bluuebunny/crossref_metadata_2025 huggingface dataset:
Foster-type criteria for Markov chains on general spaces
Authors: Fralix Brian H.
Published: July 2016

This paper establishes new Foster-type criteria for a Markov chain on a general state space to be Harris recurrent, positive Harris recurrent, or geometrically ergodic. The criteria are based on drift conditions involving stopping times rather than deterministic steps. Meyn and Tweedie (1994) developed similar criteria involving random-sized steps, independent of the Markov chain under study. They also posed an open problem of finding criteria involving stopping times. Our results essentially solve that problem. We also show that the assumption of ψ-irreducibility is not needed when stating our drift conditions for positive Harris recurrence or geometric ergodicity.
Distribution-Invariant Risk Measures, Entropy, and Large Deviations
Authors: Weber Stefan
Published: July 2016

The simulation of distributions of financial assets is an important issue for financial institutions. If risk measures are evaluated for a simulated distribution instead of the model-implied distribution, the errors in the risk measurements need to be analyzed. For distribution-invariant risk measures which are continuous on compacts, we employ the theory of large deviations to study the probability of large errors. If the approximate risk measurements are based on the empirical distribution of independent samples, then the rate function equals the minimal relative entropy under a risk measure constraint. We solve this minimization problem explicitly for shortfall risk and average value at risk.
Large Deviations Principle for Occupancy Problems with Colored Balls
Authors: Dupuis Paul , Nuzman Carl , Whiting Phil
Published: July 2016

A large deviations principle (LDP), demonstrated for occupancy problems with indistinguishable balls, is generalized to the case in which balls are distinguished by a finite number of colors. The colors of the balls are chosen independently from the occupancy process itself. There are r balls thrown into n urns with the probability of a ball entering a given urn being 1/n (i.e. Maxwell-Boltzmann statistics). The LDP applies with the scale parameter, n, tending to infinity and r increasing proportionally. The LDP holds under mild restrictions, the key one being that the coloring process by itself satisfies an LDP. This includes the important special cases of deterministic coloring patterns and colors chosen with fixed probabilities independently for each ball.
Convolution Equivalence and Infinite Divisibility: Corrections and Corollaries
Authors: Pakes Anthony G.
Published: July 2016

Corrections are made to formulations and proofs of some theorems about convolution equivalence closure for random sum distributions. These arise because of the falsity of a much used asymptotic equivalence lemma, and they impinge on the convolution equivalence closure theorem for general infinitely divisible laws.
Diffusion Approximation of State-Dependent G-Networks Under Heavy Traffic
Authors: Leite Saul C. , Fragoso Marcelo D.
Published: July 2016

This paper is concerned with the characterization of weak-sense limits of state-dependent G-networks under heavy traffic. It is shown that, for a certain class of networks (which includes a two-layer feedforward network and two queues in tandem), it is possible to approximate the number of customers in the queue by a reflected stochastic differential equation. The benefits of such an approach are that it describes the transient evolution of these queues and allows the introduction of controls, inter alia. We illustrate the application of the results with numerical experiments.
Stability of a Processor-Sharing Queue with Varying Throughput
Authors: Moyal Pascal
Published: July 2016

In this paper we present a stability criterion for processor-sharing queues, in which the throughput may depend on the number of customers in the system (such as in the case of interferences between users). Such a system is represented by a point measure-valued stochastic recursion keeping track of the remaining processing times of the customers.
Tails of Stopped Random Products: The Factoid and Some Relatives
Authors: Pakes Anthony G.
Published: July 2016

The upper tail behaviour is explored for a stopped random product ∏j=1NXj, where the factors are positive and independent and identically distributed, andNis the first time one of the factors occupies a subset of the positive reals. This structure is motivated by a heavy-tailed analogue of the factorialn!, called the factoid ofn. Properties of the factoid suggested by computer explorations are shown to be valid. Two topics about the determination of the Zipf exponent in the rank-size law for city sizes are discussed.
Transient Asymptotics of Lévy-Driven Queues
Authors: Dębicki Krzysztof , Es-Saghouani Abdelghafour , Mandjes Michel
Published: July 2016

With (Q

t
)
t
denoting the stationary workload process in a queue fed by a Lévy input process (X

t
)
t
, this paper focuses on the asymptotics of rare event probabilities of the type P(Q
0 > pB, Q

T

B

> qB) for given positive numbers p and q, and a positive deterministic function T

B
. We first identify conditions under which the probability of interest is dominated by the ‘most demanding event’, in the sense that it is asymptotically equivalent to P(Q > max{p, q}B) for large B, where Q denotes the steady-state workload. These conditions essentially reduce to T

B
being sublinear (i.e.
T

B
/B → 0 as B → ∞). A second condition is derived under which the probability of interest essentially ‘decouples’, in that it is asymptotically equivalent to P(Q > pB)P(Q > qB) for large B. For various models considered in the literature, this ‘decoupling condition’ reduces to requiring that T

B
is superlinear (i.e. T

B
/ B → ∞ as B → ∞). This is not true for certain ‘heavy-tailed’ cases, for instance, the situations in which the Lévy input process corresponds to an α-stable process, or to a compound Poisson process with regularly varying job sizes, in which the ‘decoupling condition’ reduces to T

B
/ B
2 → ∞. For these input processes, we also establish the asymptotics of the probability under consideration for T

B
increasing superlinearly but subquadratically. We pay special attention to the case T

B
= RB for some R > 0; for light-tailed input, we derive intuitively appealing asymptotics, intensively relying on sample path large deviations results. The regimes obtained can be interpreted in terms of the most likely paths to overflow.
A Characterization Related to the Equilibrium Distribution Associated with a Polynomial Structure
Authors: Bar-Lev Shaul K. , Boxma Onno , Letac Gérard
Published: July 2016

Let f be a probability density function on (a, b) ⊂ (0, ∞), and consider the class C

f
of all probability density functions of the form Pf, where P is a polynomial. Assume that if X has its density in C

f
then the equilibrium probability density x ↦ P(X > x) / E(X) also belongs to C

f
: this happens, for instance, when f(x) = Ce−λx
or f(x) = C(b − x)λ−1. We show in the present paper that these two cases are the only possibilities. This surprising result is achieved with an unusual tool in renewal theory, by using ideals of polynomials.
The probabilities of absolute ruin in the renewal risk model with constant force of interest
Authors: Konstantinides Dimitrios G. , Ng Kai W. , Tang Qihe
Published: July 2016

In this paper we consider the probabilities of finite- and infinite-time absolute ruins in the renewal risk model with constant premium rate and constant force of interest. In the particular case of the compound Poisson model, explicit asymptotic expressions for the finite- and infinite-time absolute ruin probabilities are given. For the general renewal risk model, we present an asymptotic expression for the infinite-time absolute ruin probability. Conditional distributions of Poisson processes and probabilistic techniques regarding randomly weighted sums are employed in the course of this study.
On the Convolution of Heterogeneous Bernoulli Random Variables
Authors: Xu Maochao , Balakrishnan N.
Published: July 2016

In this paper, some ordering properties of convolutions of heterogeneous Bernoulli random variables are discussed. It is shown that, under some suitable conditions, the likelihood ratio order and the reversed hazard rate order hold between convolutions of two heterogeneous Bernoulli sequences. The results established here extend and strengthen the previous results of Pledger and Proschan (1971) and Boland, Singh and Cukic (2002).
Additive Functionals for Discrete-Time Markov Chains with Applications to Birth-Death Processes
Authors: Liu Yuanyuan
Published: July 2016

In this paper we are interested in bounding or calculating the additive functionals of the first return time on a set for discrete-time Markov chains on a countable state space, which is motivated by investigating ergodic theory and central limit theorems. To do so, we introduce the theory of the minimal nonnegative solution. This theory combined with some other techniques is proved useful for investigating the additive functionals. This method is used to study the functionals for discrete-time birth-death processes, and the polynomial convergence and a central limit theorem are derived.
Multimodality of the Markov Binomial Distribution
Authors: Dekking Michel , Kong Derong
Published: July 2016

We study the shape of the probability mass function of the Markov binomial distribution, and give necessary and sufficient conditions for the probability mass function to be unimodal, bimodal, or trimodal. These are useful to analyze the double-peaking results of a reactive transport model from the engineering literature. Moreover, we give a closed-form expression for the variance of the Markov binomial distribution, and expressions for the mean and the variance conditioned on the state at time n.
Online Selection of Alternating Subsequences from a Random Sample
Authors: Arlotto Alessandro , Chen Robert W. , Shepp Lawrence A. , Steele J. Michael
Published: July 2016

We consider sequential selection of an alternating subsequence from a sequence of independent, identically distributed, continuous random variables, and we determine the exact asymptotic behavior of an optimal sequentially selected subsequence. Moreover, we find (in a sense we make precise) that a person who is constrained to make sequential selections does only about 12 percent worse than a person who can make selections with full knowledge of the random sequence.
A new characterisation property of mixed Poisson processes via Berman's theorem
Authors: Hayakawa Yu
Published: July 2016

In the literature on mixed Poisson processes, a number of characterisation properties have been studied. As a new characterisation property for mixed Poisson processes, we show that normalised event occurrence times are the order statistics of independent uniform random variables on (0,1). Berman's theorem on l

p
-isotropic sequences is applied to prove the results.
NWU property of a class of random sums
Authors: Cai Jun , Kalashnikov Vladimir
Published: July 2016

In this note, we derive an inequality for the renewal process. Then, using this inequality, together with an identity in terms of the renewal process for the tails of random sums, we prove that a class of random sums is always new worse than used (NWU). Thus, the well-known NWU property of geometric sums is extended to the class of random sums. This class is illustrated by some examples, including geometric sums, mixed geometric sums, certain mixed Poisson distributions and certain negative binomial sums.
On records and related processes for sequences with trends
Authors: Borovkov K.
Published: July 2016

We study the records and related variables for sequences with linear trends. We discuss the properties of the asymptotic rate function and relationships between the distribution of the long-term maxima in the sequence and that of a particular observation, including two characterization type results. We also consider certain Markov chains related to the process of records and prove limit theorems for them, including the ergodicity theorem in the regular case (convergence rates are given under additional assumptions), and derive the limiting distributions for the inter-record times and increments of records.
Level phase independence for GI/M/1-type Markov chains
Authors: Latouche Guy , Taylor P. G.
Published: July 2016

GI/M/1-type Markov chains make up a class of two-dimensional Markov chains. One dimension is usually called the level, and the other is often called the phase. Transitions from states in level k are restricted to states in levels less than or equal to k+1. For given transition probabilities in the interior of the state space, we show that it is always possible to define the boundary transition probabilities in such a way that the level and phase are independent under the stationary distribution. We motivate our analysis by first considering the quasi-birth-and-death process special case in which transitions from any state are restricted to states in the same, or adjacent, levels.
Cell loss probability for M/G/1 and time-slotted queues
Authors: McDonald David , Théberge François
Published: July 2016

It is common practice to approximate the cell loss probability (CLP) of cells entering a finite buffer by the overflow probability (OVFL) of a corresponding infinite buffer queue, since the CLP is typically harder to estimate. We obtain exact asymptotic results for CLP and OVFL for time-slotted queues where block arrivals in different time slots are i.i.d. and one cell is served per time slot. In this case the ratio of CLP to OVFL is asymptotically (1-ρ)/ρ, where ρ is the use or, equivalently, the mean arrival rate per time slot. Analogous asymptotic results are obtained for continuous time M/G/1 queues. In this case the ratio of CLP to OVFL is asymptotically 1-ρ.
Solution of jump parameter systems of differential and difference equations with semi-Markov coefficients
Authors: Shmerling Efraim , Hochberg Kenneth J.
Published: July 2016

We study linear jump parameter systems of differential and difference equations whose coefficients depend on the state of a semi-Markov process. We derive systems of equations for the first two moments of the random solutions of these jump parameter systems, and illustrate how moment equations can be used in examining their asymptotic stability.
A method for computing total downtime distributions in repairable systems
Authors: van der Weide J. A. M.
Published: July 2016

In this paper we derive the distribution of the total downtime of a repairable system during a given time interval. We allow dependence of the failure time and the repair time. The results are presented in the form of Laplace transforms which can be inverted numerically. We also discuss asymptotic properties of the total downtime.
Initial and final behaviour of failure rate functions for mixtures and systems
Authors: Block Henry W. , Li Yulin , Savits Thomas H.
Published: July 2016

In this paper we consider the initial and asymptotic behaviour of the failure rate function resulting from mixtures of subpopulations and formation of coherent systems. In particular, it is shown that the failure rate of a mixture has the same limiting behaviour as the failure rate of the strongest subpopulation. A similar result holds for systems except the role of strongest subpopulation is replaced by strongest min path set.
From the central limit theorem to heavy-tailed distributions
Authors: Chen Jinwen
Published: July 2016

It has been observed that in many practical situations randomly stopped products of random variables have power law distributions. In this note we show that, in order for such a product to have a power law distribution, the only random indices are the exponentially distributed ones. We also consider a more general problem, which is closely related to problems concerning transformation from the central limit theorem to heavy-tailed distributions.
Local matching of random restriction maps
Authors: Tang Mengxiang , Waterman Michael S.
Published: July 2016

Optical mapping is a new technique to generate restriction maps of DNA easily and quickly. DNA restriction maps can be aligned by comparing corresponding restriction fragment lengths. To relate, organize, and analyse these maps it is necessary to rapidly compare maps. The issue of the statistical significance of approximately matching maps then becomes central, as in BLAST with sequence scoring. In this paper, we study the approximation to the distribution of counts of matched regions of specified length when comparing two DNA restriction maps. Distributional results are given to enable us to computep-values and hence to determine whether or not the two restriction maps are related. The key tool used is the Chen-Stein method of Poisson approximation. Certain open problems are described.
Waiting times for patterns in a sequence of multistate trials
Authors: Antzoulakos Demetrios L.
Published: July 2016

Let X

n
, n ≥ 1 be a sequence of trials taking values in a given set A, let ∊ be a pattern (simple or compound), and let X

r,∊ be a random variable denoting the waiting time for the rth occurrence of ∊. In the present article a finite Markov chain imbedding method is developed for the study of X

r,∊ in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.
Optimal reward on a sparse tree with random edge weights
Authors: Khoshnevisan Davar , Lewis Thomas M.
Published: July 2016

It is well known that the maximal displacement of a random walk indexed by an m-ary tree with bounded independent and identically distributed edge weights can reliably yield much larger asymptotics than a classical random walk whose summands are drawn from the same distribution. We show that, if the edge weights are mean-zero, then nonclassical asymptotics arise even when the tree grows much more slowly than exponentially. Our conditions are stated in terms of a Minkowski-type logarithmic dimension of the boundary of the tree.
A note on the full-information Poisson arrival selection problem
Authors: Kurushima Aiko , Ano Katsunori
Published: July 2016

This note studies a Poisson arrival selection problem for the full-information case with an unknown intensity λ which has a Gamma prior density G(r, 1/a), where a>0 and r is a natural number. For the no-information case with the same setting, the problem is monotone and the one-step look-ahead rule is an optimal stopping rule; in contrast, this note proves that the full-information case is not a monotone stopping problem.
Cyclic queueing networks with subexponential service times
Authors: Ayhan H. , Palmowski Z. , Schlegel S.
Published: July 2016

For a K-stage cyclic queueing network with N customers and general service times, we provide bounds on the nth departure time from each stage. Furthermore, we analyze the asymptotic tail behavior of cycle times and waiting times given that at least one service-time distribution is subexponential.
On applications of residual lifetimes of compound geometric convolutions
Authors: Willmot Gordon E. , Cai Jun
Published: July 2016

We demonstrate that the residual lifetime distribution of a compound geometric distribution convoluted with another distribution, termed a compound geometric convolution, is again a compound geometric convolution. Conditions under which the compound geometric convolution is new worse than used (NWU) or new better than used (NBU) are then derived. The results are applied to ruin probabilities in the stationary renewal risk model where the convolution components are of particular interest, as well as to the equilibrium virtual waiting time distribution in the G/G/1 queue, an approximation to the equilibrium M/G/c waiting time distribution, ruin in the classical risk model perturbed by diffusion, and second-order reliability classifications.
Adaptive simulation using perfect control variates
Authors: Henderson Shane G. , Simon Burt
Published: July 2016

We introduce adaptive-simulation schemes for estimating performance measures for stochastic systems based on the method of control variates. We consider several possible methods for adaptively tuning the control-variate estimators, and describe their asymptotic properties. Under certain assumptions, including the existence of a ‘perfect control variate’, all of the estimators considered converge faster than the canonical rate ofn−1/2, wherenis the simulation run length. Perfect control variates for a variety of stochastic processes can be constructed from ‘approximating martingales’. We prove a central limit theorem for an adaptive estimator that converges at rateA similar estimator converges at raten−1. An exponential rate of convergence is also possible under suitable conditions.
Limiting properties of Poisson shot noise processes
Authors: Lund Robert , McCormick William P. , Xiao Yuanhui
Published: July 2016

This paper studies limiting properties of discretely sampled Poisson shot noise processes. Versions of the law of large numbers and central limit theorem are derived under very general conditions. Examples demonstrating the utility of the results are included.
Some properties of ageing notions based on the moment-generating-function order
Authors: Li Xiaohu
Published: July 2016

Classes of life distributions based on the moment-generating-function order are investigated in this paper. It is shown firstly that the class ℳ is closed under both convex linear combination and geometric compounding. Secondly, the class NBUmg (new better than used in the moment-generating-function order) is proved to be closed under increasing star-shaped transformations. Finally, the interplay between the stochastic comparison of the excess lifetime of a renewal process and the NBUmg interarrivals is studied.
The mean comparison theorem cannot be extended to the Poisson case
Authors: Večeř Jan , Xu Mingxin
Published: July 2016

In this paper, we show that the mean comparison theorem, which is valid for Brownian motion, cannot be extended to Poisson processes. A counterexample in the Poisson case for which the mean comparison theorem does not hold is provided.
A note on the ℒ-class of life distributions
Authors: Klar B.
Published: July 2016

This paper first recalls some stochastic orderings useful for studying the ℒ-class and the Laplace order in general. We use these orders to show that the higher moments of an ℒ-class distribution need not exist. Using simple sufficient conditions for the Laplace ordering, we give examples of distributions in the ℒ- and ℒα-classes. Moreover, we present explicit sharp bounds on the survival function of a distribution belonging to the ℒ-class of life distributions. The results reveal that the ℒ-class should not be seen as a more comprehensive class of ageing distributions but rather as a large class of life distributions on its own.
The delay distribution of a type k customer in a first-come-first-served MMAP[K]/PH[K]/1 queue
Authors: Van Houdt B. , Blondia C.
Published: July 2016

This paper presents an algorithmic procedure to calculate the delay distribution of a type k customer in a first-come-first-served (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements (the MMAP[K]/PH[K]/1 queue). First, we develop a procedure, using matrix analytical methods, to handle arrival processes that do not allow batch arrivals to occur. Next, we show that this technique can be generalized to arrival processes that do allow batch arrivals to occur. We end the paper by presenting some numerical examples.
Multivariate dispersive ordering of epoch times of nonhomogeneous Poisson processes
Authors: Belzunce Felíx , Ruiz José-María
Published: July 2016

In this paper we find conditions under which the epoch times of two nonhomogeneous Poisson processes are ordered in the multivariate dispersive order. Some consequences and examples of this result are given. These results extend a recent result of Brown and Shanthikumar (1998).
Losses per cycle in a single-server queue
Authors: Wolff Ronald W.
Published: July 2016

Several recent papers have shown that for the M/G/1/n queue with equal arrival and service rates, the expected number of lost customers per busy cycle is equal to 1 for every n ≥ 0. We present an elementary proof based on Wald's equation and, for GI/G/1/n, obtain conditions for this quantity to be either less than or greater than 1 for every n ≥ 0. In addition, we extend this result to batch arrivals, where, for average batch size β, the same quantity is either less than or greater than β. We then extend these results to general ways that customers may be lost, to an arbitrary order of service that allows service interruption, and finally to reneging.
The coalescent process in a population with stochastically varying size
Authors: Kaj Ingemar , Krone Stephen M.
Published: July 2016

We study the genealogical structure of a population with stochastically fluctuating size. If such fluctuations, after suitable rescaling, can be approximated by a nice continuous-time process, we prove weak convergence in the Skorokhod topology of the scaled ancestral process to a stochastic time change of Kingman's coalescent, the time change being given by an additive functional of the limiting backward size process.
Perturbed Markov chains
Authors: Solan Eilon , Vieille Nicolas
Published: July 2016

We study irreducible time-homogenous Markov chains with finite state space in discrete time. We obtain results on the sensitivity of the stationary distribution and other statistical quantities with respect to perturbations of the transition matrix. We define a new closeness relation between transition matrices, and use graph-theoretic techniques, in contrast with the matrix analysis techniques previously used.
Birth-death processes with an instantaneous reflection barrier
Authors: Chen Anyue , Liu Kai
Published: July 2016

A new structure with the special property that an instantaneous reflection barrier is imposed on the ordinary birth—death processes is considered. An easy-checking criterion for the existence of such Markov processes is first obtained. The uniqueness criterion is then established. In the nonunique case, all the honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. It is proved that honest processes are always ergodic without necessarily imposing any extra conditions. Equilibrium distributions for all these ergodic processes are established. Several examples are provided to illustrate our results.
Age-replacement policy and optimal work size
Authors: Mi Jie
Published: July 2016

Suppose that there is a sequence of programs or jobs that are scheduled to be executed one after another on a computer. A program may terminate its execution because of the failure of the computer, which will obliterate all work the computer has accomplished, and the program has to be run all over again. Hence, it is common to save the work just completed after the computer has been working for a certain amount of time, say y units. It is assumed that it takes a certain time to perform a save. During the saving process, the computer is still subject to random failure. No matter when the computer failure occurs, it is assumed that the computer will be repaired completely and the repair time will be negligible. If saving is successful, then the computer will continue working from the end of the last saved work; if the computer fails during the saving process, then only unsaved work needs to be repeated. This paper discusses the optimal work size y under which the long-run average amount of work saved is maximized. In particular, the case of an exponential failure time distribution is studied in detail. The properties of the optimal age-replacement policy are also derived when the work size y is fixed.
A necessary condition for extinction in those bisexual Galton-Watson branching processes governed by superadditive mating functions
Authors: Hull David M.
Published: July 2016

Mating functions considered by Asmussen (1980) and Daley (1968) for the class of bisexual Galton-Watson branching processes (GWBP) are shown to be superadditive. Consideration of a process that allows only sibling mating leads to a necessary condition for almost sure extinction in bisexual GWBP governed by superadditive mating functions. A simple example shows that the condition is not sufficient.
New bounds for the thermal energy storage process with stationary input
Authors: Haslett J.
Published: July 2016

The process {Xn
}, defined by Xn

+ 1 = max{Yn

+ 1 + αßX
n, ßX
n}, with αand ß in [0, 1) and {Yn
} stationary, arises in studies of solar thermal energy systems. Bounds for the stationary mean EX are given, which are more general and in some cases tighter, than those previously available.
Limit theorems for the shifting level process
Authors: Cline Daren B. H.
Published: July 2016

This paper studies the asymptotic properties of moment estimators for the general shifting level process (SLP). A law of large numbers and a weak convergence theorem are obtained under conditions involving the unobservable processes which make up SLP. Specific conditions about those underlying processes are added to give explicit results, applicable to a large class of moment estimators. Actual formulae for asymptotic variances, etc. are obtained for a simple example, the GNN model.
Quasi-stationary distribution of a two-unit warm-standby redundant system
Authors: Kalpakam S. , Hameed M. A. Shahul
Published: July 2016

This paper establishes the existence of the limit as t → ∞ of the distribution of the residual lifetime at time t of a two-unit warm-standby reliability system supported by a single repair facility, conditioned on the event that the system has not been down at any time in (0, t). The conditioned limit distribution is proved to be exponential irrespective of the distributions of the lifetimes and repair times of the individual units, provided that their Laplace transforms are rational functions of their arguments.
Coupling methods and the storage equation
Authors: Kendall Wilfrid S.
Published: July 2016

Coupling methods are applied to the study of the storage equation with stationary input. Results on the existence of statistical equilibria are obtained. Means of obtaining rate-of-convergence estimates are indicated.
Minimal distributions in a stochastic partial ordering and bounds for Gaussian processes
Authors: Eplett W. J. R.
Published: July 2016

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.
The proportional bettor's return on investment
Authors: Ethier S. N. , Tavaré S.
Published: July 2016

Suppose you repeatedly play a game of chance in which you have the advantage. Your return on investment is your net gain divided by the total amount that you have bet. It is shown that the ratio of your return on investment under optimal proportional betting to your return on investment under constant betting converges to an exponential distribution with mean as your advantage tends to 0. The case of non-optimal proportional betting is also treated.
The Palm-type grain size distribution in stationary grain models
Authors: König Dieter , Schmidt Volker
Published: July 2016

In this paper a point-process approach is given for determining the Palm-type (number-weighted) distribution of the size factor of the grains of a stationary grain model in the plane with non-overlapping, identically shaped and identically orientated convex grains starting from a suitably chosen characteristic of the grain model observed in fixed points of the plane.
The outcome of a general spatial epidemic on the line
Authors: Faddy M. J.
Published: July 2016

The general (non-spatial) stochastic epidemic is extended to allow infective individuals to move forward through a system of spatially connected locations · ··, L
1, L
2, · ·· (on the line) each containing susceptible individuals and the outcome of the epidemic in each of these locations is then considered. In the deterministic case, a (spatial) equilibrium solution and threshold behaviour are discussed. In the stochastic case, a (spatial) quasi-equilibrium behaviour (conditional on sufficient numbers of infectives present) is discussed; numerical results suggest some correspondence between this stochastic quasi-equilibrium and the deterministic equilibrium.
Time-series analysis of tides and similar motions of the sea surface
Authors: Cartwright D. E.
Published: July 2016

This survey article, most of whose results are described in greater detail in Munk and Cartwright (1966), which will hereafter be abbreviated to MC, describes methods which aim to separate the response of the sea level at a given place due to various exciting forces such as gravity, solar radiation, non-linear effects, and weather. In so doing, it provides predictors for sea level which are formally simpler and somewhat more accurate than those given by the classical methods.
The distribution of the maximum number of customers present simultaneously during a busy period for the queueing systems M/G/1 and G/M/1
Authors: Cohen J. W.
Published: July 2016

The distribution of the maximum number of customers simultaneously present during a busy period is studied for the queueing systems M/G/1 and G/M/1. These distributions are obtained by using taboo probabilities. Some new relations for transition probabilities and entrance time distributions are derived.
The spectrum of intervals of a geometric branching poisson process
Authors: Gilles D. C. , Lewis P. A. W.
Published: July 2016

A summation formula is given for computing the spectrum of intervals of a geometric branching Poisson process from the cumulant generating function of the associated counting process.
Hierarchical population models with poisson recruitment
Authors: Pollard J. H.
Published: July 2016

The analysis of an organization having various grades of employee has been undertaken in many different contexts. In 1961, for example, Young and Almond considered an institution (of undisclosed type) having six grades of staff. Gani, in 1963, considered ‘Australian Universities’ as an organization and students were graded according to their stage of study. The author used techniques similar to those of this paper in 1964 to analyze deterministically the age structure of the Australian Academy of Science. Indeed, the analyses have usually been deterministic.
Genetic correlation in the stepping stone model with non-symmetrical migration rates
Authors: Maruyama Takeo
Published: July 2016

Kimura (1953) proposed a “stepping stone” model for the study of the genetic structure of a subdivided population. In this model, it was assumed that a population consists of infinitely many colonies located at grid points of ann-dimensional lattice and that each colony exchanges individuals with neighboring colonies in each generation, and also receives immigrants as random samples from the whole population. The former type of migration has been called “short range migration” and the latter type “long range migration”. Later, Kimura and Weiss (1964), and Weiss and Kimura (1965) have obtained formulas for the genetic correlation and variance between colonies for general cases of the model, assuming that the short range migration is symmetrical in each fixed direction. The purpose of the present report is to extend the results of Kimura and Weiss to cover situations where the restriction of symmetry of short range migration is removed, so that migration rates in opposite directions need not be equal. I believe that in nature there are cases which require the model presented in this report. For example, consider a plant population distributed along a river, or on a plain where the wind at the time when the seeds are scattered is stronger in one direction than in others. For animals and plants it is often true that the centre of habitat is more densely populated than the marginal regions where the environment is less suitable for the species. In such a case the migration rate toward the outside from the centre is larger than that in the opposite direction. As in other theories of population genetics, we will assume that the size of each colony is determined by the carrying capacity of the environment and is not affected by the migration rate. Thus we assume that in each generation each colony produces many more gametes than those which contribute to the next generation, and, among those many gametes, a certain number, say 2N,are chosen from various colonies to form the individuals of a particular colony.
Use of erlangian distributions for single-server queueing systems
Authors: Kotiah T. C. T. , Thompson J. W. , Waugh W. A. O'N.
Published: July 2016

The use of Erlangian distributions has been proposed for the approximation of more general types of distributions of interarrival and service times in single-server queueing systems. Any Erlangian approximation should have the same mean and variance as the distribution it approximates, but it is not obvious what effect the various possible approximants have on the behaviour of the system. A major difference between approximants is their degree of skewness and accordingly, numerical results for various approximants are obtained for (a) the mean time spent by a customer in a simple single-server system, and (b) the mean queue length in a system with bulk service. Skewness is shown to have little effect on these quantities.
Embedding submartingales in wiener processes with drift, with applications to sequential analysis
Authors: Hall W. J.
Published: July 2016

Skorokhod (1961) demonstrated how the study of martingale sequences (and zero-mean random walks) can be reduced to the study of the Wiener process (without drift) at a sequence of random stopping times. We show how the study of certain submartingale sequences, including certain random walks with drift and log likelihood ratio sequences, can be reduced to the study of the Wiener process with drift at a sequence of stopping times (Theorem 4.1). Applications to absorption problems are given. Specifically, we present new derivations of a number of the basic approximations and inequalities of classical sequential analysis, and some variations on them — including an improvement on Wald's lower bound for the expected sample size function (Corollary 7.5).
Variances and covariances of the grade sizes in manpower systems
Authors: Vassiliou P.-C. G. , Gerontidis I.
Published: July 2016

The asymptotic behaviour of the variances and covariances of the class sizes in closed and open manpower systems is considered. Firstly, the homogeneous case is studied and a theorem is stated which provides the answer to the problem in the most general case for the homogeneous Markov-chain models in manpower systems (open systems) and social mobility models (closed systems). Secondly, the non-homogeneous problem is studied and a theorem is given where under certain conditions it is proved that the vector sequences of means, variances and covariances converge. Finally, we relate our theoretical results to examples from the literature on manpower planning.
Tail behaviour of ladder-height distributions in random walks
Authors: Grübel Rudolf
Published: July 2016

We give necessary and sufficient conditions for various results connecting the tail behaviour of a distribution with that of its right Wiener–Hopf factor.
Another approach to some Markov chain models in population genetics
Authors: Lillestøl Jostein
Published: July 2016

This paper treats the well known model of sib mating of individuals with autosomal genes; a single locus is studied. The general derivations are valid for any number of alleles. At the end of the paper, some simple examples are included; here several familiar results concerning models for two alleles are obtained as special cases of the general theory.
A survey of some recent Czechoslovak work in automatic statistical process control
Authors: Žaludová A. H. , Režný Z. , Ullrich M.
Published: July 2016

The article first reviews briefly the field of statistical process control and its relation to automatic process control. Some recent results obtained by various workers in Czechoslovakia in the field of statistical aspects of automatic process control are then presented.
On the superposition of m-dimensional point processes
Authors: Çinlar Erhan
Published: July 2016

Consider n independent vector valued point processes. Superposition is defined component by component as a natural extension of the definition for the one-dimensional case. Under proper conditions as n → ∞, it is shown that the superposed process is a many-dimensional Poisson process with independent components. The results are applied to the superposition of Markov renewal processes.
A modification of a result due to Moran
Authors: Arnold Barry C.
Published: July 2016

Let {X(n): n = 0, 1, 2, …} be a Markov chain with state space {0, 1, 2, …, N} and transition probability matrix P = (pij
).
Random selective advantages of a gene in a finite population
Authors: Jensen Louis , Pollak Edward
Published: July 2016

A problem of interest to many population geneticists is the process of change in a gene frequency. A popular model used to describe the change in a gene frequency involves the assumption that the gene frequency is Markovian. The probabilities in a Markov process can be approximated by the solution of a partial differential equation known as the Fokker-Planck equation or the forward Kolmogorov equation. Mathematically this equation iswhere subscripts indicate partial differentiation. In this equation,f(p, x; t)is the probability density that the frequency of a gene isxat timet, given that the frequency waspat timet= o. The expressions MΔXand VΔxare, respectively, the first and second moments of the change in the gene frequency during one generation. A rigorous derivation of this equation was given by Kolmogorov (1931).
Some Duality Results in the Theory of Queues
Authors: Greenberg Irwin
Published: July 2016

When the interarrival and service time distributions of a queue are interchanged a new queue is obtained which can be considered as the dual of the original. Another dual system, a dam, can also be associated with the original queue. Events defined for the original system can be transformed into events defined for the duals and conversely, and hence, probabilities obtained for one system can be extended to the others. In this paper several duality relations are derived, with particular emphasis on results pertaining to a single busy period. Examples are given, most of which refer to the M/G/1 – G/M/l queues.
A Correlated Queue
Authors: Conolly B.W. , Hadidi N.
Published: July 2016

A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.
Stability Theorems for Solutions to the Optimal Inventory Equation
Authors: Boylan S. Edward
Published: July 2016

In a previous paper, [1] it was shown that a solution, f(x) will exist for the optimal inventory equation

(where f(y − z) = f(0), y < z) provided:

1.

g(x) ≧ 0, x ≧ 0;

2.

0 < a < 1;

3.

h(x) is monotonically nondecreasing, h(0) = 0;

4.

F is a distribution function on [0, ∞);

(In [1], 1–4 were denoted collectively as (A).)

and either

5a.

g(x) is continuous for all x ≧ 0;

5b.

lim
x→∞
g(x) = ∞;

5c

h(x) is continuous for all x > 0 (Theorem 2 of [1]);

or

6.

g(x) is uniformly continuous for all x ≧ 0 (Theorem 3 of [1]).
Sufficient Conditions for a First Passage Time Process to be that of Brownian Motion
Authors: Wasan M.T.
Published: July 2016

In this paper we assign a set of conditions to a strong Markov process and arrive at a differential equation analogous to the Kolmogorov equation. However, in this case the duration variable is the net distance travelled and the state variable is a time, a situation precisely opposite to that of Brownian motion. Solving this differential equation under certain boundary conditions produces the density function of the first passage times of Brownian motion with positive drift (see [1]), with the aid of which we define a new stochastic process.
Demographic effects on the rate of genetic evolution I. constant size populations with two genotypes
Authors: Chia A.B. , Watterson G. A.
Published: July 2016

A model is constructed to describe the evolution of a monoecious haploid population in which the population of adults remains of constant size, but for which generations may be overlapping, family size distributions are to a considerable degree arbitrary, and differential survival rates may apply with respect to adult and young individuals. The eigenvalues of the transition probability matrix are obtained, and those which determine the rate of evolution are found as simple functions of mutation rates, the mean lifetime of an individual, and the variance of his viable offspring number. The model builds on, and includes as special cases, certain non-overlapping generation models of Wright (1931), Karlin and McGregor (1965), and an overlapping generation model of Moran (1958).

One of the secondary conclusions of the study is that for some models, when evolutionary rates are measured on a mean-lifetime scale, the more offspring that are produced at a mating season may slow down evolution when mutation is absent, but may speed it up if mutation is present.
The identifiability of mixtures of distributions
Authors: Tallis G. M.
Published: July 2016

This paper considers aspects of the following problem. Let F(x, θ) be a distribution function, d.f., in x for all θ and a Borel measurable function of θ. Define the mixture (Robbins (1948)),

where Φ is a d.f., then it is of interest to determine conditions under which F(x) and F(x, θ) uniquely determine Φ. If there is only one Φ satisfying (1), F is said to be an identifiable mixture. Usually a consistency assumption is used whereby it is presumed that there exists at least one solution to (1).
A characterization of stable processes
Authors: Lukacs Eugene
Published: July 2016

Let X(t) be a stochastic process whose parameter t runs over a finite or infinite n terval T. Let t
1
, t
2 ɛ T, t
1 〈 t2; the random variable X(t
2) – X(t
1) is called the increment of the process X(t) over the interval [t
1, t
2]. A process X(t) is said to be homogeneous if the distribution function of the increment X(t + τ) — X(t) depends only on the length τ of the interval but is independent of the endpoint t. Two intervals are said to be non-overlapping if they have no interior point in common. A process X(t) is called a process with independent increments if the increments over non-overlapping intervals are stochastically independent. A process X(t) is said to be continuous at the point t if plimτ→0 [X(t + τ) — X(t)] = 0, that is if for any ε > 0, limτ→0
P(| X(t + τ) — X(t) | > ε) = 0. A process is continuous in an interval [A, B] if it is continuous in every point of [A, B].
On the probability generating function of the sum of Markov Bernoulli random variables
Authors: Gani J.
Published: July 2016

A direct proof of the expression for the limit probability generating function (p.g.f.) of the sum of Markov Bernoulli random variables is outlined. This depends on the larger eigenvalue of the transition probability matrix of their Markov chain.