On Rapid Variation of Multivariate Probability Densities

Multivariate rapid variation describes decay rates of joint light tails of a multivariate distribution. We impose a local uniformity condition to control decay variation of distribution tails along different directions, and using higher-order tail dependence of copulas, we prove that a rapidly varying multivariate density implies rapid variation of the joint distribution tails. As a corollary, rapid variation of skew-elliptical distributions is established under the assumption that the underlying density generators belong to the max-domain of attraction of the Gumbel distribution.

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• 1 publication
11/09/2018

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1 Introduction and preliminaries

Multivariate regular variation has been extensively studied in the literature; see, e.g., [23], for detailed discussions and various applications in analyzing multivariate extremes. Regular variation of multivariate heavy tails of a distribution on enjoys a scaling property, that has proven useful in multivariate asymptotic theory, and in particular, such a scaling property allows to establish the conditions in terms of multivariate densities which imply regular variation of multivariate distribution tails [8, 9].

The goal of this paper is to establish a similar closure property for multivariate light tails and apply it to studying rapid variation, in the sense of de Haan [6], of skew-elliptical distributions. We develop a local uniformity condition, under which rapidly varying multivariate densities imply the rapid variation for multivariate distribution tails. In contrast to the scaling property of multivariate regular variation, multivariate rapid variation satisfies only an additive stability condition (see [16]), which lacks strong scaling homogeneity that may be needed for establishing the closure property. To overcome this difficulty, we utilize higher-order tail dependencies of copulas, introduced and studied in [12, 13, 18], and their relations to multivariate extremes. It is known that (first-order) tail dependence of copulas is equivalent to multivariate regular variation of distributions on with tail equivalent univariate marginal distributions [19]

, and together with this, our copula approach used in this paper captures the universality of multivariate dependence for distributions with regularly or rapidly varying marginal tails of different types, and further illustrates its usefulness in multivariate analysis.

Our research is motivated from the study on multivariate tail behaviors of skew-elliptical distributions, reported in [16], where tail densities of skew-elliptical distributions and their associated copulas are derived for both cases of regular and rapid variation. Since most multivariate distributions are specified directly by densities, it is important to have closure criteria in terms of densities which imply regular or rapid variation of multivariate distribution tails. Such closure criteria were obtained in [8, 9] for multivariate regular variation, and our results in this paper fill the void for multivariate rapid variation, showing further that skew-elliptical distributions are rapidly varying if their density generators belong to the max-domain of attraction of the Gumbel distribution.

The paper is organized as follows. Section 2 establishes the relation of local uniformity for the joint tails between multivariate rapid variation and its copula, and then proves the closure theorem for a rapidly varying density to ensure rapid variation of the multivariate distribution tails. Section 3 extends a result of [16] on tail densities of skew-elliptical distributions with rapidly varying density generators to the entire and establishes rapid variation of the joint tails of skew-elliptical distributions. Skew-elliptical distributions studied here belong to a general class of selection distributions [5], and these multivariate distributions can account for both skewness and heavy/light tails, and have found widespread application in various areas [1]. Section 4 concludes the paper with some remarks. In the rest of this section, some notation and preliminaries on higher-order tail dependencies of copulas are highlighted (see [13, 18] for details).

Any vector in

or is denoted by row vector and its transpose, a column vector, is denoted by . Two measurable functions and are tail equivalent, and denoted as , if as for each . These functions are said to be locally uniformly tail equivalent, and denoted as , if as locally uniformly in . Operations of vectors (sums, products, etc.) and relations (inequalities, intervals, etc.) are taken component-wise. We consider throughout this paper that any involved slowly varying function is continuous. The assumption is rather mild due to Karamata’s representation (see, e.g., [23]) that any slowly varying function can be written as the product of a continuous function and a measurable function with positive constant limit.

A copula

is a multivariate distribution with uniformly distributed univariate marginal distributions on

. Sklar’s theorem (see, e.g., [15]) states that every multivariate distribution with univariate marginal distributions can be written as for some -dimensional copula . In fact, in the case of continuous univariate margins, is unique and

 C(u1,…,ud)=F(F−11(u1),…,F−1d(ud)), (u1,…,ud)∈[0,1]d.

Let denote a random vector with distribution and , being uniformly distributed on . The survival copula is defined as follows:

 ˆC(u1,…,ud)=P(1−U1≤u1,…,1−Ud≤ud)=¯¯¯¯C(1−u1,…,1−ud) (1.1)

where is the joint survival function of . Assume throughout this paper that the density of copula exists, and that is continuous in some small open neighborhoods of and . (Its dimensionality is always clear from the context without explicit mention.)

Definition 1.1.
1. The upper tail density of with tail order is defined as follows:

 λU(w;κU):=limu→0+c(1−uwi,1≤i≤d)uκU−dℓ(u)>0, w=(w1,…,wd)∈[0,∞)d∖{0}, (1.2)

provided that the non-zero limit exists for some and for some function that is slowly varying at .

2. The lower tail density of with tail order is defined as follows:

 λL(w;κL):=limu→0+c(uwi,1≤i≤d)uκL−dℓ(u)>0, w=(w1,…,wd)∈[0,∞)d∖{0}, (1.3)

provided that the non-zero limit exists for some and for some function that is slowly varying at .

Here and hereafter is said to be slowly varying at if is slowly varying at ; that is, , , as .

Definition 1.2.
1. The upper tail dependence function with tail order is defined as follows:

 bU(w;κU):=limu→0+¯¯¯¯C(1−uwi,1≤i≤d)uκUℓ(u), w=(w1,…,wd)∈Rd+,

provided that the non-zero limit exists at any continuity point of , for and for some function that is slowly varying at .

2. The lower tail dependence function with tail order is defined as follows:

 bL(w;κL):=limu→0+C(uwi,1≤i≤d)uκLℓ(u), w=(w1,…,wd)∈Rd+,

provided that the non-zero limit exists at any continuity point of , for and for some function that is slowly varying at .

The fact that or follows from the Fréchet-Hoeffding upper bound. Clearly, (or ) for copulas with positively upper (or lower) orthant dependence. In addition, the tail densities and tail order functions are scale-homogeneous with the following scaling properties

 λU(tw;κU)=tκU−dλU(w;κU),  λL(tw;κL)=tκL−dλL(w;κL), w∈[0,∞)d∖{0} (1.4)
 bU(tw;κU)=tκUbU(w;κU),  bL(tw;κL)=tκLbL(w;κL), w∈Rd+

for any . It is seen from these scaling properties that these functions are completely determined by values of these functions at , , .

Proposition 1.3.

Assume that tail dependence functions and tail densities exist.

1. If (1.2) holds locally uniformly at , then

 λU(w;τU)=∂dbU(w;τU)∂w1⋯∂wd, w=(w1,…,wd)∈Rd+∖{0}.
2. If (1.3) holds locally uniformly at , then

 λL(w;τL)=∂dbL(w;τL)∂w1⋯∂wd, w=(w1,…,wd)∈Rd+∖{0}.

The above results are proved in [18]. Since the survival copula (see (1.1)) can be used to transform lower tail properties of into the corresponding upper tail properties of , the lower tail property (2) of Proposition 1.3 can be obtained immediately from the upper tail case. Because of this duality, we focus on upper tails only in this paper.

2 Rapid variation for multivariate densities

Let be a -dimensional distribution with density , copula and continuous marginal distributions . With proper translation and scaling sequences, marginal distributions ’s belong to the max-domain of attraction of one of the three distribution families; that is, Fréchet, Gumbel or Weibull distribution. We focus on the Gumbel case for light tails in this paper. Assume that for the upper tail case, marginal distributions are right-tail equivalent in the sense that

 fi(t)∼aif1(t), as t→∞,  1≤i≤d, (2.1)

where is the density of the -th marginal distribution , and is a constant, , with . Note that (2.1) implies the usual tail equivalence of the marginal survival functions,

 ¯¯¯¯Fi(t)∼ai¯¯¯¯F1(t), as t→∞,  1≤i≤d, (2.2)

where , , but the reverse may not be true in general.

Assume now that the marginal density , , is continuous and satisfies

 fi(t+mi(t)x)∼fi(t)e−x∼aif1(t)e−x, x∈R,  as t→∞, (2.3)

where the self-neglecting function can be taken to be a differentiable function for which its derivative converges to ([3, 7]). According to Bloom’s theorem (see [4]),

 limt→∞mi(t+mi(t)x)mi(t)=1, 1≤i≤d, (2.4)

hold locally uniformly in . All these functions are known to belong to the gamma class (see [20]). The gamma class consists of all the measurable functions , denoted by , for which there exists a measurable and positive function such that

 limt→∞g(t+m(t)x)g(t)=eαx,  x∈R.

The following two results, obtained in [20], are used in deriving our main results.

Lemma 2.1.

If , , where is self-neglecting, then

 g(t+m(t)x)g(t)→e−κx,κ>0, locally uniformly in x∈R.
Lemma 2.2.

Suppose that is self-neglecting. Then if and only if , where is slowly varying at and , .

It is worth mentioning that Omey in [20] obtained the stronger results, but the descriptions in Lemmas 2.1 and 2.2 suit our purpose.

Lemma 2.1 implies that the convergences

 limt→∞fi(t+mi(t)x)fi(t)=e−x, 1≤i≤d

hold locally uniformly in . Since the derivative of converges to , the derivative of with respect to converges to 1, locally uniformly in . Integrating both sides of (2.3) with respect to implies that

 ¯¯¯¯Fi(t+mi(t)x)∼l.u.¯¯¯¯Fi(t)e−x, x∈R,  as t→∞.

That is, is in the max-domain of attraction of the Gumbel distribution, and thus the reciprocal hazard rate can be taken as and as , .

Definition 2.3.

Suppose that has a distribution with density , that is tail equivalent in the sense of (2.1) and satisfies (2.3).

1. The tail density at is defined as

 λ(x):=limt→∞f(t+m(t)x1,…,t+m(t)xd)m−d(t)VκU(t),  x∈Rd, (2.5)

where , is self-neglecting and , provided that the non-zero limit exists.

2. The distribution is said to have a rapidly varying tail at if there exists a non-null Radon measure such that for all the -continuity points (satisfying that , where denotes the boundaries of set ),

 P(X1>t+m(t)x1,…,Xd>t+m(t)xd)VκU(t)→μ(d∏i=1(xi,+∞]), as t→∞, (2.6)

where , is self-neglecting and .

Remark 2.4.
1. The self-neglecting function in (2.5) and (2.6) can be taken as , . The function in (2.5) and (2.6) can be taken as , for some function that is slowly varying at . It follows from the Karamata representation for slowly varying functions that

 V(t+m(t)x)=¯¯¯¯F1(t+m(t)x)[ℓ(¯¯¯¯F1(t+m(t)x))]1/κU
 ∼¯¯¯¯F1(t)e−x[ℓ(¯¯¯¯F1(t)e−x)]1/κU∼¯¯¯¯F1(t)e−x[ℓ(¯¯¯¯F1(t))]1/κU=V(t)e−x,

and by Lemma 2.1, the convergence is locally uniform. Scaling functions, such as , is unique up to a constant.

2. In contrast to the tail density in multivariate regular variation [9], in (2.5) does not have a scaling property, and it satisfies the following weak additive stability condition:

 λ(x+z1)=λ(x)e−zκU, x∈Rd,

for any .

The following result is needed in analyzing multivariate tails of via our copula approach.

Proposition 2.5.

If the limit (1.2) holds locally uniformly in , then has a tail density that is related to the upper tail density of as follows: for any ,

where is the Jacobian determinant of the homeomorphic transform , .

This result was obtained in [18] on , but can be extended to using the same proof. In fact, a stronger result in which the convergence for a tail density holds locally uniformly on can be obtained as follows.

Proposition 2.6.

The limit (1.2) holds locally uniformly in if and only if the limit (2.5) holds locally uniformly in . If either of these two conditions holds, then (2.7) holds.

Proof. (1) Suppose that (1.2) holds locally uniformly in , with a slowly varying function . Consider, for any , that

 f(t+m(t)x1,…,t+m(t)xd)m−d(t)VκU(t)=c(F1(t+m(t)x1),…,Fd(t+m(t)xd))[¯¯¯¯F1(t)]κU−dℓ(¯¯¯¯F1(t))∏di=1fi(t+m(t)xi)∏di=1f1(t),

where and are described as these in Remark 2.4 (1). By the tail equivalency (2.1) and Lemma 2.1, we have

 fi(t+m(t)xi)f1(t)→aie−xi,  ¯¯¯¯Fi(t+m(t)xi)¯¯¯¯F1(t)→aie−xi,  1≤i≤d,

converge locally uniformly in . Since is ultimately continuous at , the Heine-Cantor theorem implies that for any small , there exists an such that when , for all , where is compact,

 ∣∣ ∣ ∣ ∣∣c(1−¯¯¯¯F1(t+m(t)x1)¯¯¯¯F1(t)¯¯¯¯F1(t),…,1−¯¯¯¯Fd(t+m(t)xd)¯¯¯¯F1(t)¯¯¯¯F1(t))[¯¯¯¯F1(t)]κU−dℓ(¯¯¯¯F1(t))∏di=1fi(t+m(t)xi)∏di=1f1(t)
 −c(1−aie−xi¯¯¯¯F1(t),1≤i≤d)[¯¯¯¯F1(t)]κU−dℓ(¯¯¯¯F1(t))d∏i=1aie−xi∣∣ ∣∣≤ϵ2.

Since (1.2) holds locally uniformly in , there exists an , such that as , for all ,

 ∣∣ ∣∣c(1−aie−xi¯¯¯¯F1(t),1≤i≤d)[¯¯¯¯F1(t)]κU−dℓ(¯¯¯¯F1(t))d∏i=1aie−xi−λU(aie−xi,1≤i≤d;κU)d∏i=1aie−xi∣∣ ∣∣≤ϵ2.

Therefore, as , for all , where is compact,

 ∣∣ ∣∣f(t+m(t)x1,…,t+m(t)xd)m−d(t)VκU(t)−λU(aie−xi,1≤i≤d;κU)d∏i=1aie−xi∣∣ ∣∣≤ϵ

which implies that the limit (2.5) holds locally uniformly in and (2.7) holds.

(2) Suppose that (2.5) holds locally uniformly in . Let , and obviously, where is self-neglecting. By Lemma 2.2, 222The function in Lemma 2.2 must be , by virtue of the construction used in the proof of Lemma 2.2., where is slowly varying at . Also observe that if and only if for and . Let , and consider, for ,

 c(1−uwi,1≤i≤d)uκU−dℓ(u)=c(1−¯¯¯¯F1(t)aie−xi,1≤i≤d)mκU−d(t)fκU−d1(t)ℓ(¯¯¯¯F1(t)) = c(1−¯¯¯¯F1(t)aie−xi¯¯¯¯Fi(t+m(t)xi)¯¯¯¯Fi(t+m(t)xi),1≤i≤d)m−d(t)¯¯¯¯FκU1(t)ℓ(¯¯¯¯F1(t))∏di=1f1(t)∏di=1fi(t+m(t)xi)d∏i=1fi(t+m(t)xi).

It follows from the tail equivalency and Lemma 2.1 that

 ¯¯¯¯F1(t)aie−xi¯¯¯¯Fi(t+m(t)xi)→1, 1≤i≤d,  ∏di=1f1(t)∏di=1fi(t+m(t)xi)→d∏i=1a−1iexi,

converge locally uniformly in . Since is ultimately continuous at , the Heine-Cantor theorem implies that for any small , there exists an such that when , for all , where is compact,

 ∣∣ ∣ ∣∣c(1−uwi,1≤i≤d)uκU−dℓ(u))−c(1−¯¯¯¯Fi(t+m(t)xi),1≤i≤d)m−d(t)¯¯¯¯FκU1(t)ℓ(¯¯¯¯F1(t))d∏i=1fi(t+m(t)xi)d∏i=1a−1iexi∣∣ ∣ ∣∣≤ϵ2.

Since , it follows from the local uniform convergence of (2.5) that there exists an such that as , for all , where is compact,

 ∣∣ ∣∣f(t+m(t)x1,…,t+m(t)xd)m−d(t)VκU(t)d∏i=1a−1iexi−λ(x1,…,xd)d∏i=1a−1iexi∣∣ ∣∣≤ϵ2,

where . Therefore, as , for all , where is compact,

 ∣∣ ∣∣c(1−uwi,1≤i≤d)uκU−dℓ(u))−λ(x1,…,xd)d∏i=1a−1iexi∣∣ ∣∣≤ϵ,

which implies that (1.2) holds locally uniformly on , and (2.7) holds for , .

The conditions of local uniform convergences in Proposition 2.6 are rather mild for the functions with multiplicative scaling properties. In fact, the local uniform convergences can be imposed only on a relatively compact subset of .

Proposition 2.7.

If the limit (1.2) holds locally uniformly in , , , then the limit (1.2) holds locally uniformly in .

Proof: For any , let for the Euclidean norm . Obviously, , . Consider

 limu→0+c(1−uwi,1≤i≤d)uκU−dℓ(u)=[w]κU−dlimu→0+c(1−u[w]wi/[w],1≤i≤d)(u[w])κU−dℓ(u[w])ℓ(u[w])ℓ(u).

For any compact subset , the set is a compact subset of . Since (1.2) holds locally uniformly on ,

 c(1−u[w]wi/[w],1≤i≤d)(u[w])κU−dℓ(u[w])

converges locally uniformly in . In addition, it follows from the local uniform convergence of a slowly varying function that converges to 1 locally uniformly in . It then follows from (1.4) that

 [w]κU−dlimu→0+c(1−u[w]wi/[w],1≤i≤d)(u[w])κU−dℓ(u[w])ℓ(u[w])ℓ(u)=[w]κU−dλU(w/[w];κU)=λU(w;κU),

locally uniformly in .

The other ingredient of our copula method is the integration form of Proposition 1.3.

Proposition 2.8.

Assume that tail dependence functions and tail densities of a copula exist.

1. If (1.2) holds locally uniformly in , then

 bU(w;τU)=∫w10⋯∫wd0λU(x;τU)dx, w=(w1,…,wd)∈Rd+. (2.8)
2. If (1.3) holds locally uniformly in , then

 bL(w;τL)=∫w10⋯∫wd0λL(x;τL)dx, w=(w1,…,wd)∈Rd+.

Proof. Only (1) is proved and the proof of (2) is similar. Since the limit (1.2) is locally uniform, Proposition 1.3 (1) holds.

The local uniformity of (1.2) implies that the tail density is integrable on the compact subset for any fixed small and any fixed , where . Since (1.2) converges locally uniformly and is ultimately continuous at , is continuous in , for any subset . Because is locally compact, the Heine-Cantor theorem implies that is locally uniformly continuous in , for any subset , which further implies via differentiablity of local uniform convergence that

 limvSc→wSc∂|S|bU(wS,vSc;τU)∂wS=∂|S|∂wSlimvSc→wScbU(wS,vSc;τU)=∂|S|bU(wS,wSc;τU)∂wS (2.9)

with the initial condition that , for any subset .

It follows from Fatou’s lemma that

 ∫[0,w]λU(x;τU)dx = ∫limϵ→0I{x≤w,∧di=1xi≥ϵ}λU(x;τU)dx ≤ lim