bluuebunny/crossref_metadata_2025 huggingface dataset:
Restless bandits: activity allocation in a changing world
Authors: Whittle P.
Published: July 2016

We consider a population of n projects which in general continue to evolve whether in operation or not (although by different rules). It is desired to choose the projects in operation at each instant of time so as to maximise the expected rate of reward, under a constraint upon the expected number of projects in operation. The Lagrange multiplier associated with this constraint defines an index which reduces to the Gittins index when projects not being operated are static. If one is constrained to operate m projects exactly then arguments are advanced to support the conjecture that, for m and n large in constant ratio, the policy of operating the m projects of largest current index is nearly optimal. The index is evaluated for some particular projects.
Insensitive bounds for the stationary distribution of non-reversible Markov chains
Authors: Hordijk Arie , Ridder AD
Published: July 2016

A general method is developed to compute easy bounds of the weighted stationary probabilities for networks of queues which do not satisfy the standard product form. The bounds are obtained by constructing approximating reversible Markov chains. Thus, the bounds are insensitive with respect to service-time distributions. A special representation, called the tree-form solution, of the stationary distribution is used to derive the bounds. The results are applied to an overflow model.
Asymptotic periodicity of the variances and covariances of the state sizes in non-homogeneous Markov systems
Authors: Tsaklidis G. , Vassiliou P.-C. G.
Published: July 2016

In this paper we study the asymptotic periodicity of the sequence of means, variances and covariances of the state sizes of non-homogeneous Markov systems. It is proved that under the assumption that the sequence of the extended stochastic transition matrices converge to a matrix which is an irreducible stochastic matrix of period d, and all the matrices in this sequence have the same incidence matrix, then the sequence of means, variances and covariances splits into d subsequences which converge. Finally, we discuss the application of the present results in a manpower system.
Boundary crossing for moving sums
Authors: Glaz J. , Johnson B.
Published: July 2016

Let Xi, i ≧ 1, be a sequence of independent N(0, 1) random variables and Sj,m
= Xj
+ · ·· + Xj+m

–1
, the jth moving sum. Let τ m
= inf{j ≧ 1 : Sj,m
> a} + m – 1, the boundary crossing time. Approximation in the spirit of Glaz and Johnson (1984), (1986) and Samuel-Cahn (1983) are given for Pr(τ
m
> n), E(τ
m
), and σ (τ
m
),the standard deviation of τ
m
.
Stochastic scrabble: large deviations for sequences with scores
Authors: Arratia Richard , Morris Pricilla , Waterman Michael S.
Published: July 2016

A derivation of a law of large numbers for the highest-scoring matching subsequence is given. Let Xk, Yk
be i.i.d. q=(q(i))
i∊S
letters from a finite alphabet S and v=(v(i))
i∊S
be a sequence of non-negative real numbers assigned to the letters of S. Using a scoring system similar to that of the game Scrabble, the score of a word w=i
1 · ·· im
is defined to be V(w)=v(i
1) + · ·· + v(im
). Let Vn
denote the value of the highest-scoring matching contiguous subsequence between X
1
X
2 · ·· Xn
and Y
1
Y
2
· ·· Yn. In this paper, we show that Vn/K log(n) → 1 a.s. where K ≡ K(q
,
v). The method employed here involves ‘stuttering’ the letters to construct a Markov chain and applying previous results for the length of the longest matching subsequence. An explicit form for β ∊Pr(S), where β (i) denotes the proportion of letter i found in the highest-scoring word, is given. A similar treatment for Markov chains is also included.

Implicit in these results is a large-deviation result for the additive functional, H ≡ Σ
n < τ
v(Xn
), for a Markov chain stopped at the hitting time τ of some state. We give this large deviation result explicitly, for Markov chains in discrete time and in continuous time.
On higher-dimensional analogues of the arc-sine law
Authors: Bingham N. H. , Doney R. A.
Published: July 2016

The arc-sine laws form one of the cornerstones of classical one-dimensional fluctuation theory. In higher dimensions, knowledge of fluctuation theory remains a great deal less complete. Motivated by this, we consider higher-dimensional analogues of the classical arc-sine laws.
Multivariate conditional hazard rates and the MIFRA and MIFR properties
Authors: Shaked Moshe , Shanthikumar J. George
Published: July 2016

If T = (T
1, · ··, Tn
) is a vector of random lifetimes then its distribution can be determined by a set λof multivariate conditional hazard rates. In this paper, sufficient conditions on λare found which imply that T is Block–Savits MIFRA (multivariate increasing failure rate average) or Savits MIFR (multivariate increasing failure rate). Applications for a multivariate reliability model of Ross and for load-sharing models are given. The relationship between Shaked and Shanthikumar model of multivariate imperfect repair and the MIFRA property is also discussed.
Relationships and decomposition in the delayed bernoulli feedback queueing system
Authors: König D. , Miyazawa M.
Published: July 2016

For the delayed Bernoulli feedback queue with first come–first served discipline under weak assumptions a relationship for the generating functions of the joint queue-length distribution at various points in time is given. A decomposition for the generating function of the stationary total queue length distribution has been proven. The Laplace-Stieltjes transform of the stationary joint workload distribution function is represented by its marginal distributions. The arrival process is Poisson, renewal or arbitrary stationary, respectively. The service times can form an i.i.d. sequence at each queue. Different kinds of product form of the generating function of the joint queue-length distribution are discussed.
Conditions for fixation of an allele in the density-dependent wright–Fisher models
Authors: Klebaner Fima C.
Published: July 2016

A density-dependent Wright–Fisher model is a model where the population size changes randomly depending on the genetic composition process. If population sizesMnvary without density dependence then the condition ΣMn–1= ∞ is necessary and sufficient for fixation. It is shown that the above condition is no longer necessary for fixation in the density dependent models. Another necessary condition for fixation is given. Some known results on series of functions of sums of i.i.d. random variables are generalized to weighted sums.
On the distribution of points in a poisson dirichlet process
Authors: Griffiths R. C.
Published: July 2016

A probability density function important in the Poisson Dirichlet process of population genetics is studied. An accurate computational algorithm is given for this density and for the marginal distributions of the points in the Poisson Dirichlet process. The distribution of the maximal point of the process is tabulated. Rational polynomial approximations in θ, the mutation parameter, are found for the expected values of the first three maximal points.
Divided differences and the joint distribution of linear combinations of spacings
Authors: Huffer Fred
Published: July 2016

We present an identity which can sometimes express the joint distribution of several linear combinations of uniform spacings as a sum of simpler distributions. Examples are given to show that this identity is useful in the exact computation of probabilities and expectations which arise in testing for uniformity. The identity is also used to provide a new derivation of the distribution of a linear combination of spacings.
On the dependence structure of hitting times of univariate processes
Authors: Ebrahimi Nader , Ramalingam T.
Published: July 2016

Some concepts of dependence have recently been introduced by Ebrahimi (1987) to explore the structural properties of the hitting times of bivariate processes. In this framework, the special case of univariate processes has curious features. New properties are derived for this case. Some applications to sequential inference and inequalities for Brownian motion and new better than used (NBU) processes are also provided.
A time-reversibility relationship between two Markov chains with exponential stationary distributions
Authors: Chernick M. R. , Daley D. J. , Littlejohn R. P.
Published: July 2016

The stationary non-negative Markov chains {Yn
} and {Xn
} specified by the relations

for {η n
} a sequence of independent identically distributed (i.i.d.) random variables which are independent of {Yn
}, and

for {ξ n
} a sequence of i.i.d. random variables which are independent of {Xn
}, are mutually time-reversed if and only if their common marginal distribution is exponential, relating the exponential autoregressive process of Gaver and Lewis (1980) to the exponential minification process of Tavares (1980).
A limit theorem for the number of non-overlapping occurrences of a pattern in a sequence of independent trials
Authors: Chryssaphinou O. , Papastavridis S.
Published: July 2016

A sequence of independent experiments is performed, each producing a letter from a given alphabet. Using a result by Barbour and Eagleson (1984) we prove that under general conditions the number of non-overlapping occurrences of long recurrent patterns has approximately a Poisson distribution.
A collision probability problem
Authors: Papaderou-Vogiatzaki Irini , Schneider Rolf
Published: July 2016

A random stream of convex particles in , all moving uniformly in the same direction, is given. A convex test body is brought into the stream by means of a random rigid motion, independent of the stream. The number of particles that hit the test body at a preassigned boundary set within unit time is a random variable. Its expected value is determined under different model assumptions.
A drug addiction model
Authors: Billard L. , Dayananda P. W. A.
Published: July 2016

A drug addiction process in which individuals in a closed population can become addicts or pushers is modelled. Expressions for the state probabilities and factorial moments are obtained. A generalized model is also developed. This generalized drug addiction process can be viewed as a particular case of what may be called a competing death process.
Diffusion-reaction in one dimension
Authors: Balding David
Published: July 2016

One-dimensional, periodic and annihilating systems of Brownian motions and random walks are defined and interpreted in terms of sizeless particles which vanish on contact. The generating function and moments of the number pairs of particles which have vanished, given an arbitrary initial arrangement, are derived in terms of known two-particle survival probabilities. Three important special cases are considered: Brownian motion with the particles initially (i) uniformly distributed and (ii) equally spaced on a circle and (iii) random walk on a lattice with initially each site occupied. Results are also given for the infinite annihilating particle systems obtained in the limit as the number of particles and the size of the circle or lattice increase. Application of the results to the theory of diffusion-limited reactions is discussed.
Comparisons of replacement policies
Authors: Langberg Naftali A.
Published: July 2016

For independent random lifelengths of the units in use stochastic comparisons of the number of failures and removal in [0,s] under age and block replacement policies are performed. A new concept of NBU (NWU) in sequence is introduced.
On some statistics connected with runs in Markov chains
Authors: Banjević Dragan
Published: July 2016

We consider a class of statistics connected with the history of a Markov chain before the first occurrence of certain runs. Methods for deriving generating functions, distributions and moments of the statistics considered are given. In the case of i.i.d. random variables, explicit formulas are derived.
Representation of denumerable Markov chains with multiple states by weighted circuits
Authors: Kalpazidou S.
Published: July 2016

The constructive solution to the problem of representing a strictly stationary Markov chain ζ with a countable infinity of r-sequences (i
1, i
2, · ··, ir
), r > 1, as states by a class of directed weighted circuits is given. Associating the chain ζ with its dual chain η having reversed states and the same transition law, a connection with physical laws that govern diffusion of electrical current through a directed planar network with r-series-connected nodes is shown.
On the non-closure under convolution of the subexponential family
Authors: Leslie J. R.
Published: July 2016

A distribution F of a non-negative random variable belongs to the subexponential family of distributions S if 1 – F
(2)(x) ~ 2(1 – F(x)) as x →∞. This family is of considerable interest in branching processes, queueing theory, transient renewal theory and infinite divisibility theory. Much is known about the kind of distributions that belong to S but the question of whether S is closed under convolution has remained unresolved for some time. This paper contains an example which demonstrates that S is not in fact closed.
On a non-Markovian time-dependent queueing system
Authors: Sharma S. D.
Published: July 2016

This paper studies the transient and steady-state behaviour of a continuous and discrete-time queueing system with non-Markovian type of departure mechanism. The Laplace transforms of the probability generating function of the time-dependent queue length distribution in the transient state are obtained and the probability generating function of the queue length distribution in the steady state is derived therefrom. Finally, some particular cases are discussed.
Conditional limit theorems for a left-continuous random walk
Authors: Pakes A. G.
Published: July 2016

The present work considers a left-continuous random walk moving on the positive integers and having an absorbing state at the origin. Limit theorems are derived for the position of the walk at time n given: (a) absorption does not occur until after n, or (b) absorption does not occur until after m + n where m is very large, or (c) absorption occurs at m + n. A limit theorem is given for an R-positive recurrent Markov chain on the non-negative integers with an absorbing origin and subject to condition (c) above.
Regular stochastic matrices and digraphs
Authors: Berman Elizabeth
Published: July 2016

This paper presents an algorithm to determine whether a stochastic matrix is regular. The main theorem is the following. Hypothesis: An n-by-n stochastic matrix has at least one positive entry off the main diagonal in every row and column. There is at most one row with n — 1 zeros and at most one column with n — 1 zeros. There are no j-by-k submatrices consisting entirely of zeros, where j and k are integers greater than 1, with j + k = n. Conclusion: The matrix is regular. Similar results hold for strongly connected digraphs.
Nested Monte Carlo study of random packing on the sphere
Authors: Rodgers R. P. C. , Baddeley A. J.
Published: July 2016

We consider two random sequential packing processes in which spheres of unit radius are randomly attached to the surface of a fixed unit sphere. Independent random spheres are generated and added successively, provided there is no overlap with previous spheres. In model 1, the process stops when a trial sphere intersects one of the previously-accepted spheres. In model 2, random sequential packing, any such overlapping trial sphere is discarded and the next random sphere is tried, until it is impossible to add any further spheres.

Previous workers have conjectured convincingly that no exact analytical solution is possible for this type of problem. We use Monte Carlo simulation methods to estimate transition probabilities for the two models. Because some probabilities are extremely small, a simulation using independent repetitions of the model would be inefficient. We designed a branching process of conditionally binomial trials, and performed over 108 trials on a supercomputer.
Stochastic monotonicity properties of multiserver queues with impatient customers
Authors: Bhattachary Partha P. , Ephremides Anthony
Published: July 2016

We consider multiserver queues in which a customer is lost whenever its waiting time is larger than its (possibly random) deadline. For such systems, the number of (successful) departures and the number of customers lost over a time interval are the performance measures of interest. We show that these quantities are (stochastically) monotone functions of the arrival, service and deadline processes.
Stationary increments of accumulation processes in queues and generalized semi-Markov schemes
Authors: Foley R. D. , Klutke Georgia-Ann , König Dieter
Published: July 2016

Let Tx
be the length of time to accumulate x units of a resource. In queueing, the resource could be service. We derive a sufficient condition for the process to have stationary increments where Tx
is an additive functional of a Markov process. This condition is satisfied in symmetric queues and generalized semi-Markov schemes with insensitive components. As a corollary, we show that the conditional expected response time in a symmetric queue is linear in the service requirement. A similar result holds for the conditional average residence time of an insensitive component in a GSMS.
Applications of the key renewal theorem: crudely regenerative processes
Authors: Serfozo Richard F.
Published: July 2016

Limit Statements obtainable by the key renewal theorem are of the form EXt
= v(t) + o(1), as t →∞. We show how to delineate the limit function v for processes X associated with crudely regenerative phenomena. Included are refinements of classical limit theorems for Markov and regenerative processes, limits of sums of stationary random variables, and limits for integrals and derivatives of EXt.
Useful martingales for stochastic storage processes with Lévy input
Authors: Kella Offer , Whitt Ward
Published: July 2016

We apply the general theory of stochastic integration to identify a martingale associated with a Lévy process modified by the addition of a secondary process of bounded variation on every finite interval. This martingale can be applied to queues and related stochastic storage models driven by a Lévy process. For example, we have applied this martingale to derive the (non-product-form) steady-state distribution of a two-node tandem storage network with Lévy input and deterministic linear fluid flow out of the nodes.
On the recurrence of simple random walks on some fractals
Authors: Xian-Yin Zhou
Published: July 2016

In this paper, the recurrence or transience of simple random walks on some lattice fractals is investigated. As results, we obtain that the simple random walk on the pre-Sierpinski gasket in d dimensions is recurrent for all d ≧ 2, and on the pre-Sierpinski carpet in d dimensions it is recurrent for d = 2 and transient for all d ≧ 3.
A class of bandit problems yielding myopic optimal strategies
Authors: Banks Jeffrey S. , Sundaram Rangarajan K.
Published: July 2016

We consider the class of bandit problems in which each of the n ≧ 2 independent arms generates rewards according to one of the same two reward distributions, and discounting is geometric over an infinite horizon. We show that the dynamic allocation index of Gittins and Jones (1974) in this context is strictly increasing in the probability that an arm is the better of the two distributions. It follows as an immediate consequence that myopic strategies are the uniquely optimal strategies in this class of bandit problems, regardless of the value of the discount parameter or the shape of the reward distributions. Some implications of this result for bandits with Bernoulli reward distributions are given.
Ageing properties of certain dependent geometric sums
Authors: Kováts Antal , Móri Tamás F.
Published: July 2016

We study some distribution properties of a random sum of i.i.d. non-negative random variables, where the number of terms is geometrically distributed and not independent of the summands. The results are applied to the system failure time of a one-unit system with a single spare and repair facility. In such a system when the operating unit fails it is immediately replaced by the spare and sent to the repair facility. The system continues operating until the first time when the failed unit has not yet been repaired by the failure of the operating unit. Certain ageing properties such as NBU, NWU, NBUE, NWUE, HNBUE, HNWUE, L+ and L– are shown to be inheritable from the working time of the operating unit to the system lifetime.
Exploiting Markov chains to infer queue length from transactional data
Authors: Daley D. J. , Servi L. D.
Published: July 2016

The use of taboo probabilities in Markov chains simplifies the task of calculating the queue-length distribution from data recording customer departure times and service commencement times such as might be available from automatic bank-teller machine transaction records or the output of telecommunication network nodes. For the case of Poisson arrivals, this permits the construction of a new simple exact O(n
3) algorithm for busy periods with n customers and an O(n
2 log n) algorithm which is empirically verified to be within any prespecified accuracy of the exact algorithm. The algorithm is extended to the case of Erlang-k interarrival times, and can also cope with finite buffers and the real-time estimates problem when the arrival rate is known.
Poisson approximation of intermediate empirical processes
Authors: Kaufmann E. , Reiss R.-D.
Published: July 2016

We investigate the asymptotic behaviour of empirical processes truncated outside an interval about the (1 – s(n)/n)-quantile where s(n) → ∞ and s(n)/n → 0 as the sample size n tends to ∞. It is shown that extreme value (Poisson) processes and, alternatively, the homogeneous Poisson process may serve as approximations if certain von Mises conditions hold.
Random splittings of an interval
Authors: Mountford T. S. , Port S. C.
Published: July 2016

Points are independently and uniformly distributed onto the unit interval. The first n—1 points subdivide the interval into n subintervals. For 1 we find a necessary and sufficient condition on {ln
} for the events [Xn
belongs to the ln
th largest subinterval] to occur infinitely often or finitely often with probability 1. We also determine when the weak and strong laws of large numbers hold for the length of the ln
th largest subinterval. The strong law of large numbers and the central limit theorem are shown to be valid for the number of times by time n the events [Xr
belongs to l
r th largest subinterval] occur when these events occur infinitely often.
An extension of the Cameron–Martin result
Authors: Yashin A. I.
Published: July 2016

The well-known Cameron–Martin formula allows us to calculate the mathematical expectation where Ws
is a Wiener process. This paper extends this result to the case of piecewise continuous martingales. As a particular case the mathematical expectations of a functional of generalized Ornstein– Uhlenbeck processes and pure jump processes are calculated.
Distances between censored intersections between a square lattice and a random smooth path
Authors: Wheelwright A. V. , Glasbey C. A.
Published: July 2016

A random smooth path of infinite length crossed a square lattice. Intersections with the lattice were censored if they lay within a threshold distance of a preceding uncensored intersection, defined by tracking along the path in one direction. The distribution of distances between consecutive uncensored intersections is derived.
Bivariate life distributions from Pólya's urn model for contagion
Authors: Marshall Albert W. , Olkin Ingram
Published: July 2016

Shock models based on Poisson processes have been used to derive univariate and multivariate exponential distributions. But in many applications, Poisson processes are not realistic models of physical shock processes because they have independent increments; expanded models that allow for possibly dependent increments are of interest. In this paper, univariate and bivariate Pólya urn schemes are used to derive models of shock sources. The life distributions obtained from these models form a large parametric family that includes the exponential distribution. Even in the univariate case these life distributions have not been widely used, though they form a large and flexible family. In the bivariate case, the family includes the bivariate exponential distributions of Marshall and Olkin as a special case.
Equilibrium points for three games based on the Poisson process
Authors: Dixon M. T.
Published: July 2016

An arbitrary number of competitors are presented with independent Poisson streams of offers consisting of independent and identically distributed random variables having the uniform distribution on [0, 1]. The players each wish to accept a single offer before a known time limit is reached and each aim to maximize the expected value of their offer. Rejected offers may not be recalled, but they are passed on to the other players according to a known transition matrix. This paper finds equilibrium points for two such games, and demonstrates a two-player game with an equilibrium point under which the player with the faster stream of offers has a lower expected reward than his opponent.
Optimal second-order product probability bounds
Authors: Block Henry W. , Costigan Timothy M. , Sampson Allan R.
Published: July 2016

Let P(c) = P(X
1 ≦ c
1, · ··, Xp
≦ cp
) for a random vector (X
1, · ··, Xp
). Bounds are considered of the form

where T is a spanning tree corresponding to the bivariate probability structure and di
is the degree of the vertex i in T. An optimized version of this inequality is obtained. The main result is that alwayṡ dominates certain second-order Bonferroni bounds. Conditions on the covariance matrix of a N(0,Σ) distribution are given so that this bound applies, and various applications are given.
Some results involving the maximum of Brownian motion
Authors: Doney R. A.
Published: July 2016

If X is a Brownian motion with drift and γ = inf{t > 0: Mt
= t} we derive the joint density of the triple {U, γ, Δ}, where and Δ= γ —Xγ
. In the case δ = 0 it follows easily from this that Δ has an Exp(2) distribution and this in turn implies the rather surprising result that if τ= inf{t > 0: Xt
= Mt
= t}, then Pr{τ = 0} = 0 and . We also derive various other distributional results involving the pair (X, M), including for example the distribution of ; in particular we show that, in case δ. = 1, when Pr{0 < τ < ∞} = 1, the ratio τ
+/τ has the arc-sine distribution.
Sojourn times in single-server queues by negative customers
Authors: Harrison P. G. , Pitel E.
Published: July 2016

We derive expressions for the Laplace transform of the sojourn time density in a single-server queue with exponential service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. We compare first-come first-served and last-come first-served queueing disciplines for the positive customers, combined with elimination of the last customer in the queue or the customer in service by a negative customer. We also derive the corresponding result for processor-sharing discipline with random elimination. The results show differences not only in the Laplace transforms but also in the means of the distributions, in contrast to the case where there are no negative customers. The various combinations of queueing discipline and elimination strategy are ranked with respect to these mean values.
Mean passage times for tridiagonal transition matrices and a two-parameter ehrenfest urn model
Authors: Krafft Olaf , Schaefer Martin
Published: July 2016

A two-parameter Ehrenfest urn model is derived according to the approach taken by Karlin and McGregor [7] where Krawtchouk polynomials are used. Furthermore, formulas for the mean passage times of finite homogeneous Markov chains with general tridiagonal transition matrices are given. In the special case of the Ehrenfest model they have quite a different structure as compared with those of Blom [2] or Kemperman [9].
A stochastic model of fragment formation when DNA replicates
Authors: Cowan Richard , Chiu S. N.
Published: July 2016

The double-stranded molecule, DNA, has the unique property of replication and, because of this, it is the central molecule of life. The mechanism of replication for each single strand is intricate, involving enzymes which move along each of the single strands building a complementary copy. At the frontier of this action, the events have a strong stochastic character due to the random location on the DNA of key ‘sites' where copying commences. A model of this process is analysed. The central problem of interest is the mean length of certain ‘islands' of newly replicated DNA developed at the randomly located ‘sites'. These islands, which have been observed experimentally, are called Okazaki fragments.
Forward and backward processes in bisexual models with fixed population sizes
Authors: Möhle M.
Published: July 2016

This paper introduces exchangeable bisexual models with fixed population sizes and non-overlapping generations. In each generation there are N pairs of individuals consisting of a female and a male. The N pairs of a generation produce N daughters and N sons altogether, and these 2N children form the N pairs of the next generation at random.

First the extinction of the lines of descendants of a fixed number of pairs is studied, when the population size becomes large. Under suitable conditions this structure can be approximately treated in the framework of a Galton-Watson process. In particular it is shown for the Wright-Fisher model that the discrepancy between the extinction probabilities in the model and in the approximating Galton-Watson process is of order N.

Next, the process of the number of ancestor-pairs of all pairs of a generation is analysed. Under suitable conditions this process, properly normed, has a weak limit as N becomes large. For the Wright-Fisher model this limit is an Ornstein–Uhlenbeck process (restricted to a discrete time-set). The corresponding stationary distributions of the backward processes converge to the normal distribution, as expected.
The evolution of the attainable structures of a homogeneous Markov system by fixed size
Authors: Tsaklidis George M.
Published: July 2016

In order to describe the evolution of the attainable structures of a homogeneous Markov system (HMS) with fixed size, we evaluate the volume of the sets of the attainable structures in Euclidean space as they are changing in time and we find the value of the volume asymptotically. We also estimate the evolution of the distance of two (attainable) structures of the system as it changes following the transformations of the structures; extensions are obtained concerning results from the Perron–Frobenius theory referring to Markov systems.
Homogeneous Markov chains by bounded transition matrix
Authors: Hartfiel D. J.
Published: July 2016

Let A be a stochastic matrix and ε a positive number. We consider all stochastic matrices within ε of A and their corresponding stochastic eigenvectors. A convex polytope containing these vectors is described. An efficient algorithm for computing bounds on the components of these vectors is also given. The work is compared to previous such work done by the author and by Courtois and Semai.
Archimedean copulas, exchangeability, and max-stability
Authors: Ballerini Rocco
Published: July 2016

An exchangeable sequence of random variables is constructed with all finite-dimensional distribution functions having an Archimedean copula (as defined by Schweizer and Sklar (1983)). Through a monotone transformation of this exchangeable sequence, we obtain and characterize the class of exchangeable sequences possessing the max-stable property as defined by De Haan and Rachev (1989). Several parametric examples are given.
Approximating gamma distributions by normalized negative binomial distributions
Authors: Adell José A. , De La Cal Jesús
Published: July 2016

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs
be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p
, where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.
MR/GI/1 queues by positively correlated arrival stream
Authors: Szekli R. , Disney R. L. , Hur S.
Published: July 2016

The effects of dependencies (such as association) in the arrival process to a single server queue on mean queue lengths and mean waiting times are studied. Markov renewal arrival processes with a particular transition matrix for the underlying Markov chain are used which allow us to change dependency properties without at the same time changing distributional conditions. It turns out that correlations do not seem to be pure effects, and three main factors are studied: (a) differences in the mean interarrival times in the underlying Markov renewal process, (b) intensity in the Markov renewal jump process, (c) variability in the point processes underlying the Markov renewal process. It is shown that the mean queue length can be made arbitrarily large in the class of queues with the same interarrival distributions and the same service time distributions (with fixed smaller than one traffic intensity), by making (a) large enough and (b) small enough. The existence of the moments of interest is confirmed and some stochastic comparison results for actual waiting times are shown.
On the existence of finite-dimensional filters for Markov-modulated traffic
Authors: Olivier C. , Walrand J.
Published: July 2016

A Markov-modulated Poisson process (MMPP) is a Poisson process whose rate is a finite Markov chain. The Poisson process is a simple MMPP. An MMPP/M/1 queue is a queue with MMPP arrivals, an infinite capacity, and a single exponential server. We prove that the output of an MMPP/M/1 queue is not an MMPP process unless the input is Poisson. We derive this result by analyzing the structure of the non-linear filter of the state given the departure process of the queue. The practical relevance of the result is that it rules out the existence of simple finite descriptions of queueing networks with MMPP inputs.
Non-uniqueness in probabilistic numerical identification of bacteria
Authors: Gyllenberg Mats , Koski Timo , Reilink Edwin , Verlaan Martin
Published: July 2016

In this note we point out an inherent difficulty in numerical identification of bacteria. The problem is that of uniqueness of the taxonomic structure or, in mathematical terms, the lack of statistical identifiability of finite mixtures of multivariate Bernoulli probability distributions shown here.
Understanding the Wiener–Hopf factorization for the simple random walk
Authors: Kennedy Joanne
Published: July 2016

We give a sample path proof of the well-known Wiener–Hopf identity F = G
– + G
+ – G
– ∗G
+ which relates the ladder-height distributions G
– and G
+ of a simple random walk to the step distribution F. Unlike previous approaches this direct proof is both simple and intuitive.
An application of random packing to the multiple fracture of composite materials
Authors: Kimber Alan C.
Published: July 2016

A one-dimensional packing approach is used to obtain limiting results for inter-crack distances after multiple fracture of a long brittle-matrix composite with continuous aligned fibres. The results may also be appropriate for applications of the Rényi car-parking model in which there is a reduced probability of cars parking bumper to bumper.
On the transition probabilities of the move-to-front scheme
Authors: Phatarfod R. M.
Published: July 2016

The transition probabilities of the move-to-front scheme are obtained by exploiting the connection between it and the coupon-collector's problem.
A matrix equation approach to solving recurrence relations in two-dimensional random walks
Authors: Miller James W.
Published: July 2016

The purpose of this paper is to provide explicit formulas for a variety of probabilistic quantities associated with an asymmetric random walk on a finite rectangular lattice with absorbing barriers. Quantities of interest include probabilities that a walker will exit the lattice onto some particular set of boundary states, the expected duration of the walk, and the expected number of visits to one state given a start in another. These quantities are shown to satisfy two-dimensional recurrence relations that are very similar in structure. In each case, the recurrence relations may be represented by matrix equations of the form X = AX + XB + C, where A and B are tridiagonal Toeplitz matrices. The spectral properties of A and B are investigated and used to provide solutions to this matrix equation. The solutions to the matrix equations then lead to solutions for the recurrence relations in very general cases.
Optimizing a single uncertain selection by recall of observations
Authors: Sweet Ted
Published: July 2016

This paper considers a variation of the classical secretary problem; this variation allows for the interviewer to make an offer to previously interviewed applicants and for the applicant to reject an offer. In this version, the process ends after any offer is made. Two special cases are considered, where the optimal procedure and its probability of success are given for each. Furthermore, the problem is discussed in general and conditions for optimal strategies are given.
Estimating the home range
Authors: De Haan L. , Resnick Sidney
Published: July 2016

A proposal is given for estimating the home range of an animal based on sequential sightings. We assume the given sightings are independent, identically distributed random vectors X
1,· ··, Xn
whose common distribution has compact support. If are the polar coordinates of the sightings, then is a sup-measure and corresponds to the right endpoint of the distribution . The corresponding upper semi-continuous function l(θ) is the boundary of the home range. We give a consistent estimator for the boundary l and under the assumption that the distribution of R
1 given is in the domain of attraction of an extreme value distribution with bounded support, we are able to give an approximate confidence region.
Max-infinite divisibility and multivariate total positivity
Authors: Alzaid Abdulhamid A. , Proschan Frank
Published: July 2016

The concept of max-infinite divisibility is viewed as a positive dependence concept. It is shown that every max-infinitely divisible distribution function is a multivariate totally positive function of order 2 (MTP2). Inequalities are derived, with emphasis on exchangeable distributions. Applications and examples are given throughout the paper.
Moment bounds for non-stationary dependent sequences
Authors: Kim Tae Yoon
Published: July 2016

We provide a unified approach for establishing even-moment bounds for partial sums for a class of weakly dependent random variables satisfying a stationarity condition. As applications, we discuss moment bounds for various types of mixing sequences. To obtain even-moment bounds, we use a ‘combinatorial argument' developed by Cox and Kim (1990).
Comments on the Luria–Delbrück distribution
Authors: Kemp Adrienne W.
Published: July 2016

The long-tailed Luria–Delbrück distribution arises in connection with the ‘random mutation’ hypothesis (whereas the ‘directed adaptation' hypothesis is thought to give a Poisson distribution). At timetthe distribution depends on the parameterm=gNt/(a+g) whereNtis the current population size andg/(a+g) is the relative mutation rate (assumed constant). The paper identifies three models for the distribution in the existing literature and gives a fourth model. Ma et al. (1992) recently proved that there is a remarkably simple recursion relation for the Luria–Delbrück probabilitiespnand found that asymptoticallypn≈c/n2; their numerical studies suggested thatc= 1 when the parametermis unity. Cairns et al. (1988) had previously argued and shown numerically that Pn= Σj ≧ nPj≈ m/n. Here we prove thatn(n+ 1)pn< m(1 + 11m/30) forn= 1, 2, ···, and hence prove that asnbecomes largen(n+ 1)pn, ≈m; the resultmPn≈mfollows immediately.
A formula for singular perturbations of Markov chains
Authors: Lasserre Jean B.
Published: July 2016

We give formulas for updating both the steady-state probability distribution and the fundamental matrices of a singularly perturbed Markov chain. This formula generalizes Schweitzer's regular perturbation formulas to the case of singular perturbations.
The Mk/G/∞ batch arrival queue by heterogeneous dependent demands
Authors: Falin Gennadi
Published: July 2016

Choi and Park [2] derived an expression for the joint stationary distribution of the number of customers of k types who arrive in batches at an infinite-server system of M/M/∞ type. We propose another method of solving this problem and extend the result to the case of general service times (not necessarily independent). We also get a transient solution. Our main result states that the k- dimensional vector of the number of customers of k types in the system is a certain linear function of a (2
k – 1)-dimensional vector whose coordinates are independent Poisson random variables.
A representation for the limiting random variable of a branching process with infinite mean and some related problems
Authors: Cohn Harry , Pakes Anthony G.
Published: July 2016

It is known that for a Bienaymé– Galton–Watson process {Zn
} whose mean m satisfies 1 < m < ∞, the limiting random variable in the strong limit theorem can be represented as a random sum of i.i.d. random variables and hence that convergence rate results follow from a random sum central limit theorem.

This paper develops an analogous theory for the case m = ∞ which replaces ‘sum' by ‘maximum'. In particular we obtain convergence rate results involving a limiting extreme value distribution. An associated estimation problem is considered.
Random space filling and moments of coverage in geometrical probability
Authors: Siegel Andrew F.
Published: July 2016

The moments of the random proportion of a fixed set that is covered by a random set (moments of coverage) are shown to converge under very general conditions to the probability that the fixed set is almost everywhere covered by the random set. Moments and coverage probabilities are calculated for several cases of random arcs of random sizes on the circle. When comparing arc length distributions having the same expectation, it is conjectured that if one concentrates more mass near that expectation, the corresponding coverage probability will be smaller. Support for this conjecture is provided in special cases.
A limit theorem for random coverings of a circle which do not quite cover
Authors: Kaplan Norman
Published: July 2016

We study the asymptotic behavior as α → 0 of the number of independent random arcs of length a needed to cover at least 1 − Kα (K ≧ 1) of a circle of unit circumference.
An explicit upper bound for the mean busy period in a GI/G/1 queue
Authors: Loulou Richard
Published: July 2016

In this paper, an upper bound is derived for the mean busy cycle duration in GI/G/1 queues. The bound is of the form A/(1 – ρ), where ρ is the traffic intensity and A involves three moments of the basic random variables of the queue. The proof uses a well-known result of random walk theory.
On the envelope of a Gaussian random field
Authors: Adler R. J.
Published: July 2016

For homogeneous, two-dimensional random field ξ(t), t ∈ R
2 we develop the ‘half' spectral theory sufficient to rigorously define its envelope η (t). We then specialise to the case of ξ Gaussian, which implies η is Rayleigh, and consider the mean value of a certain characteristic of the sets {t:η(t) ≧ u} (u ≧ 0). From this we deduce some qualitative information about the sample path behaviour of the Rayleigh field η .
Gaussian families and a theorem on patterned matrices
Authors: Barrett Wayne W. , Feinsilver Philip J.
Published: July 2016

In this paper we use the properties of the covariance matrix of a Gaussian Markovian family to give a probabilistic proof of a theorem about inverses of tridiagonal matrices.
Bivariate distributions as saddle points of mutual information
Authors: Kim Geung Ho , David H. T.
Published: July 2016

Fix a bivariate distribution F on X × Y, considered as a pair (α, {Fx
}), where α is a marginal distribution on X and {Fx
} is a collection of conditional distributions on Y. For essentially every (β,{Gx
}) satisfying a certain pair of moment conditions determined by (α, {F

x
}), J(β, {F

x
}) ≦ J(α, {F

x
}) ≦ J(α, {G

x
}), where J is mutual information. This relates to two sorts of extremizations of mutual information of relevance to communication theory and statistics.
Non-homogeneously paced random records and associated extremal processes
Authors: Gaver Donald P. , Jacobs Patricia A.
Published: July 2016

A study is made of the extremal process generated by i.i.d. random variables appearing at the events of a non-homogeneous Poisson process, 𝒫. If 𝒫 has an exponentially increasing rate function, then records eventually occur in a homogeneous Poisson process. The size of the latest record has a classical extreme value distribution.
Shock models with NBUE survival
Authors: Block Henry W. , Savits Thomas H.
Published: July 2016

Conditions are given on a process so that a shock model governed by the process will have nbue survival time. A related condition is given so that survival will be NBU and it is shown that a pure birth process satisfies this condition. A result of A-Hameed and Proschan is obtained as a corollary.
One-sided excursions of brownian motion, and an application
Authors: Imhof J. P.
Published: July 2016

The value of a process on C[0,∞) when it starts its first one-sided excursion of width ≧ α is related to first hitting one of two boundaries. Explicit results for brownian motion are applied to a finite dam model.
Random collision models in oriented graphs
Authors: Itoh Yoshiaki
Published: July 2016

We investigate a random collision model for competition between types of individuals in a population. There are dominance relations defined for each pair of types such that if two individuals of different types collide then after the collision both are of the dominant type. These dominance relations are represented by an oriented graph, called a tournament. It is shown that tournaments having a particular form are relatively stable, while other tournaments are relatively unstable. A measure of the stability of the stable tournaments is given in the main theorem.
Return probabilities for certain three-dimensional random walks
Authors: Daley D. J.
Published: July 2016

For some three-dimensional random walks on the cubic lattice, the probability of the walk returning to its starting point is given numerically.
On the relation between offered, critical and accepted gap distributions
Authors: Thedeen Torbjörn
Published: July 2016

We consider the situation when vehicles in a minor road try to cross a major road. The relation between the distributions of the critical, offered and accepted gaps is studied. Estimates of the critical gap distribution from observations of offered and accepted gaps are given and also studied in some simulation examples.
Sequential rank and the Pólya urn
Authors: Robbins Herbert , Whitehead John
Published: July 2016

A sequence of independent, identically distributed random variables is observed. After a sample of m has been collected attention is fixed to a particular observation. In this paper the fluctuations of the rank of this observation, as sampling continues, will be studied.

The process can be modelled by a Pólya urn scheme, and new results are obtained which are of interest in both sequential rank and Pólya urn contexts.
On the characterization of point processes with the order statistic property
Authors: Feigin Paul D.
Published: July 2016

We provide a probabilistic proof of the characterization of point processes (on the real line) with the order statistic property. The characterization is used to investigate the homogeneity of such processes and is also related to the martingale theory associated with point processes.