. Many important probabilistic models in queuing theory, insurance and finance deal with partial sums of a negative mean stationary process (a negative drift random walk), and the law of the supremum of such a process is used to calculate, depending on the context, the ruin probability, the steady state distribution of the number of customers in the system or the value at risk. When the stationary process is heavy--tailed, the corresponding ruin probabilities are high and the stationary distributions are heavy--tailed as well. If the steps of the random walk are independent, then the exact asymptotic behavior of such probability tails was described by Embrechts and Veraverbeke (1982). We show that this asymptotic behavior may be different if the steps of the random walk are not independent, and the dependence affects the joint probability tails of the stationary process. Such type of dependence can be modeled, for example, by a linear process. 1. Introduction In various applied fields...

The behaviour of averaged periodograms and cross-periodograms of a broad class of nonstationary processes is studied. The processes include nonstationary ones that are fractional of any order, as well as asymptotically stationary fractional ones, and the cross-periodogram can involve two nonstationary processes of possibly di#erent orders, or a nonstationary and an asymptotically stationary one. The averaging takes place either over the whole frequency band, or on one that degenerates slowly to zero frequency as sample size increases. In some cases it is found to make no asymptotic di#erence, and in particular we indicate how the behaviour of the mean and variance changes across the two-dimensional space of integration orders. The results employ only local-to-zero assumptions on the spectra of the underlying weakly stationary sequences. It is shown how the results can be readily applied in case of fractional cointegration with unknown integration orders. 1 1. INTRODUCTION In the analy...

This paper reviews several periodogram-based methods for estimating the long-memory parameter H in time series and suggests a way to robustify them. The high frequencies tend to bias the estimates. Using only low frequencies eliminates the bias but increases the variance. We hence suggest plotting the estimates of H as a function of a parameter which balances bias versus variance and, if the plot flattens in a central region, to use the flat part for estimating H. We apply this technique to the periodogram regression method, the Whittle approximation to maximum likelihood and to the local Whittle method. We investigate its effectiveness on several simulated fractional ARIMA series and also apply it to estimate the long-memory parameter H in computer network traffic. 1 Introduction Time series with long memory have been considered in many fields including hydrology, biology and computer networks. Unfortunately, estimating the long memory (long-range dependence) parameter H in a given d...

We study the robustness of the "standard Whittle", "local Whittle " and "aggregated Whittle" estimators by using a large number of simulated Gaussian time series with long-range dependence. We also consider what happens when the Gaussian innovations are replaced by infinite variance symmetric stable ones. The standard Whittle estimator is a parametric estimator, the local Whittle estimator is a semi-parametric one recently developed by Robinson (1995) and the aggregated Whittle estimator smoothes out the high frequencies. The goal is to estimate H; the intensity of longrange dependence. We investigate the standard deviation and bias of these estimators in order to determine when they are reliable. These estimators are then applied to real-life Ethernet data. 1 Introduction The recent discovery that long-range dependence is present in high-speed networks has given a new urgency to the development of reliable methods for estimating the intensity of long-range dependence in extremely lon...

Frequency domain statistics are studied in the presence of fractional deterministic and stochastic trends. It is shown how the behaviour of the sample variance-covariance matrix of nonstationary processes can be dominated by components corresponding to a possibly degenerating band around zero frequency. This property is used to establish the limiting distribution of the averaged periodogram matrix, of memory estimates for nonstationary series, and for frequency domain regression estimates under nonstandard conditions. Keywords: Averaged periodogram, nonstationary processes, fractional Brownian motion. 1 1. INTRODUCTION For a sequence of column vectors u t , t = 1, 2, ..., n, with real-valued elements, define the discrete Fourier transform w u (#) = 1 # 2#n n # t=1 (u t - u)e it# , where u = n -1 # n t=1 u t denotes the sample mean. Given also a vector sequence v t , t = 1, ..., n, define the cross-periodogram matrix I uv (#) = w u (#)w # v (#), (1.1) the asterisk denot...

Abstract. In this paper we study the weak convergence of the integrated periodogram indexed by classes of functions for linear and stochastic volatility processes with symmetric α-stable noise. Under suitable summability conditions on the series of the Fourier coefficients of the index functions we show that the weak limits constitute α-stable processes which have representation as infinite Fourier series with iid α-stable coefficients. The cases α ∈ (0, 1) and α ∈ [1, 2) are dealt with by rather different methods and under different assumptions on the classes of functions. For example, in contrast to the case α ∈ (0, 1), entropy conditions are needed for α ∈ [1, 2) to ensure the tightness of the sequence of integrated periodograms indexed by functions. The results of this paper are of additional interest since they provide limit results for infinite mean random quadratic forms with particular Töplitz coefficient matrices. 1.

Suppose that X t = P 1 j=0 c j Z t\Gammaj is a stationary linear sequence with regularly varying c j 's and with innovations fZ j g that have infinite variance. Such a sequence can exhibit both high variability and strong dependence. The quadratic form Qn = P n t 1 s=1 b j(t \Gamma s)X t X s plays an important role in the estimation of the intensity of strong dependence. In contrast with the finite variance case, n \Gamma1=2 (Qn \Gamma EQn ) does not converge to a Gaussian distribution. We provide conditions on the c j 's and on b j for the quadratic form Qn , adequately normalized and randomly centered, to converge to a stable law of index ff, 1 ! ff ! 2, as n tends to infinity. 1 Outline of the main ideas The main result of this paper is a limit theorem for randomly centered quadratic forms Q n = n X t;s=1 b j(t \Gamma s)X t X s ; (1.1) where fX t g is a strongly dependent linear process whose innovations have a heavy-tailed distribution. In (1.1), j is an integrable func...

Abstract. We consider some elementary functions of the components of a regularly varying random vector such as linear combinations, products, minima, maxima, order statistics, powers. We give conditions under which these functions are again regularly varying, possibly with a different index. 1.

Weak convergence of the function-indexed integrated periodogram for infinite variance processes