Modeling autocorrelation functions of long-range dependent teletraffic series based on optimal approximation in Hilbert space—A further study

This paper discusses optimal approximation of autocorrelation functions of teletraffic series by introducing a generalization of autocorrelation function form of fractional Gaussian noise (FGN). The demonstrations with real-traffic series are given.     

Prediction in a non-homogeneous Poisson cluster model

A non-homogeneous Poisson cluster model is studied, motivated by insurance applications. The Poisson center process which expresses arrival times of claims, triggers off cluster member processes which correspond to number or amount of payments. The cluster member process is an additive process. Given the past observations of the process we consider expected values of future increments and their mean squared errors, aiming at application in claims reserving problems. Our proposed process can cope with non-homogeneous observations such as the seasonality of claims arrival or the reducing property of payment processes, which are unavailable in the former models where both center and member processes are time homogeneous. Hence results presented in this paper are significant extensions toward applications.     

Characterizations of the Poisson process as a renewal process via two conditional moments

Given two independent positive random variables, under some minor conditions, it is known that fromE(X^rX+Y)=a(X+Y)^r andE(X^sX+Y)=b(X+Y)^s, for certain pairs ofr ands, wherea andb are two constants, we can characterizeX andY to have gamma distributions. Inspired by this, in this article we will characterize the Poisson process among the class of renewal processes via two conditional moments. More precisely, let {A(t), t0} be a renewal process, with {S _k, k1} the sequence of arrival times, andF the common distribution function of the inter-arrival times. We prove that for some fixedn andk, kn, ifE(S _k ^r A(t)=n)=at^r andE(S _k ^s A(t)=n)=bt^s, for certain pairs ofr ands, wherea andb are independent oft, then {A(t), t0} has to be a Poisson process. We also give some corresponding results about characterizingFto be geometric whenF is discrete.Support for this research was provided in part by the National Science Council of the Republic of China, Grant No. NSC 81-0208-M110-06.     

Fractional Poisson process

A fractional non-Markov Poisson stochastic process has been developed based on fractional generalization of the Kolmogorov–Feller equation. We have found the probability of n arrivals by time t for fractional stream of events. The fractional Poisson process captures long-memory effect which results in non-exponential waiting time distribution empirically observed in complex systems. In comparison with the standard Poisson process the developed model includes additional parameter μ. At μ=1 the fractional Poisson becomes the standard Poisson and we reproduce the well known results related to the standard Poisson process.As an application of developed fractional stochastic model we have introduced and elaborated fractional compound Poisson process.     

On generalized binomial and multinomial distributions and their relation to generalized Poisson distributions

Summary The binomial and multinomial distributions are, probably, the best known distributions because of their vast number of applications. The present paper examines some generalizations of these distributions with many practical applications. Properties of these generalizations are studied and models giving rise to them are developed. Finally, their relationship to generalized Poisson distributions is examined and limiting cases are given.     

Modeling network traffic by a cluster Poisson input process with heavy and light-tailed file sizes

We consider a cluster Poisson model with heavy-tailed interarrival times and cluster sizes as a generalization of an infinite source Poisson model where the file sizes have a regularly varying tail distribution function or a finite second moment. One result is that this model reflects long-range dependence of teletraffic data. We show that depending on the heaviness of the file sizes, the interarrival times and the cluster sizes we have to distinguish different growths rates for the time scale of the cumulative traffic. The mean corrected cumulative input process converges to a fractional Brownian motion in the fast growth case. However, in the intermediate and the slow growth case we can have convergence to a stable Lévy motion or a fractional Brownian motion as well depending on the heaviness of the underlying distributions. These results are contrary to the idea that cumulative broadband network traffic converges in the slow growth case to a stable process. Furthermore, we derive the asymptotic behavior of the cluster Poisson point process which models the arrival times of data packets and the individual input process itself.     

Regularly varying tails in a queue with discrete autoregressive arrivals of order p

We consider a discrete time single server queueing system where the service time of a customer is one slot, and the arrival process is governed by a discrete autoregressive process of order p (DAR(p)). For this queueing system, we investigate the tail behavior of the queue size and the waiting time distributions. Specifically, we show that if the stationary distribution of DAR(p) input has a tail of regular variation with index −β−1, then the stationary distributions of the queue size and the waiting time have tails of regular variation with index −β. This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment).     

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.     

Coupling methods in connection with Poisson process approximation

Motivated by a simple probabilistic model for the radioactive decay, we show thatSerfling's [1978] approach to Poisson approximation using coupling techniques can in a natural way also be applied to Poisson process approximation. This provides at the same time uniform estimations for the deviation of a Markov-Bernoulli process from the approximating Poisson process with respect to the total variation distance. An application to quasirandom input queuing models is also given.     

Poisson Distribution for a Sum of Additive Functions

We consider the limit distribution of values of a sum of additive arithmetic functions with shifted argument. The case of the Poisson limit distribution is studied. The functions considered take at most two values on the set of primes, 0 and 1, and satisfy some additional conditions. Some examples are given.      

Confidence regions for the intensity function of a cyclic Poisson process

A classical approach to constructing simultaneous confidence intervals (i.e., confidence bands or regions) for a function is via establishing a limiting process of the appropriately normalized difference between the function and its empirical estimator. In the present paper we depart from this approach and construct confidence bands for the intensity function of a cyclic Poisson process via extreme value type asymptotic results for the appropriately normalized supremum of the difference between the intensity function and its empirical estimator.      

Hitting straight lines by compound Poisson process paths

In a recent article Mallows and Nair (1989,Ann. Inst. Statist. Math.,41, 1–8) determined the probability of intersectionP{X(t)=αt for somet≥0} between a compound Poisson process {X(t), t≥0} and a straight line through the origin. Using four different approaches (direct probabilistic, via differential equations and via Laplace transforms) we extend their results to obtain the probability of intersection between {X(t), t≥0} and arbitrary lines. Also, we display a relationship with the theory of Galton-Watson processes. Additional results concern the intersections with two (or more) parallel lines. Work done in part while these authors were visiting professors at the Indian Statistical Institute, Delhi Centre, New Delhi, 110016, India. This author's investigation was supported in part by the U. S. National Science Foundation Grant No. DMS-8504319. Our coauthor and friend Prem Singh Puri died on August 12, 1989. We dedicate our contribution to this paper to his memory.     

The compound Poisson process perturbed by a diffusion with a threshold dividend strategy

In this paper, we consider the compound Poisson process perturbed by a diffusion in the presence of the so‐called threshold dividend strategy. Within this framework, we prove the twice continuous differentiability of the expected discounted value of all dividends until ruin. We also derive integro‐differential equations for the expected discounted value of all dividends until ruin and obtain explicit expressions for the solution to the equations. Along the same line, we establish explicit expressions for the Laplace transform of the time of ruin and the Laplace transform of the aggregate dividends until ruin. In the case of exponential claims, some examples are provided. Copyright © 2008 John Wiley & Sons, Ltd.     

Theorems of Large Deviations in the Approximation by the Compound Poisson Distribution

The paper is devoted to the research of large deviation probabilities in the approximation by compound Poisson law.     

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 .      

Estimating means from a non-I.I.D. Mixture of Poisson samples

In this paper, a family of estimators for estimating means when mixing two independent Poisson samples is proposed. This family is based on the probability-generating function of the Poisson distribution and is offered as an alternative to the maximum likelihood estimators, which have some drawbacks. These estimators include the method of moments estimators as a special limiting case.     

布朗运动和泊松过程共同驱动下的欧式期权定价

针对布朗运动和泊松过程共同驱动下股票价格的随机微分方程,利用It0公式和随机积分的方法,得到了该形式下欧式期权定价的模型,并给出了模型的求解.     

复合泊松过程和Meixner过程驱动下的期权定价

本文讨论了股票价格对数过程由复合泊松过程、Meixner过程驱动下的欧式看涨期权的定价问题.利用Esscher变换和风险中性Esscher测度得到了两类过程驱动下的期权定价公式,为实践者提供了理论上的参考价格.     

带干扰的双复合Poisson风险模型

对古典风险模型进行推广,主要研究保费收入过程为带干扰双复合Poisson过程的风险模型,运用鞅的方法得出了破产概率满足的Lundburg不等式.