bluuebunny/crossref_metadata_2025 huggingface dataset:
Measuring the association of a time series and a point process
Authors: Willie J. S.
Published: July 2016

We consider a bivariate stochastic process where one component is an ordinary time series and the other is a point process. In the stationary case, a useful measure of the association of the time series and the point process is provided by a conditional intensity function, ṙ
11(x;u), which gives the intensity with which events occur near time t given that the time series takes on a value x at time t + u. In this paper we consider the estimation of the function ṙ
11(x;u) and certain related functions that are also useful in partially characterizing the degree of interdependence of the time series and the point process. Histogram and smoothed histogram-type estimates are proposed and asymptotic distributions of these estimates are derived. We also discuss an application of the estimation theory to the analysis of some data from a neurophysiological study.
Covariation of a time series and a point process
Authors: Willie J. S.
Published: July 2016

Let {X(t), N(t)}, – ∞< t <∞, be a stationary bivariate stochastic process where X(t) is an ordinary time series and N(t) is an orderly point process counting the number of points in (0, t]. Suppose values of {X(t), N(t)} are available for 0< t ≦T and let σ1, σ2, ···, σ N

(T) denote the jump points of N(t) in (0, T]. For |v| < T, define m

T

12(υ)=∑ X (σj+υ)/T and μ
T

12(υ)=∑ X (σ
j
+υ)/∑1 where all summations are over indices j such that 0<σ
j
, σ
j
+υ≦T for some σj. The functions M

T

12(υ) and μ

T

12(υ) are often useful in analyzing the covariation of the time series and point process. In this paper, we shall develop some statistical properties of the functions M

T

12(υ) and μ

T

12(υ) and discuss some specific situations where it is useful to consider these functions.
Weak lumpability in finite Markov chains
Authors: Abdel-moneim Atef M. , Leysieffer Frederick W.
Published: July 2016

Criteria are given to determine whether a given finite Markov chain can be lumped weakly with respect to a given partition of its state space. These conditions are given in terms of solution classes of systems of linear equations associated with the transition probability matrix of the Markov chain and the given partition.
Strong ergodicity for continuous-time, non-homogeneous Markov chains
Authors: Scott Mark , Arnold Barry C. , Isaacson Dean L.
Published: July 2016

Characterizations of strong ergodicity for Markov chains using mean visit times have been found by several authors (Huang and Isaacson (1977), Isaacson and Arnold (1978)). In this paper a characterization of uniform strong ergodicity for a continuous-time non-homogeneous Markov chain is given. This extends the characterization, using mean visit times, that was given by Isaacson and Arnold.
A restricted occupancy problem
Authors: Fang Kai-Tai
Published: July 2016

There are m urns each containing k cells; n balls are assigned to the m urns in such a way that each cell contains at most one ball. Let Mt
be the number of urns containing exactly t balls (t = 0, 1, •••, k). In the paper, the distribution of Mt
, its moments and moment-generating function are obtained. The derivation of the last two seems to be new.
Acute triangles in the n-ball
Authors: Hall Glen Richard
Published: July 2016

Using Baddeley's [1] extension of Crofton's differential equation we derive an elementary integral formula for the probability that three randomly chosen points in the unit n-ball in ℝ
n
, with respect to Lebesgue measure, form an acute triangle. When the dimension is 2 this probability is 4/π2 − 1/8, while when the dimension is 3 it is 33/70.
Optimal stopping rules in proofreading
Authors: Yang Mark C. K. , Wackerly Dennis D. , Rosalsky Andrew
Published: July 2016

Optimal stopping rules under various conditions are obtained for a proofreader who has a probabilityp(known or unknown) of detecting a misprint in proofsheets which contain an unknown but Poisson-distributed number of misprints.
A counterexample to a conjecture on optimal list ordering
Authors: Anderson E. J. , Nash P. , Weber R. R.
Published: July 2016

A number of items are arranged in a line. At each unit of time one of the items is requested, the i th being requested with probability Pi
. We consider rules which reorder the items between successive requests in a fashion which depends only on the position in which the most recently requested item was found. It has been conjectured that the rule which always moves the requested item one closer to the front of the line minimizes the average position of the requested item. An example with six items shows that the conjecture is false.
An exponential riddle
Authors: Sengupta Bhaskar
Published: July 2016

We present a problem of finding the optimum inspection procedure for a system whose time to failure is an exponential random variable. We characterize a simple and surprising solution to the problem.
Applications of martingale theory to some epidemic models, II
Authors: Picard Philippe
Published: July 2016

We consider Weiss's and Downton's models with parametersπ, αandβdepending oninumber of susceptibles andjnumber of carriers. A martingale argument is performed when πandα /βonly depend onior, in Weiss's case, whenα /βis the product of a function ofiby a function ofj.In these cases the martingale approach proves very valuable and gives explicit results quite easily. In particular it shows that well-known relations between moments and integrals along a trajectory are still true for any stopping time and for more general models than the classic ones.
Invariance in resource-sharing systems
Authors: Courcoubetis C. , Varaiya P. , Walrand J.
Published: July 2016

A method for proving the invariance of hitting-time distributions with respect to the control strategy of a Markov chain is presented. The method is applied to resource-sharing problems. It provides a new proof of a known result and extends it. Sufficient conditions for such an invariance are given and are illustrated by examples.
Life distribution properties of devices subject to a pure jump damage process
Authors: Abdel-Hameed Mohamed
Published: July 2016

A device is subject to damage. The damage occurs randomly in time according to a pure jump process. The device has a threshold and it fails once the damage exceeds the threshold. We show that life distribution properties of the threshold right tail probability are inherited as corresponding properties of the survival probability, under suitable conditions on the parameters of the damage process. Moreover we discuss an optimal replacement problem for such devices.
Some comparability results for waiting times in single- and many-server queues
Authors: Daley D. J. , Rolski T.
Published: July 2016

It is shown that the stationary waiting time random variables W′, W″ of two M/G/l queueing systems for which the corresponding service time random variables satisfy E(S
′−x)+ ≦ E(S
″−x)+ (all x >0), are stochastically ordered as W
′≦d
W
″. The weaker conclusion, that E(W
′−x)+ ≦ E(W
″−x)+ (all x > 0), is shown to hold in GI/M/k systems when the interarrival time random variables satisfy E(x−T
′)+ ≦ E(x−T
″)+ (all x). A sufficient condition for wk
≡EW in GI/D/k to be monotonic in k for a sequence of k-server queues with the same relative traffic intensity is given. Evidence indicating or refuting possible strengthenings of some of the results is indicated.
On spàtial general epidemics and bond percolation processes
Authors: Kuulasmaa Kari , Zachary Stan
Published: July 2016

We show that a lower bound for the probability that a spatial general epidemic never becomes extinct is given by the percolation probability of an associated bond percolation process.
On the comparison of point processes
Authors: Deng Y. L.
Published: July 2016

Several different orderings for the comparison of point processes have been introduced and their relationships discussed in Whitt [9], Daley [2] and Deng [4]. It is of some interest to know whether these orderings, in general, are preserved under various operations on point processes. Some results concerning limit operations were given in Deng [4].

In the present paper, we first further introduce some convex and concave orderings for counting processes, and survey the relationships among all orderings mentioned in [9], [4] and this paper. Then we focus our attention on the study of the conditions for the preservation of orderings under the operations of superposition, thinning, shift, and random change of time.
Stationary queue-length characteristics in queues with delayed feedback
Authors: König Dieter , Schmidt Volker
Published: July 2016

A class of two-node queueing networks with general stationary ergodic governing sequence is considered. This means that, in particular, a non-Poissonian arrival process and dependent service times, as well as a non-Bernoulli feedback mechanism are admitted. A mixing condition ensures that the limiting distributions of the number of customers in the nodes observed in continuous time as well as at certain embedded epochs can be expressed by the Palm distributions of appropriately chosen marked point processes. This gives the possibility of connecting the classical concept of embedding with a general point-process approach. Furthermore, it leads to simple proofs of relationships between the limiting distributions. An example is given to illustrate how these relationships can be used to derive explicit formulas for various stationary queueing characteristics.
Lower bounds for the probability of overload in certain queueing networks
Authors: Kanter Marek
Published: July 2016

Conservative estimates for the probability that open queueing networks are not overloaded during any finite set of time points are derived. The queueing networks have infinitely many servers at each queue and general service times. A key preliminary result is that for multidimensional Markov population processes, the correspondence between initial distributions and their temporal development preserves stochastic order.
An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments
Authors: Berman Simeon M.
Published: July 2016

Let X(t), t≧0, be a Gaussian process with mean 0 and stationary increments. If the incremental variance function σ
2(t) is convex and σ
2(t) = o(t) for t → 0, then P(max[o,t]
X(s) > u) ~ P(X(t) > u) for u → ∞ and each t > 0.
On the moments of some first-passage times
Authors: Stefanov Valeri T.
Published: July 2016

Let {Xt
}
t≧0 (t may be discrete or continuous) be a random process whose finite-dimensional distributions are of exponential type. The first-passage time inf{t:Xt
≧f(t)}, where f(t) is a positive, continuous function, such that f(t)= o(t) as t↑∞, is considered. The problem of finiteness of its moments is solved for both the case that {Xt
}
t≧0 has stationary independent increments as well as the case in which no assumptions are made about stationarity and independence for the increments of the process. Applications to sequential estimation are also given.
On the compound Poisson limit for high level exceedances
Authors: Cohen Jonathan
Published: July 2016

Let ∊
1
∊
2, · ··, be a stationary sequence satisfying the weak long-range dependence condition Δ (un
(τ)) of [3] for every τ > 0, where nP(∊
1
> un
(τ))→ τ
. Assume only that P (there are j exceedances of un
(τ) by ∊
1, ∊
2, · ··, ∊
n) converges for all j with 0≦j≦υ<∞ and a given fixedτ. Then the same holds for every τ> 0. For 0≦j≦υ the limit is P(X = j) where X is compound Poisson and the multiplicity distribution is independent ofτ. These results are extended to more general levels un
and to cases where the joint distribution of the numbers of exceedances of several levels is considered. The limiting distributions of linearly normalized extreme order statistics are derived as a corollary. An application to insurance claim data is discussed.
Analysis of planar anisotropy by means of the Steiner compact
Authors: Rataj Jan , Saxl Ivan
Published: July 2016

A graphical method for the estimation of the anisotropy of planar fibre systems based on the Steiner compact set is proposed and discussed. Upper bounds for the deviation in probability of the graphical estimate of the Steiner compact are given and a consistency theorem is proved.
Autoregressive logistic processes
Authors: Arnold Barry C. , Robertson C. A.
Published: July 2016

A stochastic model is presented which yields a stationary Markov process whose invariant distribution is logistic. The model is autoregressive in character and is closely related to the autoregressive Pareto processes introduced earlier by Yeh et al. (1988). The model may be constructed to have absolutely continuous joint distributions. Analogous higher-order autoregressive and moving average processes may be constructed.
Some results on two-armed bandits when both projects vary
Authors: O'Flaherty Brendan
Published: July 2016

In the multi-armed bandit problem, the decision-maker must choose each period a single project to work on. From the chosen project she receives an immediate reward that depends on the current state of the project. Next period the chosen project makes a stochastic transition to a new state, but projects that are not chosen remain in the same state. What happens in a two-armed bandit context if projects not chosen do not remain in the same state? We derive two sufficient conditions for the optimal policy to be myopic: either the transition function for chosen projects has in a certain sense uniformly stronger stochastic dominance than the transition function for unchosen projects, or both transition processes are normal martingales, the variance of which is independent of the history of process choices.
Some remarks on Brownian motion with drift
Authors: Doney R. A. , Grey D. R.
Published: July 2016

Certain limit theorems due to Berman involve the total time spent by Brownian motion with positive drift below an independent exponentially distributed level. Imhof has calculated the density function and shown that this random variable has two interesting probabilistic properties. We give sample path arguments which explain these two facts.
Tree-dependent extreme values: the exponential case
Authors: Gupta Vijay K. , Mesa Oscar J. , Waymire E.
Published: July 2016

The length of the main channel in a river network is viewed as an extreme value statistic L on a randomly weighted binary rooted tree having M sources. Questions of concern for hydrologic applications are formulated as the construction of an extreme value theory for a dependence which poses an interesting contrast to the classical independent theory. Equivalently, the distribution of the extinction time for a binary branching process given a large number of progeny is sought. Our main result is that in the case of exponentially weighted trees, the conditional distribution of n–1/2
L given M = n is asymptotically distributed as the maximum of a Brownian excursion. When taken with an earlier result of Kolchin (1978), this makes the maximum of the Brownian excursion a tree-dependent extreme value distribution whose domain of attraction includes both the exponentially distributed and almost surely constant weights. Moment computations are given for the Brownian excursion which are of independent interest.
A construction for multi-modal processes, and a potential memory device
Authors: Whittle P.
Published: July 2016

It is shown that if parameters occurring linearly in the transition intensity of a Markov process are replaced by their respective ‘Bayesian estimates' then the new process thus generated has an equilibrium distribution which is a mixture (over parameter values) of the original parametrised equilibrium distribution.

One effectively then has an extra state dependence in that one selects from a given class of transition rules those rules which are most consistent with the value of current state. The effect of this is thus to preserve the status quo, in that unlikely transitions are made even less likely. By this means one can construct processes which show several distinct and metastable modes of behaviour, and which can serve as models for memory devices.
On a relationship between the inverse of a stationary covariance matrix and the linear interpolator
Authors: Bhansali R. J.
Published: July 2016

Let {xt} be a discrete-time multivariate stationary process possessing an infinite autoregressive representation and let ΓB(k), ΓF(k) and Γ be the block Toeplitz covariance matrices ofxB(k) = [x′–1,x′–2, · ··,x′–k]′,xF(k) = [x′1,x′2· ··x′k] andx= [·· ·x′–2,x′–1,x′0,x′1,x′2· ··]′ respectively, wherek≧ 1, is finite or infinite. Also letφm,n(j) and δm,n(u) be the coefficients ofxt+jandxt–urespectively in the linear least-squares interpolator ofxtfromxt+ 1, · ··,xt+m;xt− 1, · ··,xt–n, wherem, n≧ 0, 0 ≦j≦m, 0 ≦u≦nare integers,zt(m, n) denote the interpolation error and τ2(m, n) =E[zt(m, n)zt(m, n)′]. A physical interpretation for the components of ΓB(k)–1, ΓF(k)–1and Γ–1is given by relating these components to theφm,n(j)δm,n(u) andτ2(m, n). A similar result is shown to hold also for the estimators of ΓB(k)–land the interpolation parameters when these have been obtained from a realization of lengthTof {xt}. Some of the applications of the results are considered.
Ordering of repairable systems
Authors: Ebrahimi Nader
Published: July 2016

The problem of how to compare repairable systems has long been of interest to those who are engaged in reliability analysis. In this article we have introduced a type of comparison of two counting processes. Various properties and applications of our stochastic comparison are also studied.
On a new characterization of the exponential distribution related to a queueing system with an unreliable server
Authors: Dimitrov B. , Khalil Z.
Published: July 2016

In this paper we derive a new property of the exponential distribution, closely related to a single-server queueing system with unreliable server. We show that this new property is another characterization of the exponential law.
On the relation between partial balance and insensitivity
Authors: Fakinos D.
Published: July 2016

Direct and simple proofs are given of the equivalence of partial balance over a set of states and the fact that the corresponding equilibrium distribution is insensitive to nominal sojourn time in that set and independent of its past and remaining duration. By a counter-example it is shown that solely insensitivity is not sufficient for partial balance.
Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains
Authors: Kalpazidou S.
Published: July 2016

The asymptotic behaviour of the sequence (𝒞
n
(ω), wc,n
(ω)/n), is studied where 𝒞
n
(ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n
) until time n and wc,n
(ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ω
c
) which represents uniquely the chain (ξ n
) as a circuit chain, and ω c
is given a probabilistic interpretation.
Random mappings with an attracting center: Lagrangian distributions and a regression function
Authors: Berg Sven , Mutafchiev Ljuben
Published: July 2016

For a random mapping model with a single attracting center (Stepanov (1971)) we study the relationship between the sizes of the central tree, the adjacent points, and the free points. Joint, marginal and conditional distributions are shown to be of well-known Lagrangian type. Exact and asymptotic moment properties are investigated with the aid of Riordan's (1968) Abel identities. In particular, the relationship between the size of the central tree and that of the central component is discussed in terms of a regression function.
Families of life distributions characterized by two moments
Authors: Bhattacharjee Manish C. , Sethuraman Jayaram
Published: July 2016

We consider several classical notions of partial orderings among life distributions which have been used to describe ageing properties and tail domination. We show that if a distribution G dominates another distribution F in one of these partial orderings introduced here, and if two moments of G agree with those of F, including the moment that describes this partial ordering, then G = F. This leads to a characterization of the exponential distribution among HNBUE and HNWUE life distribution classes, and thus extends the results of Basu and Bhattacharjee (1984) and rectifies an error in that paper.
Characterization of the exponential distribution by the relevation transform
Authors: Lau Ka Sing , Rao B. L. S. Prakasa
Published: July 2016

A characterization of the exponential distribution was obtained by Grosswald et al. (1980) using the relevation transform introduced by Krakowski (1973). Here we obtain an improved version of the result in Grosswald et al. (1980).
Laplace ordering and its applications
Authors: Alzaid Abdulhamid , Kim Jee Soo , Proschan Frank
Published: July 2016

Two arbitrary life distributions F and G can be ordered with respect to their Laplace transforms. We say is Laplace-smaller than for all s > 0. Interpretations of this ordering concept in reliability, operations research, and economics are described. General preservation properties are presented. Using these preservation results we derive useful inequalities and discuss their applications to M/G/1 queues, time series, coherent systems, shock models and cumulative damage models.
On ‘stop=repeat' servicing for non-exponential queueing networks with blocking
Authors: Van Dijk Nico M.
Published: July 2016

Non-exponential queueing networks with blocking are studied under two types of blocking protocols: the ‘stop' and the ‘repeat' protocol under which services are to be either stopped or repeated respectively upon blocking. The two protocols are shown to be stationary equivalent when the system satisfies a partial balance condition. This condition is related to product-form results. The results are illustrated by closed and open network applications with reversible and nonreversible routing.
Monotonicity property of t-step maintainable structures in three-grade manpower systems: a counterexample
Authors: Guerry M. A.
Published: July 2016

In this paper the t-step maintainable regions Mt
are examined in a three-graded system under the following conditions: the total size of the system remains constant during each intermediate step, demotions do not occur and recruitment control is considered.

A counterexample, showing that the monotonicity property Mt
⊆ Mt

+1 does not exist in general, refutes the conjecture of Davies [3].
The heaps process, libraries, and size-biased permutations
Authors: Donnelly Peter
Published: July 2016

The heaps process (also known as a Tsetlin library) provides a model for a self-regulating filing system. Items are requested from time to time according to their popularity and returned to the top of the heap after use. The size-biased permutation of a collection of popularities is a particular random permutation of those popularities, which arises naturally in a number of applications and is of independent interest. For a slightly non-standard formulation of the heaps process we prove that it converges to the size-biased permutation of its initial distribution. This leads to a number of new characterizations of the property of invariance under size-biased permutation, notably what might be described as invariance under ‘partial size-biasing' of any order. Finally we consider in detail the heaps process with Poisson–Dirichlet initial distribution, exhibiting the tractable nature of its equilibrium distribution and explicitly calculating a number of quantities of interest.
Autoregressive processes and first-hit probabilities for randomized random walks
Authors: Feldman Raisa Epstein
Published: July 2016

We find the first-hit distributions of symmetric randomized random walks on ℝ with exponential lifetime using prediction formulas for Gaussian stationary autoregressive processes, associated with the random walks.
Semi-Pareto processes
Authors: Pillai R. N.
Published: July 2016

Semi-Pareto processes, of which Pareto processes form a proper sub-class, are discussed here. A semi-Pareto process has semi-Pareto inputs. Asymptotic properties of the maximum and minimum of the first n observations are examined as well as the geometric maximum and geometric minimum. A characterization of the semi-Pareto distribution is given. A canonical representation of a special class of Pareto process is also given.
Applied probability—past, present and future
Authors: Kendall David G.
Published: July 2016

After some reminiscences about the founding of the Applied Probability journals the author discusses three open problems concerned respectively with:

(i) the prehistory of applied probability

(ii) Poisson–Delauney Simplexes

(iii) voids and pancakes in the universe.
Invariant measures for Markov chains with no irreducibility assumptions
Authors: Tweedie R. L.
Published: July 2016

Foster's criterion for positive recurrence of irreducible countable space Markov chains is one of the oldest tools in applied probability theory. In various papers in JAP and AAP it has been shown that, under extensions of irreducibility such as ϕ -irreducibility, analogues of and generalizations of Foster's criterion give conditions for the existence of an invariant measure π for general space chains, and for π to have a finite f-moment ∫π (dy)f(y), where f is a general function. In the case f ≡ 1 these cover the question of finiteness of π itself.

In this paper we show that the same conditions imply the same conclusions without any irreducibility assumptions; Foster's criterion forces sufficient and appropriate regularity on the space automatically. The proofs involve detailed consideration of the structure of the minimal subinvariant measures of the chain.

The results are applied to random coefficient autoregressive processes in order to illustrate the need to remove irreducibility conditions if possible.
A generalization of the classical secretary problem: dependent arrival sequences
Authors: Reeves J. H. , Flack V. F.
Published: July 2016

This paper presents new methods for selecting the best population value under the classical secretary problem sequential selection rules. Unlike the classical case, the possible population arrival sequences considered here are not necessarily equally likely. When arrival orders in the sequence are rank dependent, the performance of the optimal solution to the standard problem (totally random arrival order assumed) is very poor, compared with procedures designed to guard against non-uniform arrival orders.
Sums and weighted sums of a gamma Markov sequence
Authors: Phatarfod Ravindra M.
Published: July 2016

We derive the Laplace transforms of sums and weighted sums of random variables forming a Markov chain whose stationary distribution is gamma. Both seasonal and non-seasonal cases are considered. The results are applied to two problems in stochastic reservoir theory.
A model for responding to promotion blockages
Authors: Kalamatianou A. G.
Published: July 2016

A model is proposed for a manpower system in which promotion probabilities are functions of the seniority structure within the grades. The aim is to investigate the adequacy of a strategy which appeared to be being used by the management of the nurses in an Athens hospital. It is shown that the strategy is capable of restoring the system to normal state but that the problem will usually recur.
Sample-path derivations of the excess, age, and spread distributions
Authors: Wolff Ronald W.
Published: July 2016

The familiar limiting distributions of excess, age, and spread are derived on sample paths, where these distributions are fractions of time. Similar results are obtained for the distribution of remaining service at a queue. For stochastic processes, where specified limits exist as finite constants with probability 1, the derived sample-path results are shown to imply that the same limiting distributions hold, where in a stochastic setting, these distributions may be defined in a more conventional way.
Cooperation in a repeated game with random payment function
Authors: Eshel Ilan , Weinshall Daphna
Published: July 2016

A model of cooperation versus defection in a sequence of games is analysed under the assumptions that the rules of the game are randomly changed from one encounter to another, that the decisions are to be made each time anew, according to the (random) rules of the specific local game, and that the result of one such game affects the ability of a player to participate and thus, cooperate in the next game. Under plausible assumptions, it is shown that all Nash solutions of the supergame determine cooperation over a non-degenerate range of rules, determining encounters of the prisoner's dilemma type.
On the general bilinear time series model
Authors: Liu Jian , Brockwell Peter J.
Published: July 2016

A sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations. The condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they consider. Under the condition specified, a solution is constructed which is shown to be causal, stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, our condition reduces to the well-known necessary and sufficient condition for existence of a causal stationary solution.
Moments Of the stationary waiting time in the Gi/Ph/1 queue
Authors: Ramaswami V. , Lucantoni D. M.
Published: July 2016

Recursive relations for computing the higher moments of the stationary waiting time distribution in a stable GI/PH/1 queue are derived. These provide an accurate and stable technique to compute these moments.
The inverses of thinned renewal processes
Authors: Yannaros Nikos
Published: July 2016

The class of inverses of a p-thinned renewal process is considered. It is shown that this class consists of renewal processes. It consists of Cox and renewal processes if and only if the given thinned process is Cox and renewal. In the non-Cox case, there exists a unique top renewal process, which by thinning generates all the possible inverses. Conditions for a renewal process to be a top process are given. Finally, a gamma renewal process is shown to be a top process when α > 1, where α is the shape parameter of the gamma distribution.
A reconciliation of two different expressions for the first-passage density of brownian motion to a curved boundary
Authors: Durbin J.
Published: July 2016

An expression for the first-passage density of Brownian motion to a curved boundary due to Daniels and Lerche is shown to give the same result as a different form due to the author. The equivalence is extended to continuous Gaussian Markov processes.
Some results concerning random arcs on the circle
Authors: Huffer Fred W.
Published: July 2016

Random arcs having random sizes are placed on a circle. Let V be the length of the uncovered portion of the circle and G be the number of uncovered gaps on the circle. Results are presented concerning the joint moments of V and G and the conditional distribution of V given G.
Slepian models for X
2-processes with dependent components with application to envelope upcrossings
Authors: Lindgren Georg
Published: July 2016

A Slepian model for the local behaviour near the level upcrossings of a x
2-process with dependent Gaussian components is presented. In case of independent components, this model is shown to take on a rather simple form, thereby simplifying earlier results by Aronowich and Adler.

The Slepian model is applied to the envelope of a stationary Gaussian process and used to approximate the probability of ‘empty' envelope upcrossings, i.e. the probability that an envelope upcrossing is not followed by a level crossing in the original process.
Ballots, queues and random graphs
Authors: Takács Lajos
Published: July 2016

This paper demonstrates how a simple ballot theorem leads, through the interjection of a queuing process, to the solution of a problem in the theory of random graphs connected with a study of polymers in chemistry. Let Γ
n
(p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where 0 < p < 1. Denote by ρ n
(s) the number of vertices in the union of all those components of Γ
n
(p) which contain at least one vertex of a given set of s vertices. This paper is concerned with the determination of the distribution of ρ n
(s) and the limit distribution of ρ n
(s) as n → ∞and ρ → 0 in such a way that np → a where a is a positive real number.
Monotonicity of throughput in non-Markovian networks
Authors: Tsoucas Pantelis , Walrand Jean
Published: July 2016

Monotonicity of throughput is established in some non-Markovian queueing networks by means of pathwise comparisons. In a series of · /GI/s/N queues with loss at the first node it is proved that increasing the waiting room and/or the number of servers increases the throughput. For a closed network of · /GI/s queues it is shown that the throughput increases as the total number of jobs increases. The technique used for these results does not apply to blocking systems with finite buffers and feedback. Using a stronger coupling argument we prove throughput monotonicity as a function of buffer size for a series of two ·/M/1/N queues with loss and feedback from the second to the first node.
Asymptotic normality of some Graph-Related statistics
Authors: Baldi Pierre , Rinott Yosef
Published: July 2016

Petrovskaya and Leontovich (1982) proved a central limit theorem for sums of dependent random variables indexed by a graph. We apply this theorem to obtain asymptotic normality for the number of local maxima of a random function on certain graphs and for the number of edges having the same color at both endpoints in randomly colored graphs. We briefly motivate these problems, and conclude with a simple proof of the asymptotic normality of certain U-statistics.
A perturbation method for solving some queues with processor sharing discipline
Authors: Sengupta Bhaskar
Published: July 2016

In this paper, we present a perturbation method of solving a second-order difference equation with variable coefficients with some additional assumptions. This method can be used to devise an algorithmic solution for the moments of sojourn times in some processor sharing queues. In particular, we examine three queues. The first has exponential service and a fairly general interrupted arrival mechanism. The second is a cyclic queue. The third is a model for a computer system in which finite and infinite sources interact.
Information flow in discrete Markov systems
Authors: Dawson D. A.
Published: July 2016

Infinite systems of interacting Markov chains are investigated. Some basic concepts of the ergodic theory of such systems are first presented. In particular, the question of the uniqueness and multiplicity of invariant probability measures is considered. In the case of one-dimensional systems, the question is studied in detail by investigating the flow of information throughout the system and some criteria for the uniqueness of invariant probability measures are obtained.
The calculation of limit probabilities for denumerable Markov processes from infinitesimal properties
Authors: Tweedie Richard L.
Published: July 2016

The problem considered is that of estimating the limit probability distribution (equilibrium distribution) πof a denumerable continuous time Markov process using only the matrix Q of derivatives of transition functions at the origin. We utilise relationships between the limit vector πand invariant measures for the jump-chain of the process (whose transition matrix we write P∗), and apply truncation theorems from Tweedie (1971) to P∗. When Q is regular, we derive algorithms for estimating πfrom truncations of Q; these extend results in Tweedie (1971), Section 4, from q-bounded processes to arbitrary regular processes. Finally, we show that this method can be extended even to non-regular chains of a certain type.
An iterated logarithm result for martingales and its application in estimation theory for autoregressive processes
Authors: Heyde C. C.
Published: July 2016

The paper begins with an iterated logarithm law of classical Hartman-Wintner form for stationary martingales. This is then used to obtain iterated logarithm results giving information on rates of convergence of estimators of the parameters in a stationary autoregressive process. In the case of an autoregression of small order, detailed rate results for each autocorrelation and for the estimators of all parameters can be obtained. A rate result for the convergence of the sample mean is given in the general case.
An approximate characterisation of optimal stopping boundaries
Authors: Whittle P.
Published: July 2016

An identity, of the type of Green's equation, is deduced for the loss function of a stopping process. This yields a set of relations determining the optimal (minimal loss) stopping boundary, which do not require simultaneous determination of the loss function in the stopping region. A no-overshoot approximation is invoked, but a bound on the magnitude of the terms neglected is obtained by appeal to a general version of Chernoff's tangency condition at an optimal boundary.
Some results for dams with Markovian inputs
Authors: Phatarfod R. M. , Mardia K. V.
Published: July 2016

The paper considers the dam problem with Markovian inputs, with special reference to the serial correlation coefficient of the input process. An input model is proposed which by giving particular values to the parameters makes the stationary distribution of the inputs one of the standard discrete distributions. The probabilities of first emptiness before overflow are first obtained by using the Markovian analogue of Wald's Identity. From these, the stationary distributions of the dam content are obtained by a duality argument. Both, finite and infinite dams are considered.
Erlang's formula and some results on the departure process for a loss system
Authors: Shanbhag D. N. , Tambouratzis D. G.
Published: July 2016

The present paper investigates the limiting distribution of the number of busy channels (queue size) and the remaining lengths of holding times at an epoch of departure for a loss system with general holding times and exponentially distributed interarrival times. Further, it is established that for this loss system in the limit an interdeparture interval length is independent of the queue size at the end of the interval and is distributed according to an exponential distribution with mean λ–1. It is also seen that in the limit interdeparture times are mutually independent.
Asymptotic behavior of near-critical multitype branching processes
Authors: Klebaner Fima C.
Published: July 2016

Sufficient conditions for survival and extinction of multitype population-size-dependent branching processes in discrete time are obtained. Growth rates are determined on the set of divergence to infinity. The limiting distribution of a properly normalized process can be generalized gamma, normal or degenerate.
On the properties of processes associated with a markov branching process
Authors: Gadag V. G. , Gupta R. P.
Published: July 2016

Consider a time-homogeneous Markov branching process. We construct reduced processes, based on whether the length of line of descent of particles of this process are (a) greater than or (b) at most equal to, τ units of time, for some fixed τ ≧ 0. We show that in both cases the reduced processes retain the branching property, but the latter does not retain the time homogeneity. We investigate finite-time and asymptotic properties of the reduced processes. Based on a realization of the original process and a realization of a reduced process, observed continuously over a time interval [0, T] for T > 0, we propose estimators for the different parameters involved, including qτ
, the probability that the original process becomes extinct before τ units of time, and f
(j)(qτ
), the jth derivative of the offspring probability generating function f(s) at q
τ when q
τ is known. We study the properties of these estimators and derive their asymptotic distributions, under the assumption that the original process is supercritical.
Asymptotic properties of intensity estimators for Poisson shot-noise processes
Authors: Liese Friedrich , Schmidt Volker
Published: July 2016

Stochastic processes {X(t)} of the form X(t) = Σ n f(t – Tn
) are considered, where {Tn
} is a stationary Poisson point process with intensity λ and f: R → R is an unknown response function. Conditions are obtained for weak consistency and asymptotic normality of estimators of λ based on long-run observations of {X(t)}.
Dynamic multivariate mean residual life functions
Authors: Shaked Moshe , Shanthikumar J. George
Published: July 2016

In this paper we introduce and study a dynamic notion of mean residual life (mrl) functions in the context of multivariate reliability theory. Basic properties of these functions are derived and their relationship to the multivariate conditional hazard rate functions is studied.

A partial ordering, called the mrl ordering, of non-negative random vectors is introduced and its basic properties are presented. Its relationship to stochastic ordering and to other related orderings (such as hazard rate ordering) is pointed out. Using this ordering it is possible to introduce a weak notion of positive dependence of random lifetimes. Some properties of this positive dependence notion are given.

Finally, using the mrl ordering, a dynamic notion of multivariate DMRL (decreasing mean residual life) is introduced and studied. The relationship of this multivariate DMRL notion to other notions of dynamic multivariate aging is highlighted in this paper.
Estimation of the density in a stationary Boolean model
Authors: Schmitt Michel
Published: July 2016

A new and practically tractable formula for estimating the intensity of the underlying Poisson process of a Boolean model is given, assuming only almost sure boundedness of the primary grain.
A distributed clustering process
Authors: Coffman E. G. , Courtois P.-J. , Gilbert E. N. , Piret Ph.
Published: July 2016

The points of a graph G will form clusters as a result of a flow process. Initially, points i of G own resources xi
which are i.i.d. random real numbers. Afterwards, resources flow between points, but always from a point to a neighbor that has accumulated a larger total resource. Thus points with small resource tend to lose it and points with large resource tend to gain. Eventually the flow stops with only two kinds of points, nulls with no resource left and absorbers with such large resource that no neighbor can take it. The final resource at an absorber is a sum of certain initial resources xi
, and the corresponding points i form one cluster. Analytical results are obtainable when G is the chain of integer points on the line. Probability distributions are derived for the distance between consecutive absorbers and the size of a cluster. Indeed these distributions do not involve the given distribution for the xi. The Laplace transform of the distribution of final resources at absorbers is derived but the distribution itself is obtained by a simulation. For general graphs G only partial results are obtained.
Minimizing the learning loss in adaptive control of Markov chains under the weak accessibility condition
Authors: Agrawal Rajeev
Published: July 2016

We consider the adaptive control of Markov chains under the weak accessibility condition with a view to minimizing the learning loss. A certainty equivalence control with a forcing scheme is constructed. We use a stationary randomized control scheme for forcing and compute a maximum likelihood estimate of the unknown parameter from the resulting observations. We obtain an exponential upper bound on the rate of decay of the probability of error. This allows us to choose the rate of forcing appropriately, whereby we achieve a o(f(n) log n) learning loss for any function as .
Some renewal-theoretic investigations in the theory of sojourn times in finite semi-Markov processes
Authors: Csenki Attila
Published: July 2016

In this note, an irreducible semi-Markov process is considered whose finite state space is partitioned into two non-empty sets A and B. Let MB
(t) stand for the number of visits of Y to B during the time interval [0, t], t > 0. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB
(t) are easily expressible. The theory is applied to a small reliability model in conjunction with a Tauberian argument to evaluate the behaviour of the first two moments of MB
(t) as t →∞.
An optimal repairable replacement model for deteriorating systems
Authors: Yeh Lam
Published: July 2016

In this paper, we study a repair replacement model for a stochastically deteriorating system. For the expected discounted reward case, we show that the optimal replacement policy is of the form ‘replace at the time of the Nth failure'.
Stochastic comparisons of density profiles for the road-hog process
Authors: Srinivasan Rengarajan
Published: July 2016

We consider the asymmetric simple exclusion process which starts from a product measure such that all the sites to the left of zero (including zero) are occupied and the right of 0 (excluding 0) are empty. We label the particle initially at 0 as the leading particle. We study the long-term behaviour of this process near large sites when the leading particle's holding time is different from that of the other particles. In particular, we assume that the leading particle moves at a slower rate than the other particles. We call this modified asymmetric simple exclusion process the road-hog process. Coupling and stochastic ordering techniques are used to derive the density profile of this process. Road-hog processes are useful in modelling series of exponential queues with Poisson and non-Poisson input process. The density profiles dramatically illustrate the flow of customers through the queues.
A note on the selection of random variables under a sum constraint
Authors: Rhee Wansoo , Talagrand Michel
Published: July 2016

Consider an i.i.d. sequence of non-negative random variables (X
1, · ··, Xn
) with known distribution F. Consider decision rules for selecting a maximum number of the subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected. Coffman et al. (1987) proved that there exists such a rule that maximizes the expected number En
(c) of variables selected, and determined the asymptotics of En
(c) for special distributions. Here we determine the asymptotics of En
(cn
) for very general choices of sequences (cn
) and of F, by showing that En
(c) is very close to an easily computable number. Our proofs are (somewhat deceptively) very simple, and rely on an appropriate stopping-time argument.
On a characterization of the exponential distribution
Authors: Van Harn K. , Steutel F. W.
Published: July 2016

The characterization of the exponential distribution as given by Dimitrov and Khalil (1990) is improved by substantially weakening the conditions.
Limit distributions for compounded sums of extreme order statistics
Authors: Beirlant Jan , Teugels Jozef L.
Published: July 2016

Let X
(1) ≦ X
(2) ≦ ·· ·≦ X
(N(t)) be the order statistics of the first N(t) elements from a sequence of independent identically distributed random variables, where {N(t); t ≧ 0} is a renewal counting process independent of the sequence of X's. We give a complete description of the asymptotic distribution of sums made from the top kt
extreme values, for any sequence kt
such that kt
→ ∞, kt
/t → 0 as t → ∞. We discuss applications to reinsurance policies based on large claims.