Results 1  10
of
93
The twoparameter PoissonDirichlet distribution derived from a stable subordinator.
, 1995
"... The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov ..."
Abstract

Cited by 364 (33 self)
 Add to MetaCart
The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to VershikShmidtIgnatov, are generalized to the twoparameter case. The sizebiased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...
Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process
, 1998
"... The asymptotic theory for the sample autocorrelations and extremes of a GARCH(1; 1) process is provided. Special attention is given to the case when the sum of the ARCH and GARCH parameters is close to one, i.e. when one is close to an infinite variance marginal distribution. This situation has been ..."
Abstract

Cited by 91 (20 self)
 Add to MetaCart
The asymptotic theory for the sample autocorrelations and extremes of a GARCH(1; 1) process is provided. Special attention is given to the case when the sum of the ARCH and GARCH parameters is close to one, i.e. when one is close to an infinite variance marginal distribution. This situation has been observed for various financial logreturn series and led to the introduction of the IGARCH model. In such a situation the sample autocorrelations are unreliable estimators of their deterministic counterparts for the time series and its absolute values, and the sample autocorrelations of the squared time series have nondegenerate limit distributions. We discuss the consequences for a foreign exchange rate series. AMS 1991 Subject Classification: Primary: 62P20 Secondary: 90A20 60G55 60J10 62F10 62F12 62G30 62M10 Key Words and Phrases. GARCH, sample autocorrelations, stochastic recurrence equation, Pareto tail, extremes, extremal index, point processes, foreign exchange rates 1 Introduc...
Regular variation of GARCH processes
 Stoch. Proc. Appl
, 2001
"... We show that the finitedimensional distributions of a GARCH process are regularly varying, i.e., the tails of these distributions are Paretolike and hence heavytailed. Regular variation of the joint distributions provides insight into the moment properties of the process as well as the dependence ..."
Abstract

Cited by 74 (14 self)
 Add to MetaCart
(Show Context)
We show that the finitedimensional distributions of a GARCH process are regularly varying, i.e., the tails of these distributions are Paretolike and hence heavytailed. Regular variation of the joint distributions provides insight into the moment properties of the process as well as the dependence structure between neighboring observations when both are large. Regular variation also plays a vital role in establishing the large sample behavior of a variety of statistics from a GARCH process including the sample mean and the sample autocovariance and autocorrelation functions. In particular, if the 4th moment of the process does not exist, the rate of convergence of the sample autocorrelations becomes extremely slow, and if the 2nd moment does not exist, the sample autocorrelations have nondegenerate limit distributions. 1This research supported by an NWO PhD grant.
Heavy Tail Modeling And Teletraffic Data
 Annals of Statistics
, 1997
"... . Huge data sets from the teletraffic industry exhibit many nonstandard characteristics such as heavy tails and long range dependence. Various estimation methods for heavy tailed time series with positive innovations are reviewed. These include parameter estimation and model identification methods ..."
Abstract

Cited by 67 (5 self)
 Add to MetaCart
. Huge data sets from the teletraffic industry exhibit many nonstandard characteristics such as heavy tails and long range dependence. Various estimation methods for heavy tailed time series with positive innovations are reviewed. These include parameter estimation and model identification methods for autoregressions and moving averages. Parameter estimation methods include those of YuleWalker and the linear programming estimators of Feigin and Resnick as well estimators for tail heaviness such as the Hill estimator and the qqestimator. Examples are given using call holding data and interarrivals between packet transmissions on a computer network. The limit theory makes heavy use of point process techniques and random set theory. 1. Introduction Classical queuing and network stochastic models contain simplifying assumptions guaranteeing the Markov property and insuring analytical tractability. Frequently interarrivals and service times are assumed to be iid and typically underlyi...
Departures from Many Queues in Series
, 1990
"... We consider a series of n singleserver queues, each with unlimited waiting space and the firstin firstout service discipline. Initially, the system is empty; then k customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribu ..."
Abstract

Cited by 60 (5 self)
 Add to MetaCart
We consider a series of n singleserver queues, each with unlimited waiting space and the firstin firstout service discipline. Initially, the system is empty; then k customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribution. We are interested in the time D(k, n) required for all k customers to complete service from all n queues. In particular, we investigate the limiting behavior of D(k, n) as n and/or k . There is a duality implying that D(k, n) is distributed the same as D(n , k) so that results for large n are equivalent to results for large k. A previous heavytraffic limit theorem implies that D(k, n) satisfies an invariance principle as n , converging after normalization to a functional of kdimensional Brownian motion. We use the subadditive ergodic theorem and a strong approximation to describe the limiting behavior of D(k n , n) where k n as n . The case of k n = xn corresponds to a hydrodyna...
Sojourn Time Asymptotics in the M/G/1 Processor Sharing Queue
 QUEUEING SYSTEMS
, 1998
"... We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index \Gamma , noninteger, iff the sojourn time distribution is regularly varying of index \Gamma . This result is derived from a new expression for the LaplaceStieltjes transform of the sojo ..."
Abstract

Cited by 56 (9 self)
 Add to MetaCart
(Show Context)
We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index \Gamma , noninteger, iff the sojourn time distribution is regularly varying of index \Gamma . This result is derived from a new expression for the LaplaceStieltjes transform of the sojourn time distribution. That expression also leads to other new properties for the sojourn time distribution. We show how the moments of the sojourn time can be calculated recursively and prove that the kth moment of the sojourn time is finite iff the kth moment of the service time is finite. In addition, we give a short proof of a heavy traffic theorem for the sojourn time distribution, prove a heavy traffic theorem for the moments of the sojourn time, and study the properties of the heavy traffic limiting sojourn time distribution when the service time distribution is regularly varying. Explicit formulas and multiterm expansions are provided for the case that the service time has a Pareto...
Limit Theory For Bilinear Processes With Heavy Tailed Noise
 Ann. Appl. Probab
, 1995
"... . We consider a simple stationary bilinear model X t = cX t\Gamma1 Z t\Gamma1 + Z t ; t = 0; \Sigma1; \Sigma2; : : : generated by heavy tailed noise variables fZ t g. A complete analysis of weak limit behavior is given by means of a point process analysis. A striking feature of this analysis is th ..."
Abstract

Cited by 53 (16 self)
 Add to MetaCart
. We consider a simple stationary bilinear model X t = cX t\Gamma1 Z t\Gamma1 + Z t ; t = 0; \Sigma1; \Sigma2; : : : generated by heavy tailed noise variables fZ t g. A complete analysis of weak limit behavior is given by means of a point process analysis. A striking feature of this analysis is that the sample correlation converges in distribution to a nondegenerate limit. A warning is sounded about trying to detect nonlinearities in heavy tailed models by means of the sample correlation function. 1. Introduction. Current efforts in time series analysis attempt to deal with data which exhibit features such as long range dependence, nonlinearity and heavy tails. There are numerous data sets from the fields of telecommunications, finance and economics which appear to be compatible with the assumption of heavytailed marginal distributions. Examples include file lengths, cpu time to complete a job, call holding times, interarrival times between packets in a network and lengths of o...
Change of structure in financial time series, long range dependence and the GARCH model
, 1999
"... Functionals of a twoparameter integrated periodogram have been used for a long time for detecting changes in the spectral distribution of a stationary sequence. The bases for these results are functional central limit theorems for the integrated periodogram having as limit a Gaussian field. In the ..."
Abstract

Cited by 46 (0 self)
 Add to MetaCart
Functionals of a twoparameter integrated periodogram have been used for a long time for detecting changes in the spectral distribution of a stationary sequence. The bases for these results are functional central limit theorems for the integrated periodogram having as limit a Gaussian field. In the case of GARCH(p; q) processes a statistic closely related to the integrated periodogram can be used for the purpose of change detection in the model. We derive a central limit theorem for this statistic under the hypothesis of a GARCH(p; q) sequence with a finite 4th moment. When applied to reallife time series our method gives clear evidence of the fast pace of change in the data. One of the straightforward conclusions of our study is the infeasibility of modeling long return series with one GARCH model. The parameters of the model must be updated and we propose a method to detect when the update is needed. Our study supports the hypothesis of global nonstationarity of the return time ser...
Regularly varying multivariate time series
 Stoch. Proc. Appl. 119
, 2009
"... Abstract: A multivariate, stationary time series is said to be jointly regularly varying if all its finitedimensional distributions are multivariate regularly varying. This property is shown to be equivalent to weak convergence of the conditional distribution of the rescaled series given that, at a ..."
Abstract

Cited by 35 (9 self)
 Add to MetaCart
(Show Context)
Abstract: A multivariate, stationary time series is said to be jointly regularly varying if all its finitedimensional distributions are multivariate regularly varying. This property is shown to be equivalent to weak convergence of the conditional distribution of the rescaled series given that, at a fixed time instant, its distance to the origin exceeds a threshold tending to infinity. The limit object, called the tail process, admits a decomposition in independent radial and angular components. Under an appropriate mixing condition, this tail process allows for a concise and explicit description of the limit of a sequence of point processes recording both the times and the positions of the time series when it is far away from the origin. The theory is applied to multivariate moving averages of finite order with random coefficient matrices.