In diverse fields such as finance, meteorological hydrorisk management, and, internet traffic operation, it is a prominent issue to model multivariate extreme events. Multivariate regular variation is a popular framework for multivariate extreme value analysis (cf. Cai et al., 2011; Weller and Cooly, 2014; Li and Hua, 2015; Einmahl et al., 2020). It describes the behavior of multivariate extremes in terms of their radii and directions (cf. Resnick, 2008): The directional behavior is specified by a probability measure on a measurable space of unit vectors, which is called spectral measure. Meanwhile the radius obeys a univariate regular variation, and furthermore the radius and direction are asymptotically independent as the former gets large. However, for efficient estimation, it is crucially required to introduce a suitable model for the spectral measure which itself has no parameter of finite dimension. In such a view, elliptical distributions have received much attention since they not only constitute a subclass of multivariate regular variation but also provide a tractable parametric model for spectral measure and tail dependence (cf. Hult and Lindskog, 2002; Klüppelberg et al., 2007; Joe and Li, 2019).

Kim (2021) derived a density function of the spectral measure of elliptical distributions and then proposed a maximum likelihood estimator based on it. However, for investigating the property of the resulting estimator, we need to specify how accurately the directional behavior at extreme level in finite sample is described by the spectral measure. This paper mainly focuses on the second order property of convergence to spectral measure for the class of elliptical distributions: Let {f_t(x) : t > 0}, and f (x) be real valued functions defined on a space X such that f_t(x) → f (x) as t → 0 for each x ∈ X. In this case, the second order property of the convergence indicates the behavior characterized by γ > 0 and a real valued function g(x) such that

By imposing a widely adopted second order (univariate) regular variation condition on the representation of elliptical distributions (cf. Hall, 1982; Hult and Lindskog, 2002), this paper derives the second order behaviors with respect to radii and directions of the multivariate extremes.

The remainder of this paper is organized as follows: Section 2 presents some preliminary definitions and theorem, and briefly reviews the second order property of univariate cases; Section 3 deals with the second order property of elliptical multivariate regular variation.

2.1. Elliptical multivariate regular variation

This section provides the preliminary definitions and theorem. First, it is necessary to clarify the topological space on which the spectral measure is defined. Let

denote the unit sphere in d-dimensional space for a positive integer d ≥ 2. is equipped with the metric that takes |s₁ – s₂| as the distance between s₁ and s₂ in , where |·| indicates the Euclidean norm. The induced topology generates a Borel σ-field on , which is denoted by hereafter.

Let X be a d-dimensional random vector, and R = |X| and S = X/R represent the radius and direction of X. Then, R and S are positive and -valued random variables, respectively. X is said to have multivariate regularly varying tail if there exist α > 0 and a probability Borel measure Λ on such that for x > 0 and with Λ(∂A) = 0,

(cf. Resnick, 2008). Taking , the above equation reduces to

i.e., x ↦ P(R > x) is (univariate) regularly varying with exponent −α as x → ∞ (Bingham et al., 1987). It means that the conditional distribution of t|X| given t|X| > 1 converges to a Pareto distribution as t → 0. Meanwhile, if x = 1 in (2.1), then it is

viz, the conditional distribution of S given t|X| > 1 approximates to Λ as t → 0. α is called the tail exponent or tail index, while Λ is done the spectral measure. Furthermore, it follows from (2.1)–(2.3) that asymptotically R and S are of conditional independence given t|X| > 1 as t → 0. These features give rise to a simulation method of multivariate extremes (Kim, 2022).

This study concentrates on the tail behavior characterized by a non-singular heavy-tailed elliptical distribution. For a comprehensive overview of elliptical distributions, refer to Hult and Lindskog (2002). Let Z = (Z₁, . . . , Z_d)′, where Z₁, . . . , Z_d are independent standard normal random variables. Then W := Z/|Z| is uniformly distributed on . In particular, if

where ∑ is a symmetric positive definite matrix, V is a positive random variable with a regularly varying tail of tail exponent α and independent of W, then, according to Theorem 3.1 of Hult and Lindskog (2002), X follows an elliptical distribution. Moreover, it exhibits a multivariate regularly varying tail, and its spectral measure can be readily proven to be

Define μ(A) := P(Z/|Z| ∈ A) for . Then, Kim (2021) shows that Λ in (2.5) has a density function with respect to μ which of form is presented in the following theorem:

2.2. The second order behavior of univariate regular variation

Before dealing with a second order behavior of multivariate regular variation, this subsection briefly reviews the univariate case (2.2) with respect to α-estimation. Let R₁, . . . , R_n be a random sample of R. For estimating α,

is popularly employed, where k < n is a positive integer and R_(k+1) indicates the (k + 1)^th largest value in R₁, . . . , R_n (cf. Hill, 1975; Hsing, 1991). Assuming that k := k_n satisfies k → ∞ and k/n → 0 as n→∞, H_n converges in probability to 1/α as n→∞. For establishing a further asymptotic property, we need to impose a more stringent condition on (2.2) which is called second order regular variation: For instance, letting L(x) := x^αP(R > x), x > 0, we assume that there exist γ > 0, c₁ > 0, and, c₂ ∈ ℝ such that

(cf. Hall, 1982). If k(k/n)γ/α→M as n→∞ for some M ≥ 0, we then have

where Z is a standard normal variable and ⇒ denotes ‘convergence in distribution’ (cf. Hsing, 1991). (2.8) is a special and prominent case of

which is the second order property of lim_t→0L(x/t)/L(1/t) = 1 for each x > 0.

For investigating the asymptotic property of the estimator of ∑ in (2.4), we need to explore the second order behavior of (2.3) (see Remark 1). Let X = V∑^1/2W be the random vector in (2.4). Moreover, it is assumed that there exist α > 0, γ > 0, a₁ > 0, and a₂ ∈ ℝ, such that

(cf. (2.8)). Let

Under (3.1), we are going to derive the behavior of Λ_t( · ) – Λ( · ) as t → 0. Let

and

Assume that ∑ is a symmetric positive definite matrix and (3.1) holds. Then,

• P(|X| > 1/t) = c₁t^α(1 + c₂t^γ + o(t^γ)) as t → 0, i.e., (2.8) holds;

• for each, Λ_t(A) – Λ(A) = t^γΓ(A) + t^γo(1) as t → 0, where the o(1)-term is uniformly negligible in.

Under the same conditions of Theorem 2, we have the followings:

• For bounded measurable function f (s) defined on,

  Λt(Af(y))-Λ(Af(y))=tγGf(y)+tγo(1)         as t→0,

  where the o(1)-term is uniformly negligible in y ∈ ℝ;

• Moreover, if, then, ∫-∞∞Gf(y)dy=a2ρ∫s∈Sd-1f(s)dΛ(γ)(s).

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT): Grant No. RS-2023-00243752.