(u)} over a finite square is derived. This expression also holds for a wide class of domains in Z
2. As an illustration, the asymptotic variance of the correlation between two first-order autonormal schemes is evaluated.
Entropy and martingales in Markov chain models
Authors: Seneta E.
Published: July 2016

The concept of entropy in models is discussed with particular reference to the work of P.A.P. Moran. For a vector-valued Markov chain {Xk
} whose states are relative-frequency (proportion) tables corresponding to a physical mixing model of a number N of particles over n urns, the definition of entropy may be based on the usual information-theoretic concept applied to the probability distribution given by the expectation . The model is used for a brief probabilistic assessment of the relationship between Boltzmann's Η-Theorem, the Ehrenfest urn model, and Poincaré's considerations on the mixing of liquids and card shuffling, centred on the property of an ultimately uniform distribution of a single particle. It is then generalized to the situation where the total number of particles fluctuates over time, and martingale results are used to establish convergence for .
Traffic light queues with dependent arrivals as a generalization to queueing theory
Authors: Ohno Katsuhisa , Mine Hisashi
Published: July 2016

This paper considers a fixed-cycle and a semi vehicular-actuated traffic light queue with strictly stationary arrivals and independent and identically distributed departure headways and lost times. These queues are reduced to a generalized model of Loynes and sufficient conditions are derived under which these queues have stationary distributions. Two typical examples of semi vehicular-actuated traffic light queues are discussed.
A characterization of the Poisson process
Authors: Samuels S. M.
Published: July 2016

Theorem: A necessary and sufficient condition for the superposition of two ordinary renewal processes to again be a renewal process is that they be Poisson processes.

A complete proof of this theorem is given; also it is shown how the theorem follows from the corresponding one for the superposition of two stationary renewal processes.
Some limit theorems for a class of network problems as related to finite Markov chains
Authors: Nakamura Masao
Published: July 2016

This paper is concerned with a class of dynamic network flow problems in which the amount of flow leaving node i in one time period for node j is the fraction pij
of the total amount of flow which arrived at node i during the previous time period. The fraction pij
whose sum over j equals unity may be interpreted as the transition probability of a finite Markov chain in that the unit flow in state i will move to state j with probability pij
during the next period of time. The conservation equations for this class of flows are derived, and the limiting behavior of the flows in the network as related to the properties of the fractions Pij
are discussed.
Optimum replacement policies with delay
Authors: Nakagawa Toshio , Osaki Shunji
Published: July 2016

Replacement theory for equipment has been investigated by several authors. This paper introduces the ‘delay’ time preparing for replacement, derives the expected cost per unit time, and discusses the optimum replacement policies under several conditions. Three special cases are discussed and numerical examples are presented.
Queues in transportation systems, II: An independently-dispatched system
Authors: Crane Michael A.
Published: July 2016

We consider a transportation system consisting of a linear network of N + 1 terminals served by S vehicles of fixed capacity. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to some terminal j, 0 ≦ j ≦ i − 1, and are served as empty units of vehicle capacity become available at i. The vehicle fleet is partitioned into N service groups, with vehicles in the ith group stopping at terminals i, i − 1,···,0. Travel times between terminals and idle times at terminals are stochastic and are independent of the customer arrival processes. Functional central limit theorems are proved for random functions induced by processes of interest, including customer queue size processes. The results are of most interest in cases where the system is unstable. This occurs whenever, at some terminal, the rate of customer arrivals is at least as great as the rate at which vehicle capacity is made available.
A note on the recurrence of yellow fever epidemics in urban populations
Authors: Radcliffe J.
Published: July 2016

An approximate expression is found for the probability density of the recurrence time between major outbreaks of a host-vector disease such as yellow fever, in which the infectious period of a host is short and the lifetime of a vector is short compared with that of a host. The typical interval between epidemics based on the modal value of this distribution is obtained. These results are the analogues of those obtained by Bartlett (1956), (1966) for the periodicity of measles epidemics in small communities.
Multidimensional age-dependent branching processes allowing immigration: The limiting distribution
Authors: Kaplan Norman
Published: July 2016

This paper continues the author's study of age-dependent branching processes allowing immigration. In this paper the multidimensional case is considered. A sufficient condition is obtained for the existence of a legitimate limiting distribution. Several corollaries are obtained, which generalize many of the results of the discrete theory and those of the one-dimensional continuous time model.
Bounds for coverage probabilities with applications to sequential coverage problems
Authors: Cooke Peter J.
Published: July 2016

This paper discusses general bounds for coverage probabilities and moments of stopping rules for sequential coverage problems in geometrical probability. An approach to the study of the asymptotic behaviour of these moments is also presented.
The effect of intermittent exposure to risk on failure time distributions
Authors: Isham Valerie
Published: July 2016

The relation is investigated between the distributions of the total time until failure and the time of exposure to risk until failure, for individuals who are at risk only intermittently during their lifetimes. A specific example is considered in the case of computer failures.
A birth and death model of neuron firing
Authors: Logan B. F. , Shepp L. A.
Published: July 2016

A simple birth-death model of particle fluctuations is studied where at each discrete time a birth and/or death may occur. We show that if the probability of a birth does not depend on the number of particles present and if births and deaths are independent, then the times between successive deaths are independent geometrically distributed random variables, which is false in the general case. Since the above properties of the times between successive neuron firings have been observed in nerve cells, the model proposed in [2] obtains added credence.
The optimal linear solution of a symmetric team control problem
Authors: Whittle P. , Rudge J.
Published: July 2016

An optimal linear control (4) is deduced for the symmetric team control problem specified by relations (1)–(3). The generating function (6) for the optimal operator is in general irrational if there is more than one agent (controller), indicating that the recursions associated with effective state estimation are no longer of finite order, as in the single-agent case.
A stochastic model for time lag in reporting of claims
Authors: Karlsson Jan-Erik
Published: July 2016

We assume that the number of claims occur according to a renewal process and treat the number of claims that occur and are reported in a certain time interval as a renewal process with random displacements. We obtain a renewal equation for the mean value function and an integral equation for the Laplace transform of the distribution of the claims that are reported. We also give asymptotic expressions for the mean value function and calculate the generating function in the case where the renewal process is a Poisson process. This matter is a part of the IBNR-problem in insurance mathematics.