Tail Dependence for Regularly Varying Time Series

Abstract

We use tail dependence functions to study tail dependence for regularly varying (RV) time series. First, tail dependence functions about RV time series are deduced through the intensity measure. Then, the relation between the tail dependence function and the intensity measure is established: they are biuniquely determined. Finally, we obtain the expressions of the tail dependence parameters based on the expectation of the RV components of the time series. These expressions are coincided with those obtained by the conditional probability. Some simulation examples are demonstrated to verify the results we established in this paper.

1. Introduction

Copula is a useful tool for handling multivariate distributions with given univariate margins. A copula 𝐶 is a distribution function, defined on the unit cube [0,1]^𝑑, with uniform one-dimensional margins 𝑈_𝑖. For any (𝑢₁,…,𝑢_𝑑)∈[0,1]^𝑑, 𝐶(𝑢₁,…,𝑢_𝑑)=𝑃{𝑈₁≤𝑢₁,…,𝑈_𝑑≤𝑢_𝑑}; the survival copula is 𝐶(𝑢₁,…,𝑢_𝑑)=𝑃{𝑈₁≥1−𝑢₁,…,𝑈_𝑑≥1−𝑢_𝑑}, the joint survival function of copula 𝐶 is 𝐶(𝑢₁,…,𝑢_𝑑)=𝐶(1−𝑢₁,…,1−𝑢_𝑑). Given a copula 𝐶, let𝐹𝑡₁,…,𝑡_𝑑=𝐶𝐹₁𝑡₁,…,𝐹_𝑑𝑡_𝑑,where𝑡₁,…,𝑡_𝑑∈ℝ^𝑑, (1.1) then 𝐹 is a multivariate distribution with univariate margins 𝐹₁,…,𝐹_𝑑. On the other hand, given a distribution 𝐹 with margins 𝐹₁,…,𝐹_𝑑, there exists a copula 𝐶 such that (1.1) holds. And copula 𝐶 is unique if 𝐹₁,…,𝐹_𝑑 are all continuous (Sklar [1], Nelsen [2]).

In generally, copula forms a natural way to describe the dependence between series when making abstraction of their marginal distributions. Overviews of the probabilistic and statistical properties of copula are to be found in [1–6].

Tail dependence plays an important role among dependence measures due to its ability to describe dependence among extreme values (Frahm et al. [7], Resnick [8, 9], and Nikoloulopoulos et al. [10]) which is introduced by Joe [4]. The issue of tail dependence is mainly for heavy tailed phenomena, heavy tailed phenomenon in fractal time series. It is extensively studied and applied in insurance, risk management, traffic management and engineering management, and so forth. [11–27].

Researchers find various multivariate distributions with heavy tails to describe the extremal or tail dependence, see, Pisarenko and Rodkin [13], Hult and Lindskog [28], and Fang et al. [29]. Many interesting tail quantities have been derived via standard methods: coefficients of tail dependence [30–37] and tail dependence copulas (Charpentier and Segers [38]).

In this paper, we are interested in the tail behavior of the time series 𝑋₁,…,𝑋_𝑡 which have the form:𝐗=𝑋₁,…,𝑋_𝑡=𝑅𝑍₁,…,𝑅𝑍_𝑡, (1.2) where the scale variable 𝑅 is independent of random vector (𝑍₁,…,𝑍_𝑡). And 𝐗 is multivariate regularly varying with distribution function 𝐹 having copula 𝐶.

This distribution is a generalized class, including, for example, multivariate Pareto and multivariate elliptical distribution as special ones. Especially, the multivariate 𝑡 distribution is included in it. As an example, we will justify the results through multivariate 𝑡 copula.

In order to analyze the tail dependence behavior of (1.2), we first study the tail dependence functions via intensity measure. Then using the relation between tail dependence parameter and the tail dependence functions, we explore the explicit representations of the tail dependence parameters.

The outline of this paper is as follows. After some preliminaries about multivariate regularly varying series and dependence functions in Section 2, detailed results for the tail dependence functions are discussed in Section 3, the expressions of tail dependence parameters for RV time series are demonstrated in Section 4, and multivariate 𝑡 distribution is demonstrated as an example in Section 5.

Throughout, (𝑋₁,…,𝑋_𝑡) is a random vector with joint distribution function 𝐹 and copula 𝐶. Minima and maxima will be denoted by ∧ and ∨, respectively. The Cartesian product ∏^𝑡_𝑖=1[𝑎_𝑖,𝑏_𝑖] is denoted by [𝐚,𝐛] for any 𝐚,𝐛∈𝑅^𝑡.

2. Preliminaries

Definition 2.1. The 𝑡-dimensional random vector 𝐗 is said to be regularly varying with index 𝛼≥0 if there exists a random vector Θ with values in 𝕊^𝑡−1 a.s., where 𝕊^𝑡−1 denotes the unit sphere in 𝑅^𝑑 with respect to the norm |⋅|, such that, for all 𝑢>0, 𝑃{|𝐗|>𝑢𝑥,𝐗/|𝐗|∈⋅}𝑃{|𝐗|>𝑥}^𝑣−→𝑢^−𝛼𝑃{Θ∈⋅}, (2.1) as 𝑥→∞. The symbol −→^𝑣 stands for vague convergence on 𝕊^𝑡−1; vague convergence of measures is treated in detail in Kallenberg [39]. The distribution of Θ is referred to as the spectral measure of 𝐗. For further information on multivariate regular variation we refer to Resnick [8, 9].
In fact, (2.1) is equivalent to the following expression 𝑛𝑃𝑎_𝑛⁻¹𝐗∈⋅−→^𝑣𝜇(⋅), (2.2) where 𝜇 is an intensity measure or Radon measure on 𝑅/{0} and {𝑎_𝑛} is a sequence an of nonnegative numbers.

From the Definition 2.1, we can see that the regularly varying distribution is connected with intensity measure 𝜇. The following lemma yields the explicit relation between them which can be found in [8].

Lemma 2.2. Let random vector 𝐗 be regularly varying with index 𝛼≥0 and distribution function 𝐹, then it is equivalent to the following.(1)There exists an intensity measure 𝜇 on 𝑅^𝑡/{0}, such that for every Borel set 𝐵⊂𝑅^𝑡/{0} bounded away from the origin that satisfies 𝜇(𝜕𝐵)=0, lim_𝑢→∞𝑃{𝐗∈𝑢𝐵}𝑃𝐗|}(𝐵){|>𝑢=𝜇, (2.3) with the homogeneous condition 𝜇(𝑢𝐵)=𝑢^−𝛼𝜇(𝐵).(2)There exists an intensity measure 𝜇 on 𝑅^𝑡/{0}, such that lim_𝑢→∞1−𝐹𝑢𝑥₁,𝑢𝑥₂,…,𝑢𝑥_𝑡()=𝑃[]1−𝐹𝑢,𝑢,…,𝑢𝐗/𝑢∈0,𝐱^𝑐𝑃{𝐗/𝑢∈[0,1]^𝑐}=𝜇[0,𝐱]^𝑐, (2.4) for all continuous points 𝐱 of 𝜇. According to Lemma 2.2, one notices that for any nonnegative multivariate regularly varying random vector 𝐗, its nondegenerate univariate margins X_𝑖 have regularly varying right tails and with the same index of 𝐗 also, that is, 𝐹_𝑖(𝑥)=𝑃𝐗_𝑖>𝑥=𝑥^−𝛼𝐿_𝑖(𝑥),𝑥>0, (2.5) where 𝐿_𝑖(𝑥) is a slowly varying function.

Lemma 2.3 (Breiman [40]). Let 𝜉 and 𝜂 be two independent nonnegative random variables, 𝜂 be regularly varying with index 𝛼. If there exists a 𝛾>𝛼, such that 𝐸𝜉^𝛾<∞, then 𝑃𝜉𝜂>𝑥∼𝐸𝜉^𝛼𝑃𝜂>𝑥. (2.6) The multivariate version of the Lemma belongs to Basrak et al. [41]. It is said that, if 𝐗 is regularly varying in the sense of (2.2), 𝐀 is a random 𝑡×𝑡 matrix, independent of 𝐗, with 0<𝐸‖𝐀‖^𝛾<∞ for some 𝛾>𝛼, then 𝑛𝑃𝑎_𝑛⁻¹𝐀𝑋∈⋅−→^𝑣𝜇(⋅)∶=𝐸𝜇∘𝐀⁻¹(⋅), (2.7) where −→^𝑣 denotes vague convergence on ℝ^𝑡/{0}.

Definition 2.4 (Kluppelberg et al. [42]). Let 𝐹 be the distribution function of random vector 𝐗 with continuous margins 𝐹_𝑖,1≤𝑖≤𝑡 and copula 𝐶. For any 𝐰=(𝑤₁,𝑤₂,…,𝑤_𝑡)∈𝑅^𝑡₊, the lower dependence function is defined as 𝑙(𝐰;𝐶)=lim_𝑥→0+𝐶𝑥𝑤₁,𝑥𝑤₂,…,𝑥𝑤_𝑡𝑥, (2.8) and the upper dependence function is defined as 𝑢(𝐰;𝐶)=lim_𝑥→0+𝐶1−𝑥𝑤₁,1−𝑥𝑤₂,…,1−𝑥𝑤_𝑡𝑥. (2.9) The upper exponent function is defined as 𝑢^∗(𝐰;𝐶)=_{∅≠𝑆⊂𝐼}(−1)^|𝑆|−1𝑢_𝑆𝑤_𝑆;𝐶_𝑆, (2.10) where 𝑢_𝑆(𝑤_𝑆;𝐶_𝑆)=lim_𝑥→0+𝐶(1−𝑥𝑤_𝑗,∀𝑗∈𝑆)/𝑥.
From the definition, we can verify the elementary properties listed in Proposition 2.5 of the tail dependence function. We denote 𝜏_𝐽=lim_𝑥→1−𝑃{𝐹_𝑗(𝑋_𝑗)>𝑥,∀𝑗∉𝐉∣𝐹_𝑖(𝑋_𝑖)>𝑥,∀𝑖∈𝐉} and 𝜉_𝐽=lim_𝑥→0+𝑃{𝐹_𝑗(𝑋_𝑗)<𝑥,∀𝑗∉𝐉∣𝐹_𝑖(𝑋_𝑖)<𝑥,∀𝑖∈𝐉} are the upper tail and lower dependence parameters of 𝐗, respectively, where 𝐽 is a nonempty subset of 𝐼={1,…,𝑡}. 𝐶_𝐽 is the margin of 𝐶 with component indexes in 𝐽.

Proposition 2.5. (1) For any 1≤𝑖,𝑗≤𝑡, 𝜏_𝑖𝑗=𝑢1,1;𝐶_𝑖𝑗;𝜉_𝑖𝑗=𝑙1,1;𝐶_𝑖𝑗, (2.11) where 𝐶_𝑖𝑗 is the margin copula of 𝑋_𝑖,𝑋_𝑗.(2)For any nonempty 𝐽⊂𝐼,𝜏_𝐽=𝑢(1,1,…,1;𝐶)𝑢1,1,…,1;𝐶_𝐽;𝜉_𝐽=𝑙(1,1,…,1;𝐶)𝑙1,1,…,1;𝐶_𝐽, (2.12)(3)𝑢(𝐰;𝐶)=lim_𝑥→0+𝐶𝑥𝑤₁,𝑥𝑤₂,…,𝑥𝑤_𝑡𝑥=𝑙𝐰;𝐶. (2.13)

Proof. (1) According to the definition of 𝜏_𝑖𝑗, we get 𝜏_𝑖𝑗=lim_𝑥→1−𝑃𝐹_𝑗𝑋_𝑗>𝑥∣𝐹_𝑖𝑋_𝑖>𝑥=lim_𝑥→0+𝑃𝐹_𝑗𝑋_𝑗>1−𝑥,𝐹_𝑖𝑋_𝑖>1−𝑥𝑃𝐹_𝑖𝑋_𝑖>1−𝑥=lim_𝑥→0+𝐶_𝑖𝑗(1−𝑥,1−𝑥)𝑥=𝑢1,1;𝐶_𝑖𝑗; (2.14) similarly, 𝜉_𝑖𝑗=lim_𝑥→0+𝑃𝐹_𝑗𝑋_𝑗<𝑥∣𝐹_𝑖𝑋_𝑖<𝑥=lim_𝑥→0+𝑃𝐹_𝑗𝑋_𝑗<𝑥,𝐹_𝑖𝑋_𝑖<𝑥𝑃𝐹_𝑖𝑋_𝑖<𝑥=lim_𝑥→0+𝐶_𝑖𝑗(𝑥,𝑥)𝑥=𝑙1,1;𝐶_𝑖𝑗. (2.15)(2)Note that𝜏_𝐽=lim_𝑥→1−𝑃𝐹_𝑗𝑋_𝑗>𝑥,∀𝑗∈𝐼𝑃𝐹_𝑖𝑋_𝑖>𝑥,∀𝑖∈𝐉=lim_𝑥→0+𝐶(1−𝑥,…,1−𝑥)/𝑥𝐶_𝐽(1−𝑥,…,1−𝑥)/𝑥 (2.16) combined with (2.9), the first part is determined. The second part can be verified similarly.(3)We can obtained the proof only paying attention to 𝐶(𝑢₁,…,𝑢_𝑡)=𝐶(1−𝑢₁,…,1−𝑢_𝑡).
From the proposition, the upper tail dependence function of copula 𝐶 is the lower one of its survival copula 𝐶. And in most fractal time series, from the point of view of either theory or applications, people only need to understand the right tail of the data, so we focus on the upper tail function 𝑢(𝐰;𝐶) and coefficient 𝜏_𝐽 in the following.
We first study the upper tail dependence function of multivariate regularly varying time series in (1.2) using the intensity measure.

3. The Upper Tail Dependence Function for RV Time Series

Theorem 3.1. Let 𝐗₁,…,𝑋_𝑡 be RV time series with regularly varying index 𝛼, distribution function 𝐹, copula 𝐶, and the stochastic representation as (1.2). If the margins are tail equivalent as 𝑥→∞, then the upper tail dependence function can be written as 𝑢()=𝜇∏𝐰;𝐶^𝑡𝑖=1𝑤_𝑖^−1/𝛼,∞𝜇[1,∞]×𝑅^𝑡−1, (3.1) and the upper exponent function can be written as 𝑢^∗()=𝜇∏𝐰;𝐶^𝑡𝑖=10,𝑤_𝑖^−1/𝛼c𝜇[0,1]×𝑅^𝑡−1c. (3.2)

Proof. For any 𝐰=(𝑤₁,…,𝑤_𝑡)∈𝑅^𝑡₊, 𝑢(𝐰;𝐶)=lim_𝑥→0+𝑃𝐹_𝑖𝑋_𝑖≤𝑥𝑤_𝑖,∀𝑖∈𝐼𝑃𝐹₁𝑋₁≤𝑥=lim_𝑥→0+𝑃𝑋_𝑖>𝐅_𝑖⁻¹𝑥𝑤_𝑖,∀𝑖∈𝐼𝑃𝑋₁>𝐅₁⁻¹(𝑥). (3.3) Since every margin 𝐹_𝑖 is regularly varying with the same index 𝛼, we obtain that 𝐹_𝑖𝑦=𝐿_𝑖𝑦𝑦^𝛼,𝑦>0, (3.4) where 𝐿_𝑖(𝑦) is slowing varying function. So for any 𝑤_𝑖>0, as 𝑥→0⁺, 𝐹_𝑖𝑤_𝑖^1/𝛼𝑦=𝐿_𝑖𝑤_𝑖^1/𝛼𝑦𝑤_𝑖𝑦^𝛼=1𝑤_𝑖⋅𝐿_𝑖𝑤_𝑖^1/𝛼𝑦𝐿_𝑖𝑦⋅𝐿_𝑖𝑦𝑦^𝛼=1𝑤_𝑖⋅ℎ_𝑖𝑤_𝑖,𝑦⋅𝐹_𝑖𝑦, (3.5) where ℎ_𝑖(𝑤_𝑖,𝑦)=𝐿_𝑖(𝑤_𝑖^1/𝛼𝑦)/𝐿_𝑖(𝑦)→1 as 𝑦→∞. So the equation becomes 𝐹_𝑖𝑤_𝑖^1/𝛼𝑦=1𝑤_𝑖⋅ℎ_𝑖𝑤_𝑖,𝑦⋅𝐹_𝑖𝑦, (3.6) in other words, 𝑤_𝑖^1/𝛼𝑦=𝐅_𝑖⁻¹1𝑤_𝑖𝐹_𝑖𝑦ℎ_𝑖𝑤_𝑖,𝑦. (3.7) Now we let 𝐹_𝑖(𝑦)=𝑥𝑤_𝑖, then 𝑤_𝑖^1/𝛼𝐅_𝑖⁻¹𝑥𝑤_𝑖=𝐅_𝑖⁻¹𝑥ℎ_𝑖𝑤_𝑖,𝐅_𝑖⁻¹𝑥𝑤_𝑖, (3.8) so, 𝐅_𝑖⁻¹(𝑥𝑤_𝑖)=𝑤_𝑖^−1/𝛼𝐅_𝑖⁻¹(𝑥ℎ_𝑖(𝑤_𝑖,𝐅_𝑖⁻¹(𝑥𝑤_𝑖))).
As 𝑥→0⁺, ℎ_𝑖(𝑤_𝑖,𝐅_𝑖⁻¹(𝑥𝑤_𝑖))→1, so we get that 𝐅_𝑖⁻¹𝑥𝑤_𝑖≈𝑤_𝑖^−1/𝛼𝐅_𝑖⁻¹(𝑥). (3.9)
And since the margins are equivalent, that is, 𝐹_𝑖(𝑦)/𝐹₁(𝑦)→1 as 𝑦→∞. We have 𝐅_𝑖⁻¹(𝑥)/𝐅₁⁻¹(𝑥)→1 as 𝑥→0⁺ (Resnick [8]). So for sufficient small 𝑥, 𝐅_𝑖⁻¹(𝑥)≈𝐅₁⁻¹(𝑥), and 𝑧=𝐅₁⁻¹(𝑥), combining (3.3) and (2.3), we obtain that 𝑢(𝐰;𝐶)=lim_𝑥→0+𝑃𝑋_𝑖>𝑤_𝑖^−1/𝛼𝐅_𝑖⁻¹(𝑥),∀𝑖∈𝐼𝑃𝑋₁>𝐅₁⁻¹(𝑥)=lim_𝑧→∞𝑃𝑋_𝑖>𝑤_𝑖^−1/𝛼𝑧,∀𝑖∈𝐼𝑃𝑋₁=𝜇∏>𝑧_𝑡𝑖=1𝑤_𝑖^−1/𝛼,∞𝜇[1,∞]×𝑅^𝑡−1. (3.10)
In order to calculate 𝑢^∗(𝐰;𝐶), we recall the inclusion-exclusion formula, it says that 𝑃∩_𝑖∈𝐼𝐴_𝑖=_{∅≠𝑆⊂𝐼}(−1)^|𝑆|−1𝑃∪_𝑗∈𝑆𝐴_𝑗 (3.11) is valid for any finite set 𝐼 and arbitrary events 𝐴_𝑖, where 𝑖∈𝐼.
Using this formula, (2.10) becomes 𝑢^∗(𝐰;𝐶)=lim_𝑥→0+𝑃𝐹_𝑗𝑋_𝑗>1−𝑥𝑤_𝑗,∃𝑗∈𝐼𝑥=lim_𝑥→0+𝑃𝐹_𝑗𝑋_𝑗≤𝑥𝑤_𝑗,∃𝑗∈𝐼𝑃𝐹₁𝑋₁≤𝑥=lim_𝑥→0+𝑃𝑋_𝑗>𝐅_𝑗⁻¹𝑥𝑤_𝑗,∃𝑗∈𝐼𝑃𝑋₁>𝐅₁⁻¹(𝑥.) (3.12) By using the same method of (3.3), the following equation holds: 𝑢^∗(𝐰;𝐶)=lim_𝑧→∞𝑃𝑋_𝑖>𝑤_𝑖^−1/𝛼𝑧,∃𝑖∈𝐼𝑃𝑋₁=𝜇∏>𝑧^𝑡𝑖=10,𝑤_𝑖^{−1/𝛼𝑐}𝜇[1,∞]×𝑅^𝑡−1. (3.13)

Corollary 3.2. Under the same conditions as Theorem 3.1, the following result holds 𝜇[1,∞]×𝑅^𝑡−1=1𝑢^∗().1,…,1;𝐶 (3.14)

Proof. By (2.4), one can see that 𝜇([0,1]^𝑐)=1. So we can get the result immediately by letting all 𝑤_𝑖=1,1≤𝑖≤𝑡 in (3.2).
According to Theorem 3.1 and Corollary 3.2, we can represent the intensity measure through the tail dependence function as the following Corollary.

Corollary 3.3. Under the same conditions as Theorem 3.1, one has 𝜇=𝑢𝑤([𝐰,∞])^1−𝛼,…,𝑤_𝑡^−𝛼;𝐶𝑢^∗(),𝜇[]1,…,1;𝐶0,𝐰^𝑐=𝑢^∗𝑤₁^−𝛼,…,𝑤_𝑡^−𝛼;𝐶𝑢^∗().1,…,1;𝐶 (3.15)

4. The Upper Tail Dependence Parameters for Regularly Varying Time Series

According to Proposition 2.5 and Theorem 3.1, we can express the tail dependence parameters by their tail dependence functions. In this section, we will deduce the upper tail dependence parameters of time series with multivariate varying distribution in (1.2) by this method. Hereafter, we let 𝜇 be the intensity measure of 𝐑=(𝑅,𝑅,…,𝑅) with copula 𝐶^𝑅. Where 𝑅 is regularly varying at ∞ with index 𝛼, with survival function 𝐹_𝑅(𝑟)=𝐿(𝑟)/𝑟^𝛼, and 𝐿(⋅) is a slowly varying function. So for any nonnegative vector 𝐰=(𝑤₁,…,𝑤_𝑡), we have𝜇[𝟎,𝐰]^𝑐;𝐶^𝑅=lim_𝑟→∞𝑃𝑅>𝑟∧^𝑡_𝑖=1𝑤_𝑖𝑃{𝑅>𝑟}=lim_𝑟→∞𝐹_𝑅𝑟⋀^𝑡_𝑖=1𝑤_𝑖𝐹_𝑅(𝑟), (4.1) by inserting 𝐹_𝑅(𝑟∧^𝑡_𝑖=1𝑤_𝑖)=𝐿(𝑟⋀^𝑡_𝑖=1𝑤_𝑖)/(𝑟⋀^𝑡_𝑖=1𝑤_𝑖)^𝛼 and 𝐹_𝑅(𝑟)=𝐿(𝑟)/𝑟^𝛼 into the representation, then,𝜇[𝟎,𝐰]^𝑐;𝐶^𝑅=1⋀^𝑡_𝑖=1𝑤_𝑖^𝛼=^𝑡_𝑖=1𝑤_𝑖^−𝛼. (4.2) Similarly, we have,𝜇[𝐰,∞];𝐶^𝑅=^𝑡_𝑖=1𝑤_𝑖^−𝛼. (4.3) Consequently, we get the main result as follows.

Theorem 4.1. Let 𝑋₁,…,𝑋_𝑡 be regularly varying time series with the same regularly varying index 𝛼 and the stochastic representation given in (1.2), the margins are tail equivalent as 𝑥→∞. If there exists a 𝛾>𝛼 holds for 0<𝐸(𝑍^𝛾_𝑖+)<∞, then the upper tail dependence parameter of 𝑋₁,…,𝑋_𝑡 is 𝜏_𝐽=E⋀^𝑡_𝑖=1𝑍^𝛼_𝑖+/𝐸𝑍^𝛼_𝑖+𝐸⋀_𝑖∈𝐽𝑍^𝛼_𝑖+/𝐸𝑍^𝛼_𝑖+. (4.4)

Proof. We first calculate the tail dependence function of 𝐗=(𝑅𝑍₁,…,𝑅𝑍_𝑡). In the following, let 𝐶^𝑋 and 𝐶^𝑌 be the copula of 𝐗 and 𝐘, respectively. Denote 𝑌₁,…,𝑌_𝑡^𝑇=𝐀(𝑅,…,𝑅)^𝑇, (4.5) where 𝑍𝐀=diag¹⁺𝐸𝑍^𝛼₁₊^1/𝛼𝑍,…,^𝑡+𝐸𝑍^𝛼₁₊^1/𝛼. (4.6)
Note that 𝑌_𝑖=(𝑍_𝑖+/(𝐸(𝑍^𝛼_i+))^1/𝛼)𝑅 is strictly increasing transformation of 𝑋_𝑖>0,forall𝑖∈𝐼, and the tail dependence function and the parameter are all copula properties. Hence 𝐘 and 𝐗 have the same tail dependence functions. By Lemma 2.3, one can see that the marginal variables 𝑌_𝑖 of vector 𝐘 are tail equivalent and regularly varying with the same index as 𝐗 as 𝑥→∞. Denote the intensity measures of 𝐘 and 𝐑 by 𝜇(⋅) and 𝜇(⋅), respectively. According to (2.7), 𝜇(⋅)=𝐸𝜇𝐀⁻¹⋅. (4.7) Now by (4.6), we see that, 𝐀⁻¹𝐸𝑍=diag^𝛼1+1/𝛼𝑍₁₊𝐸𝑍,…,^𝛼t+1/𝛼𝑍_𝑡+, (4.8) combining this with (4.3), for any nonnegative 𝐰, we obtain the intensity measure given by 𝜇([𝐰,∞])=𝐸𝜇𝐀⁻¹[𝐰,∞]=𝐸𝜇^𝑡_𝑖=1𝐸𝑍^𝛼_i+^1/𝛼𝑍_𝑖+𝑤_𝑖,∞;𝐶^𝑅=𝐸^𝑡_𝑖=1𝑍^𝛼_𝑖+𝐸𝑍^𝛼_𝑖+𝑤_𝑖^−𝛼. (4.9) Hence, we have 𝜇^𝑡_𝑖=1𝐰_𝑖^−𝛼,∞=𝐸^𝑡_𝑖=1𝑍^𝛼_𝑖+𝐸𝑍^𝛼_𝑖+𝑤_𝑖. (4.10) Substituting this measure into (3.1), we get the upper tail dependence function of vector 𝐘 as follows: 𝑢𝐰;𝐶^𝑌=𝐸^𝑡_𝑖=1𝑍^𝛼_𝑖+𝐸𝑍^𝛼_𝑖+𝑤_𝑖. (4.11) Since 𝐘 and 𝐗 have the same tail dependence functions, we have 𝑢𝐰;𝐶^𝑋=𝐸^𝑡_𝑖=1𝑍^𝛼_𝑖+𝐸𝑍^𝛼_𝑖+𝑤_𝑖. (4.12) By (2) in Proposition 2.5, we obtain the upper tail dependence parameters of vector 𝐗.

5. Examples

Let 𝐙 in (1.2) be 𝐙=𝐴(𝑈₁,…,𝑈_𝑛), where 𝐴 is a 𝑡×𝑛 matrix with 𝐴𝐴^𝑇=Σ, and Σ is a 𝑡×𝑡 semidefinite matrix, 𝑈=(𝑈₁,…,𝑈_𝑛) is uniformly distributed on the unit sphere (with respect to Euclidean distance) in 𝑅^𝑛. We know that 𝐗 conforms to an elliptical contoured distribution (Fang et al. [43]). The tail dependence of the elliptical contoured distribution has been discussed in Schmidt [33]. Here we select the 𝑡 distribution to display our results in Theorem 4.1 as a special case.

If 𝐗∼𝕥_𝑛(𝝁,𝚺,𝜈), then 𝐗 has the stochastic representation ([43]):√𝐗=𝝁+𝜈√𝑆𝐙, (5.1) where 𝑆∼𝜒²_𝜈 and 𝐙∼ℕ_𝑛(𝟎,𝚺) are independent, 𝝁∈𝑅^𝑛.

Let 𝑅=𝜈/𝑆. Then 𝑅²∼𝐼𝐺(𝜈/2,𝜈/2) and 𝑅 is regularly varying with index 𝜈 at ∞. So the vector (𝑋₁,…,𝑋_𝑛) is regularly varying according to Schmidt [33].

For the upper tail dependence that only relies on the tail behavior of the random vector, we can focus, without loss of generality, on the random vector 𝐗 with zero mean vector. Furthermore, since the strictly increasing transformation of (𝑋₁,…,𝑋_𝑛) does not change the copula, Δ^−1/2𝐗 has the same copula as 𝐗, where 𝚺=(𝜎_𝑖𝑗) and Δ=diag(𝜎₁₁,𝜎₂₂,…,𝜎_𝑛𝑛). Thus Δ^−1/2𝐗∼𝕥_𝑛(0,Δ^−1/2𝚺Δ^−1/2,𝜈). It is evident that Δ^−1/2𝚺Δ^−1/2 becomes the correlation matrix of the random vector. Consequently, we may assume that the covariance matrix 𝚺 is the correlation matrix. In this situation, all 𝑍^′_𝑖𝑠 have the same margins as 𝑁(0,1). So 𝐸(𝑍^𝜈_𝑖+) are all equal for any 1≤𝑖≤𝑛. Under these assumptions, using (4.4), we get the upper tail dependence parameter of 𝐭_𝑛(0,𝚺,𝜈) as𝜏_𝐽=𝐸⋀^𝑛_𝑖=1𝑍^𝜈_𝑖+𝐸⋀_𝑖∈𝐽𝑍^𝜈_𝑖+. (5.2) This is coincided to the one obtained in Shi and Lin [34].

6. Simulations

In Section 4, we obtain the expressions of the tail dependence indexes about RV time series in (1.2). In Section 5, we display our result in the multivariate 𝑡 distribution as example. In this Section, we will illustrate these results by some Monte Carlo simulated numerical examples. Given that 𝑦⁽¹⁾,𝑦⁽²⁾,…,𝑦^(𝑚) be generated from the multivariate normal distribution 𝐍_𝑛(𝟎,𝝆), then the upper tail dependence indices of 𝐭_𝑛(𝝁,𝚺,𝜈) can be estimated by𝜏_𝐽=∑^𝑚_𝑘=1⋀^𝑛_𝑗=1𝑦_𝑗^(𝑘)𝜈𝐼𝑦^(𝑘)>0or𝑦^(𝑘)<0∑^𝑚_𝑘=1⋀_𝑖∈𝐽𝑦_𝑖^(𝑘)𝜈𝐼𝑦^(𝑘)>0or𝑦^(𝑘).<0 (6.1)

We estimate the upper tail dependence parameter of 3-dimensional 𝑡 distribution under autoregressive of order 1 (AR(1)), exchangeable(EX), Toeplitz(TOEP), and unstructured(UN) correlation structure, respectively. For each correlation matrix, we first generate 80,000 pseudorandom vectors, then use (5.2) to estimate tail dependence parameter for different 𝜈. Specifically, we do the following simulations.₁=,1−0.30.09−0.31−0.30.09−0.31²=1−0.3−0.3−0.31−0.3−0.3−0.31. (6.2)

Let 𝐉={2} and {1,2}, respectively. The corresponding upper tail dependence parameters are denoted by 𝜏₂ and 𝜏₁₂. 𝚺₁ and 𝚺₂ are under AR(1) and EX correlation structure, respectively, the simulated values of 𝜏₂, 𝜏₁₂ about different 𝜈 are computed and plotted in Figure 1. 𝚺₃ and 𝚺₄ are under TOEP and UN correlation structure, the corresponding results are demonstrated in Figure 2.

(a)

(b)

Figure 1 

The estimation of 𝜏2, 𝜏12 under AR(1) (the left one) and EX (the right one) correlation structure.

(a)

(b)

Figure 2 

The estimation of 𝜏2, 𝜏12 under TOEP (the left one) and UN (the right one) correlation structure.

From the two figures, in spite of the correlation structure, 𝜏_𝐽 decreased and approached 0 quickly as 𝜈 increased to ∞, which is the tail dependence index for multivariate normal copula.

Many researchers try to discuss the monotonicity of the tail dependence parameter about the regular varying index. Embrechts et al. [11] proved that the tail dependence of the bivariate 𝑡 distribution is decreasing about the regular varying index 𝜈, and demonstrated that the tail dependence parameter 𝜏₁ is decreasing in 𝜈 by numerical results. But From the right graph in Figure 2., these conclusions are not always correct when 𝑡≥3.₃=,1−0.30.5−0.31−0.30.5−0.31⁴=10.30.50.310.70.50.71. (6.3)

7. Conclusion

In the paper, we mainly study tail dependence of RV time series in (1.2). We use tail dependence function and intensity measure to express tail dependence parameters. Using tail dependence function, we do not need to consider the explicit representation of the copula. We first discuss the tail dependence function of the RV time series due to the propositions of the regularly varying function, connecting the biuniquely determined property between the tail dependence function and the intensity measure. Then we calculate the explicit formula of the upper tail dependence parameter about the RV time series under some conditions. In fact, we can obtain the extreme upper tail dependence index (Shi and Lin [34]) very similarly to Theorem 4.1, for concise, we omit it here.

Copula of continuous variables is invariant under strictly increasing transformation (Nelsen [2]). In order to obtain the tail dependence function of random vector 𝐗, we shift to solve that of 𝐘 in (4.5), which is just a strictly increasing transformation of 𝐗.

At last, we select the 𝑡 distribution as a special case to display our result, they are coincided to the one given in [34]. The monotonicity of the tail dependence parameters about the regular varying index is still an open problem. Under what constraints the tail dependence parameters will be deceasing in the variation index? We are still interested in the problem. We will discuss it in the following work in details. In engineering application, when we confront fractal time series and seasonal data, we can model the tail dependence property via the tail dependence function if the data is consistent with the constraint conditions in our work.

Acknowledgment

The authors are very grateful to the referees for their comments and suggestions on the earlier version of the paper, which led to a much improved paper. This project is supported by MSRFSEU 3207011102, NSFC 11171065, and NSFJS BK2011058.