Some applications of the Archimedean copulas in the proof of the almost sure central limit theorem for ordinary maxima

Definition 1.1

A d- dimensional function C: [0, 1]^d → [0, 1], d ≥ 2, defined on the unit cube [0, 1]^d, is a d-dimensional copula if C is a joint cdf of a d-dimensional random vector with uniform- [0, 1] marginals, i.e.,

where all of the r.v.’s V_i, i = 1, 2, ..., d, have an uniform-[0, 1] cdf.

The theoretical groundwork for an area concerning the applications of copulas has been laid in the papers by Sklar [18]-[19], where the following celebrated claim has been stated among some other valuable results.

Theorem 1.2

(Sklar’s theorem). For a given multivariate (joint) cdf F of a random vector (X₁, ...,X_d) with marginal cdfs F₁, ..., F_d, d ≥ 2, there exists a unique copula C satisfying

This copula is unique if the F'_is, i = 1, ..., d, are continuous.

Conversely, for a given copula C: [0, 1]^d → [0, 1] and the marginal cdfs F₁, ..., F_d, relation (2) defines a multivariate distribution of(X₁, ..., X_d) with margins X_i having the cdfs F_i, i = 1, ..., d, respectively.

In view of Sklar’s theorem, we may treat a copula as a structure describing the dependence between the coordinates of the random vector (X₁, ..., X_d. Indeed, (2) means that C couples the marginal cdfs F_i into the joint cdf F. Simultaneously, due to Sklar’s proposition, we are also able to decouple the dependence structure into the corresponding marginals.

In our investigations leading to the proof of the ASCLT for some order statistics, we are concerned with a special class of copulas, commonly known as the Archimedean copulas. Before we define the Archimedean copula, we will introduce the notion of copula’s generator.

Definition 1.3

Suppose that d ≥ 2 and Ψ: [0, 1] → [0, ∞] is a strictly decreasing, convex function satisfying the conditions Ψ(0) = ∞ and Ψ (1) = 0. Let for v_i ∈ [0, 1], i = 1, ..., d,

The function Ψ is called a generator of C^Ψ.

If d ≥ 3, C^Ψ is, on the whole, not a copula. However, the following statement from Kimberling [20] gives a necessary and sufficient condition under which C^Ψ is a copula for all d ≥ 2.

Theorem 1.4

Choose d ≥ 2. The function C^Ψ (v₁, ..., v_d) in (3) is a copula iff a generator Ψ has an inverse Ψ^—1, which is completely monotonic on [0, ∞], i.e.,

We are now in a position to define the class of Archimedean copulas.

Definition 1.5

IfΨ^—1 is completely monotonic on [0, ∞], we say that C^Ψ given by (3) is the so-called Archimedean copula.

In our research, we study the situation when the investigated sequence of r.v.’s (X_i) is a stochastic process defined as follows. Namely, we assume that, for every i ∈ ℕ, a r.v. X_i has a marginal cdf F of the continuous type and that, for any sequence (t₁, t₂, ..., t_n), of natural numbers, the n-dimensional distribution of (x_t1, x_t2, ..., x_tn) is defined by a certain Archimedean copula C^Ψ_n = C^Ψ having a generator Ψ_n = Ψ, not depending on n. It means that, for any (x₁, x₂, ..., x_n) ∈ ℝⁿ,

where the mapping Ψ: [0, 1] →[0,∞] - called a generator of C^Ψ - is a strictly decreasing, convex function, satisfying Ψ (0) = ∞ and Ψ (1) = 0, whose inverse function Ψ^—1 is completely monotonic on [0, ∞], i.e., ( − 1 ) j d j d z j Ψ − 1 ( z ) ≥ 0 for all j ∈ ∕ and any z ∈ [0; ∞].

Then, it can be shown that, there exists a r.v. Θ > 0 such that Ψ^—1 is the Laplace transform of Θ, i.e.,

where - here and throughout the whole paper - E denotes the expected value with respect to Θ.

We assume that, for any x ∈ ℝ and θ ∈ supp Θ,

where G satisfies

It is known (see Marshall and Olkin [21] and Frees and Valdez [22]) that under the assumptions above, x₁, ...,x_n are conditionally independent given Θ.

The purpose of our note is to prove the ASCLT for (M_n) - an appropriate sequence of the maxima among X₁, ...., X_n, n ∈ ℕ. Before we give the statement of our main result, we will introduce some additional conditions and notations. Thus, we also assume that, for some numerical sequence (u_n):

and that

The rest of our paper is structured as follows. In Section 2, we formulate our main result, which is the ASCLT for the ordinary maxima (M_n) obtained from the processes of identically distributed r.v.’s of the continuous type, such that the corresponding multidimensional distributions are determined by the Archimedean copula. In Section 3, some auxiliary results necessary for the proof of the established ASCLT are stated and proved. The complete proof of our ASCLT is given in Section 4. Additionally, in Section 5, some application of the basic claim is depicted.

Theorem 2.1

Suppose that: (X_i) is a stationary sequence of identically distributed r.v.’s of the continuous type, with a common cdf F, and that, for any fixed n ∈ ℕ, the family of r.v.’s (X₁, ..., X_n) has the Archimedean copula C^Ψ.

Assume in addition that (X_i), a numerical sequence (u_n) and a generator Ψ of C^Ψ satisfy the conditions in (6)-(10), respectively. Then,

Lemma 3.1

Let the natural numbers m,n satisfy the condition 1 ≤ m < n and M_n (M_m,n) denote a sequence of maxima among X₁, ..., X_n (X_m+1, ..., X_n). Then, under the assumptions of Theorem 2.1,

proof

First, observe that

where Θ; is a r.v. satisfying (5)-(6).

Due to (6) and the fact that X₁, ..., X_n are conditionally independent given Θ, we obtain:

Consequently, by (11)-(13), we get

This and the relation that z^n—m — zⁿ ≤ m/n, for all z ∈ [0, 1], imply

which completes the proof of Lemma 3.1.☐

Our second auxiliary result is the following lemma.

Lemma 3.2

Let the natural numbers m, n satisfy the condition 1 ≤ m < n and M_m (M_{m, n}) denote a sequence of maxima among X₁, ...,X_m (X_m+1, ..., X_n). Then, under the assumptions of Theorem 2.1,

proof

Clearly, we have

Since (6) holds and X₁, ..., X_n are conditionally independent given Θ, we obtain:

By (14)-(16) and the relation in (12), we get

Thus, it is easy to check that

Therefore, we may write that

The properties that: z^n—m — zⁿ ≤ m/n, if z ∈ [0,1], and 0 ≤ {G(u_m)^Θ} ≤ 1 immediately imply

Furthermore, it follows from the Cauchy-Schwarz inequality that

Hence, using (7) and (5), we have

Applying the fact that |z — xy| ≤ |z — x||y| + |y — 1||z, for any x, y and z, together with the properties that 0 ≤ Ψ^—1 ≤ 1 and Ψ^—1 (0) = 1, we may write

The last relation and the fact that Ψ^—1 is a Lipschitz function yield

where L > 0 denotes an appropriate Lipschitz constant.

By (20), (21) and assumption (9), we obtain that there exists β ≥ 2 such that

Therefore, putting γ := β/2, we obtain

Combining (18), (19) and (23), we get a desired result from Lemma 3.2.  ☐

The following lemma will also be needed for the proof of our main result.

Lemma 3.3

Under the assumptions of Theorem 2.1 on (X_i), (u_n) and Ψ, we have

where τ is such as in (8).

proof

Firstly, we will show that assumption (9) yields the condition

Observe that, for any j ≥ 2,

where the last relation follows from the fact that X₁, ..., X_n are conditionally independent given Θ.

Consequently, we obtain

As in addition, a r.v. | P (X_j > u_n|Θ) — EP (X_j > u_n|Θ)| is bounded above by 2, we may write that

Furthermore, using assumption (6), we obtain

Thus, by (25)-(26), we get

Due to (7) and the property that 1— exp (— x) ≤ x, for any x ∈ ℝ, we have

In view of (27)-(28), we obtain

Derivation (29), assumption (10) and condition (9) yield

By virtue of (30), we have

Combining (31) with (8), we obtain

Therefore, putting C₂ := C₁ ∨ Cμ we get

Thus, in view of (32),

Employing (33), we immediately obtain a desired relation in (24).

In the second stage of our proof, we will show that assumption (9) implies the following property: for any integers i₁ < ... < i_p < j₁ < ... < j_p' ≤ n, for which j₁ — i_p ≥ l_n, we have

where:

and α_n,l_n → 0 as n → ∞, for some sequence l_n = o(n).

Let us notice that, since |z — xy| ≤ | z — x||y| + |y — 1||z, for any x, y and z, we may write as follows

This and property (4) yield:

Thus, in view of (35)-(38), we get

This and the fact that Ψ^—1 is a Lipschitz function imply

where L > 0 stands for an appropriate Lipschitz constant.

On the other hand, due to (9), we obtain

Therefore,

which yields (34).

Finally, since the conditions in (24) and (34) hold, a desired convergence straightforwardly follows from Theorem 5.3.1 in Leadbetter et al. [23].  ☐

Proof of Theorem 2.1

First, we will show that the following property holds

In view of Lemma 3.1 in Csáki and Gonchigdanzan [6], in order to prove (40), it is enough to show that

for some ɛ > 0.

Let

Then,

It is clear that

Thus, it remains to estimate the second component Σ₂ in (43). Observe that

which implies

Thus, by Lemmas 3.1 and 3.2, there exists a positive constant C₁ such that

It follows from (45) and a definition of Σ₂ in (43) that

Obviously, we have

where the penultimate relation follows from the fact that ∑ n = m + 1 N 1 n δ + 1 ≤ 1 δ 1 m δ for any δ > 0.

Furthermore, using the property mentioned in the previous line, we immediately obtain

In view of (46)-(48), we get

Combining (43), (44) and (49), we have

Thus, the relation in (41) is fulfilled and (40) holds true.

Finally, the convergence in (40), Lemma 3.3 and the regularity property of logarithmic mean imply the result established in Theorem 2.1. ☐

Theorem 5.1

Let:

Suppose that (X_i) is a stationary, standard normal sequence satisfying (9) and that: for any fixed n ∈ ℕ, the random vector (X₁, ..., X_n) has the Gumbel copula C^Ψ with a generator of the form Ψ (t) = (—ln t)^α for some α ≥ β, where β is such as in (9), and that the conditions in (6)-(7) are satisfied. Then, the claim of Theorem 2.1 holds true with u_n := u_n (x) = a_nx + b_n and τ = e^—x, i.e., for any x ∈ ℝ,

proof

Put u_n := u_n (x) = a_nx + b_n, where a_n, b_n are defined as in (50) and x is a fixed real number. It may be checked that (see, e.g., the derivations in Leadbetter et al. [23])