Index Of Regular Variation Research Articles

Large deviations of [formula omitted]-blocks of regularly varying time series and applications to cluster inference

In the regularly varying time series setting, a cluster of exceedances is a short period for which the supremum norm exceeds a high threshold. We propose to study a generalization of this notion considering short periods, or blocks, with ℓp−norm above a high threshold. Our main result derives new large deviation principles of extremal ℓp−blocks, which guide us to define and characterize spectral cluster processes in ℓp. We then study cluster inference in ℓp to motivate our results. We design consistent disjoint blocks methods to infer features of cluster processes. Our inferential setting promotes the use of large empirical quantiles from the ℓp−norm of blocks as threshold levels which eases implementation and also facilitates comparison for different p>0. Our approach highlights the advantages of cluster inference based on extremal ℓα−blocks, where α>0 is the index of regular variation of the series. We focus on inference of important indices in extreme value theory, e.g., the extremal index.

Cluster based inference for extremes of time series

We introduce a new type of estimator for the spectral tail process of a regularly varying time series. The approach is based on a characterizing invariance property of the spectral tail process, which is incorporated into the new estimator via a projection technique. We show uniform asymptotic normality of this estimator, both in the case of known and of unknown index of regular variation. In a simulation study the new procedure shows a more stable performance than previously proposed estimators.

Diameter of P.A. random graphs with edge‐step functions

In this work we prove general bounds for the diameter of random graphs generated by a preferential attachment model whose parameter is a function f:N→[0,1] that drives the asymptotic proportion between the numbers of vertices and edges. These results are sharp when f is a regularly varying function at infinity with strictly negative index of regular variation −γ. For this particular class, we prove a characterization for the diameter that depends only on −γ. More specifically, we prove that the diameter of such graphs is of order 1/γ with high probability, although its vertex set order goes to infinity polynomially. Sharp results for the diameter for a wide class of slowly varying functions are also obtained.

Multivariate matrix Mittag–Leffler distributions

We extend the construction principle of multivariate phase-type distributions to establish an analytically tractable class of heavy-tailed multivariate random variables whose marginal distributions are of Mittag–Leffler type with arbitrary index of regular variation. The construction can essentially be seen as allowing a scalar parameter to become matrix-valued. The class of distributions is shown to be dense among all multivariate positive random variables and hence provides a versatile candidate for the modelling of heavy-tailed, but tail-independent, risks in various fields of application.

Preferential Attachment Random Graphs with Edge-Step Functions

We analyze a random graph model with preferential attachment rule and edge-step functions that govern the growth rate of the vertex set, and study the effect of these functions on the empirical degree distribution of these random graphs. More specifically, we prove that when the edge-step function f is a monotone regularly varying function at infinity, the degree sequence of graphs associated with it obeys a (generalized) power-law distribution whose exponent belongs to (1, 2] and is related to the index of regular variation of f at infinity whenever said index is greater than $$-1$$ . When the regular variation index is less than or equal to $$-1$$ , we show that the empirical degree distribution vanishes for any fixed degree.

Spectral tail processes and max-stable approximations of multivariate regularly varying time series

A regularly varying time series as introduced in Basrak and Segers (2009) is a (multivariate) time series such that all finite dimensional distributions are multivariate regularly varying. The extremal behavior of such a process can then be described by the index of regular variation and the so-called spectral tail process, which is the limiting distribution of the rescaled process, given an extreme event at time 0. As shown in Basrak and Segers (2009), the stationarity of the underlying time series implies a certain structure of the spectral tail process, informally known as the “time change formula”. In this article, we show that on the other hand, every process which satisfies this property is in fact the spectral tail process of an underlying stationary max-stable process. The spectral tail process and the corresponding max-stable process then provide two complementary views on the extremal behavior of a multivariate regularly varying stationary time series.

Weak and one-sided strong laws for random variables with infinite mean

Let aj be positive weight constants and Xj be independent non-negative random variables (j=1,2,…) and Sn(a)=∑i=1naiXi. If the Xj have the same relatively stable distribution, then under mild conditions there exist constants bn→∞ such that W¯n(a)=bn−1Sn(a)→p1, i.e., a weak law of large numbers holds. If the weights comprise a regularly varying sequence, then under some additional technical conditions, this outcome can be strengthened to a strong law if and only if the index of regular variation is −1. This paper addresses a case where the Xj are not identically distributed, but rather the tail probability P(ajXj>x) is asymptotically proportional to aj(1−F(x)), where F is a relatively stable distribution function. Here the weak law holds but the strong law does not: under typical conditions almost surely lim infn→∞W¯n(a)=1 and lim supn→∞W¯n(a)=∞.

Subsampling Inference for the Autocorrelations of GARCH Processes*

AbstractWe provide self-normalization for the sample autocorrelations of power GARCH(p, q) processes whose higher moments might be infinite. To validate the studentization, whose goal is to match the growth rate dependent on the index of regular variation of the process, we substantially extend existing weak-convergence results. Since asymptotic distributions are non-pivotal, we construct subsampling-based confidence intervals for the autocorrelations and cross-correlations, which are shown to have satisfactory empirical coverage rates in a simulation study. The methodology is further applied to daily returns of CAC40 and FTSA100 indices and their squares.

On relative stability and weighted laws of large numbers

Weighted laws of large numbers are established for components which are independent copies of a positive relatively stable law and the weights comprise a regularly varying sequence. The index of regular variation of the weights must be at least −1 for a weak law and be exactly −1 for a strong law. Consideration is given to the special case where the truncated moment function is proportional to the logarithm, a case arising from quotients of independent random variables and quotients of successive order statistics. Closure of the class of positive relatively stable laws under independent multiplication (i.e, Mellin convolution) is reprised and tail equivalences are extended.

Extremal solutions to a system of n nonlinear differential equations and regularly varying functions

The strongly increasing and strongly decreasing solutions to a system of n nonlinear first order equations are here studied, under the assumption that both the coefficients and the nonlinearities are regularly varying functions. We establish conditions under which such solutions exist and are (all) regularly varying functions, we derive their index of regular variation and establish asymptotic representations. Several applications of the main results are given, involving n-th order nonlinear differential equations, equations with a generalized ϕ-Laplacian, and nonlinear partial differential systems.

On multidimensional Mandelbrot cascades

Let Z be a random variable with values in a proper closed convex cone , A a random endomorphism of C and N a random integer. We assume that Z, A, N are independent. Given N independent copies of we define a new random variable . Let T be the corresponding transformation on the set of probability measures on C, i.e. T maps the law of Z to the law of . If the matrix has dominant eigenvalue 1, we study existence and properties of fixed points of T having finite non-zero expectation. Existing one-dimensional results concerning T are extended to higher dimensions. In particular we give conditions under which such fixed points of T have multidimensional regular variation in the sense of extreme value theory and we determine the index of regular variation.

Functional limit theorems for renewal shot noise processes with increasing response functions

We consider renewal shot noise processes with response functions which are eventually nondecreasing and regularly varying at infinity. We prove weak convergence of renewal shot noise processes, properly normalized and centered, in the space D[0,∞) under the J1 or M1 topology. The limiting processes are either spectrally nonpositive stable Lévy processes, including the Brownian motion, or inverse stable subordinators (when the response function is slowly varying), or fractionally integrated stable processes or fractionally integrated inverse stable subordinators (when the index of regular variation is positive). The proof exploits fine properties of renewal processes, distributional properties of stable Lévy processes and the continuous mapping theorem.

Topological regular variation: I. Slow variation

Motivated by the Category Embedding Theorem, as applied to convergent automorphisms (Bingham and Ostaszewski (in press) [11]), we unify and extend the multivariate regular variation literature by a reformulation in the language of topological dynamics. Here the natural setting are metric groups, seen as normed groups (mimicking normed vector spaces). We briefly study their properties as a preliminary to establishing that the Uniform Convergence Theorem (UCT) for Baire, group-valued slowly-varying functions has two natural metric generalizations linked by the natural duality between a homogenous space and its group of homeomorphisms. Each is derivable from the other by duality. One of these explicitly extends the (topological) group version of UCT due to Bajšanski and Karamata (1969) [4] from groups to flows on a group. A multiplicative representation of the flow derived in Ostaszewski (2010) [45] demonstrates equivalence of the flow with the earlier group formulation. In companion papers we extend the theory to regularly varying functions: we establish the calculus of regular variation in Bingham and Ostaszewski (2010) [13] and we extend to locally compact, σ-compact groups the fundamental theorems on characterization and representation (Bingham and Ostaszewski (2010) [14]). In Bingham and Ostaszewski (2009) [15], working with topological flows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be specified using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure.

The Index Theorem of topological regular variation and its applications

We develop further the topological theory of regular variation of [N.H. Bingham, A.J. Ostaszewski, Topological regular variation: I. Slow variation, LSE-CDAM-2008-11]. There we established the uniform convergence theorem (UCT) in the setting of topological dynamics (i.e. with a group T acting on a homogenous space X ), thereby unifying and extending the multivariate regular variation literature. Here, working with real-time topological flows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be specified using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure.

Extreme values statistics for Markov chains via the (pseudo-) regenerative method

This paper is devoted to the study of specific statistical methods for extremal events in the markovian setup, based on the regenerative method and the Nummelin technique. Exploiting ideas developed in [53], the principle underlying our methodology consists in generating first (a random number l of) approximate pseudo-renewal times 1, 2, . . . , l for a sample path X1, . . . , Xn drawn from a Harris chain X with state space E, from the parameters of a minorization condition fulfilled by its transition kernel, and computing then submaxima over the approximate cycles thus obtained: max1+1i2 f(Xi), . . . , max1+l−1il f(Xi) for any measurable function f : E ! R. Estimators of tail features of the sample maximum max1in f(Xi) are then constructed by applying standard statistical methods, tailored for the i.i.d. setting, to the submaxima as if they were independent and identically distributed. In particular, the asymptotic properties of extensions of popular inference procedures based on (conditional) maximum likelihood theory, such as the Hill's method for the index of regular variation, are thoroughly investigated. Using the same approach, here we also consider the problem of estimating the extremal index of the sequence {f(Xn)}n2N under suitable assumptions. Eventually, practical issues related to the application of the methodology we propose are discussed and preliminary simulation results are displayed.

Estimation of the memory parameter of the infinite-source Poisson process

Long-range dependence induced by heavy tails is a widely reported feature of internet traffic. Long-range dependence can be defined as the regular variation of the variance of the integrated process, and half the index of regular variation is then referred to as the Hurst index. The infinite-source Poisson process (a particular case of which is the $M/G/\infty$ queue) is a simple and popular model with this property, when the tail of the service time distribution is regularly varying. The Hurst index of the infinite-source Poisson process is then related to the index of regular variation of the service times. In this paper, we present a wavelet-based estimator of the Hurst index of this process, when it is observed either continuously or discretely over an increasing time interval. Our estimator is shown to be consistent and robust to some form of non-stationarity. Its rate of convergence is investigated.

Assessing confidence intervals for the tail index by Edgeworth expansions for the Hill estimator

We establish Edgeworth expansions for the distribution function of the standardized Hill estimator for the reciprocal of the index of regular variation of the tail of a distribution function. The expansions are used to derive expansions for coverage probabilities of confidence intervals for the tail index based on the Hill estimator.

Regular variation of order 1 nonlinear AR-ARCH models

We prove both geometric ergodicity and regular variation of the stationary distribution for a class of nonlinear stochastic recursions that includes nonlinear AR-ARCH models of order 1. The Lyapounov exponent for the model, the index of regular variation and the spectral measure for the regular variation all are characterized by a simple two-state Markov chain.

Assessing Confidence Intervals for the Tail Index by Edgeworth Expansions for the Hill Estimator

We establish Edgeworth expansions for the distribution function of the centered and normalized Hill estimator for the reciprocal of the index of regular variation of the tail of a distribution function.The expansions are used to derive expansions for coverage probabilities of confidence intervals for the tail index based on the Hill estimator.

Asymptotically best linear unbiased tail estimators under a second-order regular variation condition

For regularly varying tails, estimation of the index of regular variation, or tail index, γ , is often performed through the classical Hill estimator, a statistic strongly dependent on the number k of top order statistics used, and with a high asymptotic bias as k increases, unless the underlying model is a strict Pareto model. First, on the basis of the asymptotic structure of Hill's estimator for different k-values, we propose “asymptotically best linear (BL) unbiased” estimators of the tail index. A similar derivation on the basis of the log-excesses and of the scaled log-spacings is performed and the adequate weights for the largest log-observations are provided. The asymptotic behaviour of those estimators is derived, and they are compared with other alternative estimators, both asymptotically and for finite samples. As an overall conclusion we may say that even asymptotic equivalent estimators may exhibit very diversified finite sample properties.