bluuebunny/crossref_metadata_2025 huggingface dataset:
On platoon formation on two-lane roads
Authors: Barzily Zeev , Rubinovitch Michael
Published: July 2016

An approximate model for the study of platoon formation on two-lane highways is discussed in detail. The model assumes that the two-lane highway is divided in each traffic direction into alternating road sections of fixed lengths. The passing in one type of section is unrestricted and the passing in the other one is prohibited. It is assumed that there are slow and fast vehicles on the highway and that inputs follow independent Poisson processes. The results include the distribution of the number of vehicles in a platoon and the average speed of a typical fast vehicle.
The front velocity of the simple epidemic
Authors: Wierman John C.
Published: July 2016

The propagation rate of the one-dimensional stochastic simple epidemic converges almost surely to a front velocity for the epidemic. Percolation methods are used in the proof.
Rates of Poisson convergence for U-statistics
Authors: Brown T. C. , Silverman B. W.
Published: July 2016

Poisson limit theorems for U-statistics are studied. A general rate of convergence is obtained; this rate is improved for the special case where the U-statistic arises from the consideration of distances between uniformly distributed points in a well-behaved plane region.
On best fractional linear generating function bounds
Authors: Hwang Tea-Yuan , Wang Nae-Sheng
Published: July 2016

Under weak conditions, this paper provides a best lower and a best upper bounding fractional linear generating function for any probability generating function when they have the same mean. These bounds can be used to obtain bounds for the expectation and the percentiles of the extinction-time distribution of a Galton-Watson branching process and other parameters of interest. For the special case of the four points probability generating function, the best bounds obtained are better than the bounds derived by Agresti (1974).
Extinction probability for critical age-dependent branching processes with generation dependence
Authors: Fearn Dean H.
Published: July 2016

The limiting behavior of the probability of extinction of critical age-dependent branching processes with generation dependence is obtained using Goldstein's methods. Regularity conditions on the mean and variance of the birth distributions are assumed. Also the lifespan distribution is assumed to satisfy suitable regularity conditions.
Non-linear time series models for non-linear random vibrations
Authors: Ozaki T.
Published: July 2016

Non-linear time series models for non-linear vibrations are presented. Some typical behaviour of non-linear vibrations generated from Duffing's equation or van der Pol's equation are explained through the models.
Approximations to densities in geometric probability
Authors: Solomon H. , Stephens M. A.
Published: July 2016

Many random variables arising in problems of geometric probability have intractable densities, and it is very difficult to find probabilities or percentage points based on these densities. A simple approximation, a generalization of the chi-square distribution, is suggested, to approximate such densities; the approximation uses the first three moments. These may be theoretically derived, or may be obtained from Monte Carlo sampling.

The approximation is illustrated on random variables (the area, the perimeter, and the number of sides) associated with random polygons arising from two processes in the plane. Where it can be checked theoretically, the approximation gives good results. It is compared also with Pearson curve fits to the densities.
Transition probability matrices for correlated random walks
Authors: Nain R. B. , Sen Kanwar
Published: July 2016

For correlated random walks a method of transition probability matrices as an alternative to the much-used methods of probability generating functions and difference equations has been investigated in this paper. To illustrate the use of transition probability matrices for computing the various probabilities for correlated random walks, the transition probability matrices for restricted/unrestricted one-dimensional correlated random walk have been defined and used to obtain some of the probabilities.
An autoregressive model for multilag Markov chains
Authors: Pegram G. G. S.
Published: July 2016

By assembling the transition matrix of a finite discrete Markov chain from overlays of matrices which are defined only by the serial correlation coefficients and marginal distribution of the chain to be modelled, a considerable saving is made in the number of parameters required to define a multilag Markov chain. This parsimony is achieved without detriment to the marginal distribution or serial correlation structure of the modelled chain. Applications to daily precipitation sequences and reservoir reliability are outlined to demonstrate the model's versatility.
A limit theorem for sample maxima and heavy branches in Galton–Watson trees
Authors: O'brien G. L.
Published: July 2016

Let Yn
be the maximum of n independent positive random variables with common distribution function F and let Sn
be their sum. Then converges to zero in probability if and only if is slowly varying. This result implies that in a supercritical Galton-Watson process which does not become extinct, there cannot be a sequence {τ n
} of particles, each descended from the preceding one, such that the fraction of all particles which are descendants of τ n
does not converge to zero as n →∞. Weakly m-adic trees, which behave to some extent like sample Galton-Watson trees, can have such sequences of particles.
Inference for the diffusion branching process
Authors: Brown B. M. , Hewitt J. I.
Published: July 2016

For the diffusion branching process, we consider a method of inference that is essentially sequential in nature. The method allows us to simplify the natural sufficient statistics involved, and we are able to get their distributions quite easily by translating our problem into a standard problem in Brownian motion. Under certain circumstances, we are left with a complete sufficient statistic whose distribution belongs to an exponential family, and can therefore derive minimum variance unbiased estimators, etc.
Some mean first-passage time approximations for the Ornstein-Uhlenbeck process
Authors: Thomas Marlin U.
Published: July 2016

This paper describes an accurate method of approximating the mean of the first-passage time distribution for an Ornstein-Uhlenbeck process with a single absorbing barrier. The accuracy of the approximation is demonstrated through some numerical comparisons.
A secretary problem with uncertain employment
Authors: Smith M. H.
Published: July 2016

A ‘Secretary Problem’ with no recall but which allows the applicant to refuse an offer of employment with a fixed probability 1 – p, (0 < p < 1), is considered. The optimal stopping rule and the maximum probability of employing the best applicant are derived.
An ergodic theorem for evolution in a random environment
Authors: Norman M. Frank
Published: July 2016

Let w
1, w
12 and w
2 be the fitnesses of genotypes A
1
A
1, A
1
A
2 and A
2
A
2 in an infinite diploid population, and let pn
be the A
1, gene frequency in the nth generation. If fitness varies independently from generation to generation, then pn
is a Markov process with a continuum of states. If E[In(wi
/w
12)] < 0 for i = 1, 2, then there is a unique stationary probability, and the distribution of pn
converges to it as n → ∞.
The law of the iterated logarithm for Brownian sheets
Authors: Park W. J.
Published: July 2016

Strassen-type law of the iterated logarithm for Brownian sheets presented by Pyke [7] is proved by using recent results of Kuelbs and Lepage [4]: the law of the iterated logarithm for Brownian motion in a Banach space and some applications are given.
On the first-exit time problem for temporally homogeneous Markov processes
Authors: Tuckwell Henry C.
Published: July 2016

Using an integral equation of Darling and Siegert in conjunction with the backward Kolmogorov equation for the transition probability density function, recurrence relations are derived for the moments of the time of first exit of a temporally homogeneous Markov process from a set in the phase space. The results, which are similar to those for diffusion processes, are used to find the expectation of the time between impulses of a Stein model neuron.
A convergence theorem for parities in a birth-and-death process
Authors: Saunders I. W.
Published: July 2016

The parity of an individual in a population process is the number of offspring that individual has produced. If Nk
(t) is the number of individuals of parity k at time t in a birth-and-death process and we show that, conditional on N(t)→ ∞,
A stochastic model for polymer degradation
Authors: Srinivasan S. K. , Mehata K. M.
Published: July 2016

The stochastic model for breaking of molecular segments proposed by Bithell is analysed and some results relating to the distribution of the number of fragments are obtained by using a slightly more general model which allows multiple ruptures. The product density technique is employed to derive the mean and mean square number of segments at any time t and the number of segments with length greater than y at time of production.
Physical nearest-neighbour models and non-linear time-series. II Further discussion of approximate solutions and exact equations
Authors: Bartlett M. S.
Published: July 2016

The approximate two- and three-dimensional solutions for spatial correlations, using the non-linear time-series approach for nearest-neighbour systems developed in my previous paper, are further discussed.

Orthogonal expansions for the correlation functions are also developed which determine with this approach, though so far only in principle, the exact solutions.
The rate of convergence in limit theorems for service systems with finite queue capacity
Authors: Tomko Joseph
Published: July 2016

The paper deals with the asymptotic analysis of waiting time distribution for service systems with finite queue capacity. First an M/M/m system is considered and the rate of approximation is given. Then the case of the M/G/1 system is studied for traffic intensity ρ > 1. In the last section a condition is given under which an estimate can be derived for the remainder term in central limit theorems for randomly stopped sums.
Traffic intersection control and zero-switch queues under conditions of Markov chain dependence input
Authors: Lehoczky J. P.
Published: July 2016

A model for a vehicle controlled intersection based on a zero-switch queueing system is introduced and analyzed under Markov chain dependence input. Conditions for asymptotic stability are derived. Under these conditions the moments of the cycle lengths are derived. The expected area beneath the sample paths given by the queue sizes is computed. This area represents total delay to all cars, and is a measure of the effectiveness of the control device.
A model for interaction of a Poisson and a renewal process and its relation with queuing theory
Authors: Råde Lennart
Published: July 2016

This paper discusses the response process when a Poisson process interacts with a renewal process in such a way that one or more points of the Poisson process eliminate a random number of consecutive points of the renewal process. A queuing situation is devised such that the c.d.f. of the length of the busy period is the same as the c.d.f. of the length of time intervals of the renewal response process. The Laplace-Stieltjes transform is obtained and from this the expectation of the time intervals of the response process is derived. For a special case necessary and sufficient conditions for the response process to be a Poisson process are found.
An optimal isolation policy for an epidemic
Authors: Abakuks Andris
Published: July 2016

Policies of isolating infectives in the general stochastic epidemic are considered. With costs assigned to the infection and isolation of individuals, an optimal policy is found, which at any stage minimises the expected future cost. An optimal policy is also found for the general deterministic epidemic and the two policies are compared. Finally, some numerical examples are provided.
On dams with Markovian inputs
Authors: Pakes A. G.
Published: July 2016

Some recent work on discrete time dam models has been concerned with special cases in which the input process is a Markov chain whose transition probabilities, p
ij
, are given by where A(·) and B(·) are probability generating functions (p.g.f.'s). In this paper we obtain some results for the general situation. The convergence norm of the matrix [p
ij
xj] is found and the results are used to obtain the p.g.f. of the first emptiness time. Distributions of the dam content are obtained and conditions are found for the existence of their limits. The p.g.f. of this distribution is so complicated that its identification in any special case is extremely difficult, or even impossible. Thus useful approximations are needed; we obtain a ‘heavy traffic’ limit theorem which suggests that under certain circumstances the limiting distribution can be approximated by an exponential distribution.
Traffic noise as a filtered Markov renewal process
Authors: Marcus Allan H.
Published: July 2016

Traffic noise depends significantly on statistical properties of highway flow. The noise heard by an off-highway observer is the sum of the contributions of all the vehicles on the highway. We assume that vehicle spacings on a single-lane infinite straight road form a Markov renewal process (MRP) with N states or vehicle types. The noise impact is then a two-sided filtered MRP. We give explicit formulae for the mean and variance in the case N = 2 and exponential headways. Jewell's (1965) study (cars vs. trucks in the southbound curb lane of U.S. 40) gives parameters of the MPR for a numerical example. Even with only a modest degree of truck clustering and variability in vehicle spacings and noise emission, the variation in noise level is much greater than usually predicted. The coefficient of variation of arithmetic noise intensity is greater than unity at distances from the roadway less than 700 feet. The logarithmic noise level parameters L
50 and L
10 usually computed in engineering practice may be in error by 2 to 3 decibels if these sources of variability are ignored.
Almost sure stability of maxima
Authors: Resnick Sidney I. , Tomkins R. J.
Published: July 2016

For random variables {Xn, n≧ 1} unbounded above setMn= max {X1,X2, …,Xn}. When do normalizing constantsbnexist such thatMn/bn→1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for alland in this casebn∼F–1(1 – 1/n) Necessary and sufficient conditions for lim supn→∞,Mn/bn= l >1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn= 1 a.s. except whenl= 1. When theXnare r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.
Simpler conditions for Wald equations
Authors: Brown B. M. , Eagleson G. K.
Published: July 2016

If T is a stopping time on a martingale {St
}, the mth order moments of T and ST (for integers m ≧ 2) are related by an identity known as a Wald equation of order m. Wald equations hold under certain conditions, which are simplified in this paper by showing that among the set of previously known conditions, some are implied by the others.
The divided pooled output of several strictly-periodic point processes
Authors: Proudfoot A. D. , Lampard D. G.
Published: July 2016

The coefficients of variation and serial correlation are given for the inter-event intervals of the process obtained by selecting every kth event of the pooled output of several periodic point processes. Relevance to physiological research is explained.
Labelling experiments and phase-age distributions for multiphase branching processes
Authors: Brockwell P. J. , Kuo W. H.
Published: July 2016

A supercritical age-dependent branching process is considered in which the lifespan of each individual is composed of four phases whose durations have joint probability density f(x
1, x
2, x
3, x
4). Starting with a single individual of age zero at time zero we consider the asymptotic behaviour as t→ ∞ of the random variable Z
(4) (a
0,…, a

n
, t) defined as the number of individuals in phase 4 at time t for which the elapsed phase durations Y
01,…, Y
04,…, Yi

1,…, Yi

4,…, Yn

4 of the individual itself and its first n ancestors satisfy the inequalities Yij
≦ aij
, i = 0,…, n, j = 1,…, 4. The application of the results to the analysis of cell-labelling experiments is described. Finally we state an analogous result which defines (conditional on eventual non-extinction of the population) the asymptotic joint distribution of the phase and elapsed phase durations of an individual drawn at random from the population and the phase durations of its ancestors.
Optimal issuing policies under stochastic field lives
Authors: Brown Mark , Solomon Herbert
Published: July 2016

We consider a stockpile in which each item has a stochastic field life with known distribution, this distribution changing in time in a known deterministic fashion. The problem is to order the items for issuance to the field with the objective of maximizing the total field life. Under monotone likelihood ratio assumptions on the field life distributions, and convexity and concavity assumptions on the change in time of the distributions, we show that LIFO and FIFO policies possess a strong optimality property.
Expectations and variances of stopping variables in sequential selection processes
Authors: Henke M.
Published: July 2016

A sequential stochastic decision process with independent random variables is considered in which the decision maker selects a chance with a certain probability at each time period or at random times. If the decision maker has selected m chances, the process has to be stopped. The expectation and the variance of the stopping variable are determined for a finite and an infinite decision horizon.
An extension of a theorem concerning an interesting Markov chain
Authors: Hendricks W. J.
Published: July 2016

In a single-shelf library of N books we suppose that books are selected one at a time and returned to the kth position on the shelf before another selection is made. Books are moved to the right or left as necessary to vacate position k. The probability of selecting each book is assumed to be known, and the N! arrangements of the books are considered as states of an ergodic Markov chain for which we find the stationary distribution.
A note on functions of Markov processes with an application to a sequence of χ2 statistics
Authors: Cameron Murray A.
Published: July 2016

A sufficient condition for a function of a Markov process to be Markovian is obtained by considering a reverse process of the original Markov process. An application of this result provides a simple derivation of the joint distribution of a sequence of Pearson χ
2 statistics previously obtained by Zaharov, Sarmanov and Sevast'ianov (1969).
Recognizing the maximum of a random sequence based on relative rank with backward solicitation
Authors: Yang Mark C. K.
Published: July 2016

The classical secretary problem is generalized to admit stochastically successful procurement of previous interviewees, but each has a certain probability of refusing the offer. A general formula for solving this problem is obtained. Two special cases: constant probability of refusing and geometric probability of refusing are discussed in detail. The optimal stopping rules in these two cases turn out to be simple.
The generalised state-dependent queue: the busy period Erlangian
Authors: Conolly B. W.
Published: July 2016

A continued fraction representation is presented of the Laplace transform of the generating function of the fundamental joint probability and density of busy period length measured in customers served and duration in time. The setting is the single server Erlang queueing system where the parameters of negative exponentially distributed arrival and service times have a general dependence on instantaneous system state.
Analysis of transient behaviour of certain processes with return to a central state
Authors: Buckholtz Peter G. , Campbell L. Lorne , Milbourne Ross D. , Wasan M. T.
Published: July 2016

In economics, cash management problems may be modelled by birth-death processes which reset to central states when a boundary is reached. The nature of the transient behaviour of the probability distribution of such processes symmetric about a central state is investigated. A diffusion approximation of such processes is given and the transient probability behaviour derived from the diffusion equation.
Series expansions of probability generating functions and bounds for the extinction probability of a branching process
Authors: Daley D. J. , Narayan Prakash
Published: July 2016

In the Taylor series expansion about s = 1 of the probability generating function f(s) of a non-negative integer-valued random variable with finite nth factorial moment the remainder term is proportional to another p.g.f. This leads to simple proofs of other power series expansions for p.g.f.'s, including an inversion formula giving the distribution in terms of the moments (when this can be done). Old and new inequalities for the extinction probability of a branching process are established.
Optimality conditions for a Markov decision chain with unbounded costs
Authors: Robinson D. R.
Published: July 2016

It is known that when costs are unbounded satisfaction of the appropriate dynamic programming ‘optimality' equation by a policy is not sufficient to guarantee its average optimality. A ‘lowest-order potential' condition is introduced which, along with the dynamic programming equation, is sufficient to establish the optimality of the policy. Also, it is shown that under fairly general conditions, if the lowest-order potential condition is not satisfied there exists a non-memoryless policy with smaller average cost than the policy satisfying the dynamic programming equation.
On a probabilistic analogue of the Fibonacci sequence
Authors: Heyde C. C.
Published: July 2016

One of the earliest population models to be studied gives rise to the Fibonacci sequence and has a history dating back more than 750 years. A stochastic version of the model is discussed in this paper, its basic defining property being E(Xn | Xn

−1, · ··, X
0) = Xn

−1 + Xn

−2 a.s. The process {Xn
} mimics many of the standard properties of the Fibonacci sequence. In particular, under mild additional conditions, a.s. as n → where α is the ‘golden ratio'
First-order stochastic chemical reactions and oscillations in the variance
Authors: Hertzberg Richard C. , Gallucci Vincent F.
Published: July 2016

The general solution of a Markov model for first-order kinetics is developed as a sum of independent, multinomially distributed random processes. Fluctuations in the mean and variance functions are discussed and shown to be unrelated in time during the early phase of the reaction. Numerical examples are presented for two- and three-component systems.
Moment computations for subcritical branching processes
Authors: Lange Kenneth , Boehnke Michael , Carson Richard
Published: July 2016

The present paper considers methods for computing moments connected with subcritical multitype branching processes. By stressing the interplay between moment and cumulant structure, it is possible to prove simultaneously the existence of the moments. The kinds of moments considered are those arising in (1) times to extinction, (2) ultimate numbers of particles starting from a single particle, and (3) equilibrium numbers of particles for subcritical processes experiencing immigration.
Dispersive distributions, and the connection between dispersivity and strong unimodality
Authors: Lewis Toby , Thompson J. W.
Published: July 2016

Two continuous distributions, G, H so related that any two quantiles of H are more widely separated than the corresponding quantiles of G may be said to be ‘ordered in dispersion'; Saunders and Moran have given examples. It is shown here that distributions F (called ‘dispersive' distributions) exist, e.g. the exponential, such that if G, H are ordered in dispersion then so also are the convolutions F ∗G, F ∗H. The class of dispersive distributions is determined, and shown to coincide with the class of strongly unimodal distributions.
Cluster shock models
Authors: Thall Peter F.
Published: July 2016

The survival distribution of a device subject to a sequence of shocks occurring randomly over time is studied by Esary, Marshall and Proschan (1973) and by A-Hameed and Proschan (1973), (1975). The present note treats the case in which shocks occur according to a homogeneous Poisson cluster process. It is shown that if[the device surviveskshocks] =zk, 0 <z< 1, then the device exhibits a decreasing failure rate. A DFR preservation theorem is proved for completely monotonic. A counterexample to the IFR preservation theorem is given in whichis strictly IFR while the failure rate is initially decreasing and then increasing.
Extending the correlation structure of exponential autoregressive–moving-average processes
Authors: McKenzie Ed
Published: July 2016

Some recent constructions for the generation of dependent sequences of identically distributed negative exponential random variables with specific correlation structures are generalized. This is achieved by attributing a correlation structure to the binary sequence which controls the generation of the exponentials. The procedure causes the autocorrelation function of the exponential sequence to copy that of the binary sequence and thus be extended to include negative values and other values beyond the usual range.
Sticky Brownian motion as the limit of storage processes
Authors: Harrison J. Michael , Lemoine Austin J.
Published: July 2016

The paper considers a modified storage processwith state space [0,∞). Away from the origin,Wbehaves like an ordinary storage process with constant release rate and finite jump intensity A. In state 0, however, the jump intensity falls toIt is shown thatWcan be obtained by applying first a reflection mapping and then a random change of time scale to a compound Poisson process with drift. When these same two transformations are applied to Brownian motion, one obtains sticky (or slowly reflected) Brownian motionW∗on [0,∞). ThusW∗is the natural diffusion approximation forW, and it is shown thatWconverges in distribution toW∗under appropriate conditions. The boundary behavior ofW∗is discussed, its infinitesimal generator is calculated and its stationary distribution (which has an atom at the origin) is computed.
Weak convergence of the simple birth-and-death process
Authors: Ivan Constantin
Published: July 2016

The existence of a weak limit birth-and-death process on the natural integers for the simple birth-and-death process conditional on non-extinction up to time t as t→∞ is proved. Starting from the latter a new weak limiting procedure yields a diffusion Markov process on the positive infinite semi-axis.
Normal approximations for distributions arising in the stochastic approach to chemical reaction kinetics
Authors: Dunstan F. D. J. , Reynolds J. F.
Published: July 2016

Earlier stochastic analyses of chemical reactions have provided formal solutions which are unsuitable for most purposes in that they are expressed in terms of complex algebraic functions. Normal approximations are derived here for solutions to a variety of reactions. Using these, it is possible to investigate the level at which the classical deterministic solutions become inadequate. This is important in fields such as radioimmunoassay.
Monte Carlo simulation of the renewal function
Authors: Brown Mark , Solomon Herbert , Stephens Michael A.
Published: July 2016

The problem of Monte Carlo estimation of M(t) = EN(t), the expected number of renewals in [0, t] for a renewal process with known interarrival time distribution F, is considered. Several unbiased estimators which compete favorably with the naive estimator, N(t), are presented and studied. An approach to reduce the variance of the Monte Carlo estimator is developed and illustrated.
Non-linear threshold autoregressive models for non-linear random vibrations
Authors: Ozaki Tohru
Published: July 2016

Time series models for non-linear random vibrations are discussed from the viewpoint of the specification of the dynamics of the damping and restoring force of vibrations, and a non-linear threshold autoregressive model is introduced. Typical non-linear phenomena of vibrations are demonstrated using the models. Stationarity conditions and some structural aspects of the model are briefly discussed. Applications of the model in the statistical analysis of real data are also shown with numerical results.
A note on filtered Markov renewal processes
Authors: Andersen Per Kragh
Published: July 2016

A Markov renewal theorem necessary for the derivation of the moment formulas for a filtered Markov renewal process stated by Marcus (1974) is proved and its applications are outlined.
Ruin excursions, the G/G/∞ queue, and tax payments in renewal risk models
Authors: Albrecher Hansjörg , Borst Sem C. , Boxma Onno J. , Resing Jacques
Published: July 2016

In this paper we investigate the number and maximum severity of the ruin excursion of the insurance portfolio reserve process in the Cramér–Lundberg model with and without tax payments. We also provide a relation of the Cramér–Lundberg risk model with the G/G/∞ queue and use it to derive some explicit ruin probability formulae. Finally, the renewal risk model with tax is considered, and an asymptotic identity is derived that in some sense extends the tax identity of the Cramér– Lundberg risk model.
Characterization of tails through hazard rate and convolution closure properties
Authors: Bardoutsos Anastasios G. , Konstantinides Dimitrios G.
Published: July 2016

We use the properties of the Matuszewska indices to show asymptotic inequalities for hazard rates. We discuss the relation between membership in the classes of dominatedly or extended rapidly varying tail distributions and corresponding hazard rate conditions. Convolution closure is established for the class of distributions with extended rapidly varying tails.
A large deviation principle for Minkowski sums of heavy-tailed random compact convex sets with finite expectation
Authors: Mikosch Thomas , Pawlas Zbyněk , Samorodnitsky Gennady
Published: July 2016

We prove large deviation results for Minkowski sums S

n
of independent and identically distributed random compact sets where we assume that the summands have a regularly varying distribution and finite expectation. The main focus is on random convex compact sets. The results confirm the heavy-tailed large deviation heuristics: ‘large’ values of the sum are essentially due to the ‘largest’ summand. These results extend those in Mikosch, Pawlas and Samorodnitsky (2011) for generally nonconvex sets, where we assumed that the normalization of S

n
grows faster than n.
On the prediction of fractional Brownian motion
Authors: Gripenberg Gustaf , Norros Ilkka
Published: July 2016

Integration with respect to the fractional Brownian motionZwith Hurst parameteris discussed. The predictoris represented as an integral with respect toZ,solving a weakly singular integral equation for the prediction weight function.
A note on the point processes of rare events
Authors: Hüsler J. , Schmidt M.
Published: July 2016

We discuss the limits of point processes which are generated by a triangular array of rare events. Such point processes are motivated by the exceedances of a high boundary by a random sequence since exceedances are rare events in this case. This application relates the problem to extreme value theory from where the method is used to treat the asymptotic approximation of these point processes. The presented general approach extends, unifies and clarifies some of the various conditions used in the extreme value theory.
Time series models with univariate margins in the convolution-closed infinitely divisible class
Authors: Joe Harry
Published: July 2016

A unified way of obtaining stationary time series models with the univariate margins in the convolution-closed infinitely divisible class is presented. Special cases include gamma, inverse Gaussian, Poisson, negative binomial, and generalized Poisson margins. ARMA time series models obtain in the special case of normal margins, sometimes in a different stochastic representation. For the gamma and Poisson margins, some previously defined time series models are included, but for the negative binomial margin, the time series models are different and, in several ways, better than previously defined time series models. The models are related to multivariate distributions that extend a univariate distribution in the convolution-closed infinitely divisible class. Extensions to the non-stationary case and possible applications to modelling longitudinal data are mentioned.
Economies of scale in queues with sources having power-law large deviation scalings
Authors: Duffield N. G.
Published: July 2016

We analyse the queue QL
at a multiplexer with L sources which may display long-range dependence. This includes, for example, sources modelled by fractional Brownian motion (FBM). The workload processes W due to each source are assumed to have large deviation properties of the form P[W
t/a(t) > x] ≈ exp[– v(t)K(x)] for appropriate scaling functions a and v, and rate-function K. Under very general conditions lim
L→x

L
–1 log P[QL > Lb] = – I(b), provided the offered load is held constant, where the shape function I is expressed in terms of the cumulant generating functions of the input traffic. For power-law scalings v(t) = tv, a(t) = ta
(such as occur in FBM) we analyse the asymptotics of the shape function lim
b→x

b
–u/a
(I(b) – δbv/a
) = vu
for some exponent u and constant v depending on the sources. This demonstrates the economies of scale available though the multiplexing of a large number of such sources, by comparison with a simple approximation P[QL
> Lb] ≈ exp[−δLbv/a
] based on the asymptotic decay rate δ alone. We apply this formula to Gaussian processes, in particular FBM, both alone, and also perturbed by an Ornstein–Uhlenbeck process. This demonstrates a richer potential structure than occurs for sources with linear large deviation scalings.
Networks of queues with batch services and customer coalescence
Authors: Chao Xiuli , Pinedo Michael , Shaw Dequan
Published: July 2016

Consider a queueing network with batch services at each node. The service time of a batch is exponential and the batch size at each node is arbitrarily distributed. At a service completion the entire batch coalesces into a single unit, and it either leaves the system or goes to another node according to given routing probabilities. When the batch sizes are identical to one, the network reduces to a classical Jackson network. Our main result is that this network possesses a product form solution with a special type of traffic equations which depend on the batch size distribution at each node. The product form solution satisfies a particular type of partial balance equation. The result is further generalized to the non-ergodic case. For this case the bottleneck nodes and the maximal subnetwork that achieves steady state are determined. The existence of a unique solution is shown and stability conditions are established. Our results can be used, for example, in the analysis of production systems with assembly and subassembly processes.
An inequality for pairs of martingales and its application to fractal image coding
Authors: Dekking F. M.
Published: July 2016

We prove a monotonicity property for a function of general square integrable pairs of martingales which is useful in fractal-based algorithms for compression of image data.
On a random vibration model
Authors: Huang Dawei , Spencer N. M.
Published: July 2016

A random vibration model is investigated in this paper. The model is formulated as a cosine function with a constant frequency and a random walk phase. We show that this model is second-order stationary and can be rewritten as a vector-valued AR(1) model as well as a scalar ARMA(2, 1) model. The linear innovation sequence of the AR(1) model is shown to be a martingale difference sequence while the linear innovation sequence of the ARMA(2, 1) model is only an uncorrelated sequence. A non-linear predictor is derived from the AR(1) model while a linear predictor is derived from the ARMA(2, 1) model. We deduce that the non-linear predictor of this model has less mean square error than that of the linear predictor. This has significance, for example, for predicting seasonal phenomena with this model. In addition, the limit distributions of the sample mean, the finite Fourier transforms and the autocovariance functions are derived using a martingale approach. The limit distribution of autocovariance functions differs from the classical result given by Bartlett's formula.
Ageing first-passage times of Markov processes: a matrix approach
Authors: Li Haijun , Shaked Moshe
Published: July 2016

Using a matrix approach we discuss the first-passage time of a Markov process to exceed a given threshold or for the maximal increment of this process to pass a certain critical value. Conditions under which this first-passage time possesses various ageing properties are studied. Some results previously obtained by Li and Shaked (1995) are extended.
Stationary M/G/1 excursions in the presence of heavy tails
Authors: Asmussen Søren , Klüppelberg Claudia
Published: July 2016

It is shown that the stationary excursions above level x for the stationary M/G/1 queue with the service time distribution belonging to a certain class of subexponential distributions are asymptotically of two types as x →∞: either the excursion starts with a jump from a level which is O(1) and the initial excess over x converges to ∞, or it starts from a level of the form x – O(1) and the excess has a proper limit distribution. The two types occur with probabilities ρ, resp. 1 – ρ.
A note on single server loss systems with a superposition of inputs
Authors: Willie Helmut
Published: July 2016

Explicit formulas for the time congestion and the call blocking probability are derived in a single server loss system whose total input consists of a finite superposition of independent general stationary traffic streams with exponentially distributed service times. The results are used for studying to what extent two arrival processes with coinciding customer-stationary state distributions are similar or even identical, and whether an arrival process with coinciding customer-stationary and time-stationary state distributions is of the Poisson type.
Time-dependent solution and optimal control of a bulk service queue
Authors: Selim Shokri Z.
Published: July 2016

We consider the queueing system denoted by M/MN
/1/N where customers are served in batches of maximum size N. The model is motivated by a traffic application. The time-dependent probability distribution for the number of customers in the system is obtained in closed form. The solution is used to predict the optimal service rates during a finite time horizon.
On the L
2-convergence of a superadditive bisexual Galton-Watson branching process
Authors: González M. , Molina M.
Published: July 2016

In this paper the L
2-convergence of a superadditive bisexual Galton–Watson branching process is studied. Necessary and sufficient conditions for the convergence of the suitably normed process are given. In the final section, a result about one of the most important bisexual models is proved.
Detailed probabilistic analysis of the integrated three-valued telegraph signal
Authors: Di Matteo Ilaria , Orsingher Enzo
Published: July 2016

In this paper the integrated three-valued telegraph process is examined. In particular, the third-order equations governing the distributions , (where N(t) denotes the number of changes of the telegraph process up to time t) are derived and recurrence relationships for them are obtained by solving suitable initial-value problems. These recurrence formulas are related to the Fourier transform of the conditional distributions and are used to obtain explicit results for small values of k. The conditional mean values (where V(0) denotes the initial velocity of motions) are obtained and discussed.
Consistency of Hill's estimator for dependent data
Authors: Resnick Sidney , Stărică Cătălin
Published: July 2016

Consider a sequence of possibly dependent random variables having the same marginal distribution F, whose tail 1−F is regularly varying at infinity with an unknown index − α < 0 which is to be estimated. For i.i.d. data or for dependent sequences with the same marginal satisfying mixing conditions, it is well known that Hill's estimator is consistent for α−1 and asymptotically normally distributed. The purpose of this paper is to emphasize the central role played by the tail empirical process for the problem of consistency. This approach allows us to easily prove Hill's estimator is consistent for infinite order moving averages of independent random variables. Our method also suffices to prove that, for the case of an AR model, the unknown index can be estimated using the residuals generated by the estimation of the autoregressive parameters.
On constant tail behaviour for the limiting random variable in a supercritical branching process
Authors: Hambly B. M.
Published: July 2016

We examine a family of supercritical branching processes and compute the density of the limiting random variable, W, for their normalized population size. In this example the left tail of W decays exponentially and there is no oscillation in this tail as typically observed. The branching process is embedded in the n-adic rational random walk approximation to Brownian motion on [0, 1]. This connection allows the explicit computation of the density of W.
On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes
Authors: Di Crescenzo A. G. , Giorno V. , Nobile A. G. , Ricciardi L. M.
Published: July 2016

The method earlier introduced for one-dimensional diffusion processes [6] is extended to obtain closed form expressions for the transition p.d.f.'s of two-dimensional diffusion processes in the presence of absorbing boundaries and for the first-crossing time p.d.f.'s through such boundaries. Use of such a method is finally made to analyse a two-dimensional linear process.
Pathological behaviour in loss networks
Authors: Hunt P. J.
Published: July 2016

Hunt and Kurtz [9] consider a loss network as the number of circuits and the offered traffics become large. They prove a functional law of large numbers for such a network and illustrate their results with some simple examples. In this paper we apply their results to slightly more complicated examples to illustrate other, and sometimes surprising, behaviour of the loss networks in heavy traffic. The networks we consider operate under somewhat unusual routing rules but this is to enable us to produce the behaviour in networks with only a few links. In larger, real-world networks it is likely that much more ‘natural' and intuitively appealing routing rules could produce similar undesirable behaviour.
A simple derivation of exact reliability formulas for linear and circular consecutive-k-of-n: F systems
Authors: Peköz Erol A. , Ross Sheldon M.
Published: July 2016

Exact reliability formulas for linear and circular consecutive-k-of-n: F systems are derived in the case of equal component reliabilities.
Moments of the first-passage time of a Wiener process with drift between two elastic barriers
Authors: Dominé Marco
Published: July 2016

The first-passage problem for the one-dimensional Wiener process with drift in the presence of elastic boundaries is considered. We use the Kolmogorov backward equation with corresponding boundary conditions to derive explicit closed-form expressions for the expected value and the variance of the first-passage time. Special cases with pure absorbing and/or reflecting barriers arise for a certain choice of a parameter constellation.
The evolution and measurement of a population of pairs
Authors: Jakeman Eric , Phayre Sean , Renshaw Eric
Published: July 2016

The statistical properties of a population of immigrant pairs of individuals subject to loss through emigration are calculated. Exact analytical results are obtained which exhibit characteristic even–odd effects. The population is monitored externally by counting the number of emigrants leaving in a fixed time interval. The integrated statistics for this process are evaluated and it is shown that under certain conditions only even numbers of individuals will be observed.
The evolution of the attainable structures of a continuous time homogeneous Markov system with fixed size
Authors: Tsaklidis George M.
Published: July 2016

In order to describe the evolution of the attainable structures of a continuous time homogeneous Markov system (HMS) with fixed size, we evaluate the volume of the sets of the attainable structures in Euclidean space in the course of time, and we find the value of the volume asymptotically. Then, using the concept of the volume of the attainable structures, we provide a method to evaluate the ‘age' of the system in continuous and discrete time. We also estimate the evolution of the distance of two (attainable) structures of the system as it changes following the transformations of the structures.
On improvements of the order of approximation in the Poisson limit theorem
Authors: Borovkov K. , Pfeifer D.
Published: July 2016

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n
2). The general case is discussed in terms of operator semigroups.
An analysis of a modified M/G/1 queue using a martingale technique
Authors: Roughan Matthew
Published: July 2016

We consider a variation of the M/G/1 queue in which, when the system contains more than k customers, it switches from its initial general service distribution to a different general service distribution until the server is cleared, whereupon it switches back to the original service distribution. Using a technique due to Baccelli and Makowski we define a martingale with respect to an embedded process and from this arrive at a relationship between the process and a modified Markov renewal process. Using this an analysis of the stationary behaviour of the queue is possible.
Limit theorems for a branching process with disasters
Authors: Athreya K. B. , Kaplan N.
Published: July 2016

A Bellman–Harris process is considered where the population is subjected to disasters which occur at random times. Each particle alive at the time of a disaster survives it with probability p. In the situation when explosion can occur, several limit theorems are proven. In particular, we prove that the age-distribution converges to the same stable distribution as the Bellman-Harris process and that the population size continues to be asymptotically exponential.
On optimal policies and martingales in dynamic programming
Authors: Rieder Ulrich
Published: July 2016

A martingale approach to a dynamic program with general state and action spaces is taken. Several necessary and sufficient conditions are given for a policy to be optimal. The results comprehend and modify different criteria of optimality given for dynamic programming problems. Finally, two applications are stated.
The distribution of the distance between random points
Authors: Alagar Vangalur S.
Published: July 2016

This paper considers the distribution of distance between random points and shows how the distribution can be found when the points are chosen uniformly and independently in a hypersphere or in two adjacent unit squares. The value of a powerful extension of the classical Crofton technique is illustrated here for solving such geometric probability problems. This method is quite different from those employed by Hammersley and Oser.