A note on the stationarity of a threshold first-order bilinear process

In the present note we study the threshold first-order bilinear model

X(t)=aX(t−1)+(b₁1_{X(t−1)<c}+b₂1_{X(t−1)c})X(t−1)e(t−1)+e(t), tεN

where {e(t), tεN} is a sequence of i.i.d. absolutely continuous random variables, X(0) is a given random variable and a, b₁, b₂ and c are real numbers. Under suitable conditions on the coefficients and lower semicontinuity of the densities of the noise sequence, we provide sufficient conditions for the existence of a stationary solution process to the present model and of its finite moments of order p.     

Space-time properties of a storage process

In a recent paper, Stadje analyzed the space-time properties of some storage processes. We give a short probabilistic proof of these results. This revised version was published online in June 2006 with corrections to the Cover Date.     

Prediction for discrete time series

Let {X_n} be a stationary and ergodic time series taking values from a finite or countably infinite set Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times _n along which we will be able to estimate the conditional probability P(=x|X₀,...,) from data segment (X₀,...,) in a pointwise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (among others, all stationary and ergodic Markov chains are included in this class) then almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then _n is upperbounded by a polynomial, eventually almost surely.Mathematics Subject Classification (2000): 62G05, 60G25, 60G10     

Testing the goodness of fit of a mixed Poisson process

In the literature on the statistical analysis of point processes certain tests for homogeneous Poisson processes are proposed, which in fact are tests for mixed Poisson processes. Some conclusions from this fact are drawn.     

复合泊松过程及其在保险风险中的若干应用

本文利用齐次泊松过程的可加性,研究了复合泊松过程的可加性及其性质。作为应用,讨论了单个理赔额服从指数分布的复合泊松风险模型在第n次索赔时发生负盈余的概率。     

SOME FUNCTIONAL LIMIT THEOREMS FOR THE INFINITE SERIES OF OU PROCESSES

This paper obtains functional modulus of continuity and Strassen's functional LIL of the infinite series of independent Ornstein-Uhlenbeck processes, which also imply the Levy's exact modulus of continuity and LIL of this process respectively.     

On tails of stationary measures on a class of solvable groups

Let G be a subgroup of GL(R,d) and let (Qn,Mn) be a sequence of i.i.d. random variables with values in Rd?G and law μ. Under some natural conditions there exists a unique stationary measure ν on Rd of the process Xn=MnXn−1+Qn. Its tail properties, i.e. behavior of as t tends to infinity, were described some over thirty years ago by H. Kesten, whose results were recently improved by B. de Saporta, Y. Guivarc'h and E. Le Page. In the present paper we study the tail of ν in the situation when the group G₀ is Abelian and Rd is replaced by a more general nilpotent Lie group N. Thus the tail behavior of ν is described for a class of solvable groups of type NA, i.e. being semi-direct extension of a simply connected nilpotent Lie group N by an Abelian group isomorphic to Rd. Then, due to A. Raugi, (N,ν) can be interpreted as the Poisson boundary of (NA,μ).     

Boundary Modified Contact Processes

We introduce a one dimensional contact process for which births to the right of the rightmost particle and to the left of the leftmost particle occur at rate _e (where e is for external). Other births occur at rate _i (where i is for internal). Deaths occur at rate 1. The case _e= _i is the well known basic contact process for which there is a critical value _c>1 such that if the birth rate is larger than _c the process has a positive probability of surviving. Our main motivation here is to understand the relative importance of the external birth rates. We show that if _e1 then the process always dies out while if _e>1 and if _i is large enough then the process may survive. We also show that if _i< _c the process dies out for all _e. To extend this notion to d>1 we introduce a second process that has an epidemiological interpretation. For this process each site can be in one of three states: infected, a susceptible that has never been infected, or a susceptible that has been infected previously. Furthermore, the rates at which the two types of susceptible become infected are different. We obtain some information about the phase diagram about this case as well.     

On the problem of testing whether a mixed poisson process is homogeneous

In 2 recent paper Albrecht proposes a test procedure for testing whether a point process is homogeneous, also taking care of alternatives when the process is a mixed Poisson process. In the present note it is argued that it seems impossible to find a meaningful test criterion for testing whether a mixed Poisson process is homogeneous.     

Some results for secondary processes generated by a poisson process

We obtain the moment structure of a general class of random variables generated by a Poisson process. We then apply these relationships to several applied probability models. Among these are queues, counter models and low density traffic flow.     

Tomography models of random convex polygons

Combinatorial integral geometry possesses some results that can be interpreted as belonging to the field of Geometric Tomography. The main purpose of the present paper is to present a case of parallel X-ray approach to tomography of random convex polygons. However, the Introduction reviews briefly some earlier results by the author that refer to reconstruction of (non-random) convex domains by means of a point X-ray. The main tool in treating the parallel X-rays is disintegrated Pleijel identity, or rather, its averaged version, whose derivation is represented in complete detail. The paper singles out a class of random polygons called tomography models, that offer essential advantages for the analysis. The definition of a tomography model is given in terms of stochastic independence. Fortunately, random translation-invariant Poisson processes of lines in IR ² suggest a class of examples. We recall that each such line process is determined by its rose of directions ρ(?). For rather general ρ(?), the number weighted typical polygon in the polygonal partition of the plane generated by the corresponding Poisson line process happens to be a tomography model. For general tomography models, a differential equation is derived for the Laplace transform for parallel X-rays, that rises several interesting computational problems.     

A physical basis for Krein's prediction formula

A prediction problem of the following variety is considered. A stationary random process w(·) of known spectrum is observed over |t|?a. Using these observed values, w(b) is to be predicted for some b with |b|>a. We present a physical interpretation of a solution to this problem due to Krein, which used the theory of inverse Sturm-Liouville problems. Our physical model involves a nonuniform lossless transmission line excited at one end by white noise. The signal at the other end is the process w(t), and the prediction is found by calculating as intermediate quantities the voltage and current stored on the line at t=0. These quantities are spatially uncorrelated and provide a spatial representation at t=0 of the innovations of w(t) over |t|?a.     

Some Simple Remarks on an Autoregressive Scheme and an Implied Problem

Nonminimum phase autoregressive schemes are considered in the context of the problem of determining a stationary process that satisfies a possibly nonlinear autoregressive system of equations. It's noted that in most solutions there is an implicit assumption that the independent random variables generating the process are independent of the past of the process. This is not true of the non-minimum phase schemes. A simple scheme that has a characteristic polynomial with roots inside and outside the unit disc in the complex plane is discussed.     

Weak ergodicity of stationary pairwise independent processes

It is proven that a stationary process of pairwise independent random variables with values in a separable metric space is weakly ergodic, i.e. each random variable is independent of the system of invariant sets of the process. An example shows that a process of identically distributed pairwise independent random variables is in general, however, not weakly ergodic.

     

机器不可靠且需求过程为k个泊松过程叠加的生产存贮系统

本文对机器不可靠的二阀值控制存贮水平的生产存贮系统进行了研究，一旦机器发生故障，会立即得到修理以使它重新工作；对产品的需求假设是k个复合泊松到达过程的叠加，需求量为独立同分布的随机变量。我们得到了存贮水平稳态分布存在的充分条件、分布函数的显性表达式，以及在稳态下系统运行的一些性能指标。     

Expansions for Joint Laplace Transform of Stationary Waiting Times in (max,+)-linear Systems with Poisson Input

We give a Taylor series expansion for the joint Laplace transform of stationary waiting times in open (max,+)-linear stochastic systems with Poisson input. Probabilistic expressions are derived for coefficients of all orders. Even though the computation of these coefficients can be hard for certain systems, it is sufficient to compute only a few coefficients to obtain good approximations (especially under the assumption of light traffic). Combining this new result with the earlier expansion formula for the mean stationary waiting times, we also provide a Taylor series expansion for the covariance of stationary waiting times in such systems.It is well known that (max,+)-linear systems can be used to represent stochastic Petri nets belonging to the class of event graphs. This class contains various instances of queueing networks like acyclic or cyclic fork-and-join queueing networks, finite or infinite capacity tandem queueing networks with various types of blocking, synchronized queueing networks and so on. It also contains some basic manufacturing models such as kanban networks, assembly systems and so forth. The applicability of this expansion technique is discussed for several systems of this type.     

Singular ergodic control for multidimensional Gaussian–Poisson processes

Singular control for multidimensional Gaussian–Poisson processes with a long-run (or ergodic) and a discounted criteria is discussed. The corresponding Hamilton–Jacobi–Bellman equations are discussed. Complete details on the proofs and further extensions are left for future works.     

Thresholds of moving average of stationary processes for given on target significant levels

A Markov chain is associated to a finite order one-sided moving average of a discrete time stationary Gaussian process. A method is developed to specify thresholds for given on target significant levels in the sense that in the long run the probability that the moving average process lies in [L _i , L _i+1), will be π _i , i = 0,. . . ,m. Special inputs, AR(1) and MA(1) are treated in details. This article extends the work of Soltani et al. in (Commun Stat Theory Methods 36(14):2595–2606) where the inputs were assumed to be i.i.d.; and a single threshold was considered. This research was supported by Kuwait University, Research Administration, Research Grant No.[SS08/06].     

Penalized maximum likelihood estimation for a function of the intensity of a Poisson point process

Let be an unknown 2 times differentiable function and consider M to be an α- homogeneous Poisson process on Graf(f). The goal is to estimate f having a sample of the inhomogeneous Poisson process N constructed by dislocating each point of M perpendicularly to Graf(f) by a normal random variable with zero mean and constant variance σ². The exact formulas for the mean measure and the intensity function of N are obtained. Then, the function f is estimated directly using a hybrid spline approach to penalized maximum likelihood. Simulation results indicate the procedure to be consistent as and .