A self-correcting point process

Suppose a point process is attempting to operate as closely as possible to a deterministic rate ρ, in the sense of aiming to produce ρt points during the interval (0,t] for all t. This can be modelled by making the instantaneous rate of t of the process a suitable function of n-ρt, n being the number of points in [0, t]. This paper studies such a self-correcting point process in two cases: when the point process is Markovian and the rate function is very general, and when the point process is arbitrary and the rate function is exponential. In each case it is shown that as t→∞ the mean number of points occuring in (0, t] is ρt+O(1) while the variance is bounded further, in the Markov case all the absolute central moments are bounded. An application to the outputs of stationary D/M/s queues is given.     

On a Markov process generated by non-decreasing concave functions

A discrete-time Markov process on [0, ∞) is considered. The process is generated by selecting at each time, in an independent and stationary way, a concave non-decreasing function. Sufficient conditions for the existence of a unique stationary limiting distribution are given.     

On a model of sequential point patterns

A finite point process motivated by the cooperative sequential adsorption model is proposed. Analytical properties of the point process are considered in details. It is shown that the introduced point process is useful for modeling both aggregated and regular point patterns. A possible scheme of maximum likelihood estimation of the process parameters is briefly discussed. V. Shcherbakov is on leave from Laboratory of Large Random Systems, Faculty of Mathematics and Mechanics, Moscow State University, Moscow.     

Inference in a model with at most one slope-change point

In this paper the problem of slope-change point in linear regression model is discussed with the help of the theory of Gaussian process. The distribution of the estimators of the change point proposed in this paper can be approximated by the first type of extremal distribution. Based on this fact, the detection and interval estimation of a change-point in various situations are discussed.     

On the jackknife statistics for eigenvalues and eigenvectors of a correlation matrix

This paper deals with some problems of eigenvalues and eigenvectors of a sample correlation matrix and derives the limiting distributions of their jackknife statistics with some numerical examples.     

The coalescent point process of multi-type branching trees

We define a multi-type coalescent point process of a general branching process with countably many types. This multi-type coalescent fully describes the genealogy of the (quasi-stationary) standing population providing types along ancestral lineages of all individuals in the standing population. We show that the coalescent process is a functional of a certain Markov chain defined by the planar embedding of the multi-type branching process.     

A point process approach to inventory models

The aim of the present paper is to make use of the modern theory of point processes to study optimal solutions for single‐item inventory. Demand for goods is assumed to occur according to a compound Poisson process and production occurs continuously and deterministically between times of demand, such that the inventory evolves according to a Markov process in continuous time. The aim is to propose a way of finding optimal production schemes by minimizing a certain expected loss over some finite period. There are holding/production costs depending on the stock level, and random penalty amounts will occur due to excess demand which is assumed backlogged. For simplicity we will not incorporate fixed costs. We give some numerical illustrations. Copyright © 2000 John Wiley & Sons, Ltd.     

On the exponential process associated with a CARMA-type process

We study the correlation decay and the expected maximal increments of the exponential processes determined by continuous-time autoregressive moving average (CARMA)-type processes of order (p, q). We consider two background driving processes, namely fractional Brownian motions and Lévy processes with exponential moments. The results presented in this paper are significant extensions of those very recent works on the Ornstein–Uhlenbeck-type case (p = 1, q = 0), and we develop more refined techniques to meet the general (p, q). In the concluding section, we discuss the perspective role of exponential CARMA-type processes in stochastic modelling of the burst phenomena in telecommunications and the leverage effect in financial econometrics.     

On the Asymptotic Behavior of the Storage Process Fed by a Markov Modulated Brownian Motion

Consider a storage model fed by a Markov modulated Brownian motion. We prove that the stationary distribution of the model exits and that the running maximum of the storage process over the interval [0, t] grows asymptotically like log t as t→∞.     

Filtering the histories of a partially observed marked point process

We consider a situation in which the evolution of an ‘underlying’ marked point process is of interest, but where this process is not directly observable. Instead, we assume that another marked point process, which is fully determined by the underlying process, can be observed. The problem is then the estimation, at any given time t, of the underlying development so far, given the corresponding observations. The solution, in the sense of a conditional distribution of the underlying pre-t history, is shown to satisfy a recursive filter formula. Sufficient conditions for the uniqueness of the solution are given. Two non-trivial examples are considered in detail.     

Immigration process in catalytic medium

The longtime behavior of the immigration process associated with a catalytic super-Brownian motion is studied. A large number law is proved in dimension d≤3 and a central limit theorem is proved for dimension d=3.     

On testing of parameters in modulated power law process

The modulated power law process (MPLP) is often used to model failure data from repairable system, when both renewal type behaviour and time trends are present. The MPLP allows for the failure rate of a system to be affected by the failure and repair. Since the MLEs of the estimates do not have closed form expressions, they have to be approximated, and hence deriving a test procedure will be difficult. Black and Rigdon (1996) have proposed asymptotic MLEs and asymptotic likelihood ratio tests for the parameters which also do not have closed form expressions and hence are not easy for application. In this paper, we derive a closed form expression for the test statistics which is simple and easy to apply for testing (i) H₀: β=1 versus H₁: β≠1 when κ is known and (ii) H₀: (β=1 and κ=1) versus H₁: (β≠1 or κ≠1). The simulation study for percentiles and powers are given. We also compare the performance of the test with that of Black and Rigdon's (1996) test. Some numerical examples are also provided to illustrate the testing procedures. Copyright © 2001 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment

In this paper, the stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment is studied. It is shown that there exist 3 equilibria. The sufficient conditions for local asymptotic stability of the infection‐free equilibrium and no‐immune equilibrium are given. We also discussed the local stability of positive equilibrium and the existence of Hopf bifurcation. Moreover, the direction and stability of Hopf bifurcation is obtained by using standard form theory and the center manifold theorem. Finally, numerical simulations are performed to verify the theoretical conclusions.     

Optimal order-replacement policy for a phase-type geometric process model with extreme shocks

A system is subject to random shocks that arrive according to a phase-type (PH) renewal process. As soon as an individual shock exceeds some given level the system will break down. The failed system can be repaired immediately. With the increasing number of repairs, the maximum shock level that the system can withstand will be decreasing, while the consecutive repair times after failure will become longer and longer. Undergoing a specified number of repairs, the existing system will be replaced by a new and identical one. The spare system for the replacement is available only by sending a purchase order to a supplier, and the duration of spare system procurement lead time also follows a PH distribution. Based on the number of system failures, a new order-replacement policy (also called $(K\text{,}N)$ policy) is proposed in this paper. Using the closure property of the PH distribution, the long-run average cost rate for the system is given by the renewal reward theorem. Finally, through numerical calculation, it is determined an optimal order-replacement policy such that the long-run expected cost rate is minimum.     

Estimator of a change point in single index models

This paper considers the problem of change point in single index models. In order to obtain asymptotically valid confidence intervals for the estimation of the change point, the convergence rate and asymptotic distribution of the change point estimate is studied. Some simulation results are presented which show that the numerical performance of our estimator is satisfactory.     

具有线性脉冲的周期捕食系统的持久性

研究具有HollingIV功能性反应和脉冲的周期捕食食饵系统．找到了影响该系统动力学行为的阈值Ro．证明了当Ro〈1时,该系统的食饵灭绝周期解是局部渐近稳定的;当R0〉1时,该系统的食饵灭绝周期解变得不稳定且食饵将一致持久．     

On the maximal multiplicity of block sizes in a random set partition

We study the asymptotic behavior of the maximal multiplicity M_n = M_n(σ) of the block sizes in a set partition σ of [n] = {1,2,…,n}, assuming that σ is chosen uniformly at random from the set of all such partitions. It is known that, for large n, the blocks of a random set partition are typically of size W = W(n), with We^W = n. We show that, over subsequences {n_k}_{k ≥ 1} of the sequence of the natural numbers, , appropriately normalized, converges weakly, as k→∞, to , where Z₁ and Z₂ are independent copies of a standard normal random variable. The subsequences {n_k}_{k ≥ 1}, where the weak convergence is observed, and the quantity u depend on the fractional part f_n of the function W(n). In particular, we establish that . The behavior of the largest multiplicity M_n is in a striking contrast to the similar statistic of integer partitions of n. A heuristic explanation of this phenomenon is also given.     

Stationary patterns and stability in a tumor-immune interaction model with immunotherapy

This paper examines a diffusive tumor-immune system with immunotherapy under homogeneous Neumann boundary conditions. We first investigate the large-time behavior of nonnegative equilibria and then explore the persistence of solutions to the time-dependent system. In particular, we present the sufficient conditions for tumor-free states. We also determine whether nonconstant positive steady-state solutions (i.e., a stationary pattern) exist for this coupled reaction-diffusion system when the parameter of the immunotherapy effect is small. The results indicate that this stationary pattern is driven by diffusion effects. For this study, we employ the comparison principle for parabolic systems and the Leray-Schauder degree.     

A reflected diffusion process in a regime-switching environment

This paper provides steady-state analysis of a reflected diffusion process governed by a regime-switching environment. We characterize differential equations satisfied by the limiting densities for overlapped and non-overlapped cases of reflecting boundary positions. Also we provide closed-form solutions for some specific cases and propose numerical methods for general cases.     

Weak convergence of random polygonal lines to a Gaussian process

We obtain a limit theorem of convergence in distribution for random polygonal lines defined by sums of independent random variables with replacements. In a particular case, the limit is the Gaussian Ornstein-Uhlenbeck process.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 33–44, January–March, 2005.Translated by V. Mackeviius