PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 78(92) (2005), 93–105

A CHARACTERIZATION OF ABSOLUTELY MONOTONIC (Δ) FUNCTIONS OF A FIXED ORDER

Patricia Mariela Morillas

Communicated by Stevan Pilipović

Abstract. Absolutely monotonic (Δ) function of order n are characterized in terms of n-dimensional totally increasing functions. Applications to n-copulas are presented.

1. Introduction

Let n be a non-negative integer and let c, d be two real numbers such that c < d. Let ϕ be a real function defined in [c, d]. Let t ∈ [c, d] and h > 0 be such that t + nh ∈ [c, d]. The difference of order n and step h of ϕ in t is defined by

Δ_{h}^{0}ϕ(t) = ϕ(t), Δ_{h}^{n}ϕ(t) = Δ_{h}^{n−1}ϕ(t + h) − Δ_{h}^{n−1}ϕ(t).

Equivalently,

Δ_{h}^{n}ϕ(t) = ∑_{k=0}^{n}(−1)^{n−k}\binom{n}{k}ϕ(t + kh).

If Δ_{h}^{k}ϕ(t) ⩾ 0 for all t ∈ [c, d], k = 0, …, n and h > 0 such that t + kh ⩽ d, then ϕ is said to be absolutely monotonic (Δ) of order n (abbreviated: ϕ is AM(Δ, n)). The function ϕ is said to be absolutely monotonic (abbreviated: ϕ is AM) if it is AM(Δ, n) for all n.

AM functions were first introduced by Bernstein [1]. He proved that they are necessarily analytic. A function is absolutely monotonic on the negative real axis, if and only if it can be represented there by a Laplace–Stieltjes integral with nondecreasing determining function (see e.g., [2] and [15]). AM functions appeared first in the construction of n-dimensional distribution functions in connection with Archimedean n-copulas (see [7]). They were used then in more general multivariate models: mixtures of powers (see [8] and [4]) and mixtures of max-infinitely divisible distributions (see [5]). Absolutely monotonic (D) functions of order n (abbreviated: AM(D, n) functions), i.e., non-negative functions with non-negative derivatives up to and including the n-th, appeared recently with certain transformation of ncopulas (see [9]). If a function is AM(D, n) then it is AM(Δ, n), but the converse is not true. For example, a non differentiable convex increasing and non-negative function is AM(Δ, 2) but not AM(D, 2).

2000 Mathematics Subject Classification: Primary 26A48, 26A51, 62H10.

Key words and phrases: Absolutely monotonic function, totally increasing function, multivariate distribution, copulas.

This work was partially supported by CONICET, Argentina.

In this paper, a characterization of AM(Δ, n) functions is obtained, and it is shown that for constructing Archimedean n-copulas and the transformed n-copulas considered in [9] we can use AM(Δ, n) functions instead of AM(D, n) functions. In this manner, the multivariate models that can be obtained with these methods is enlarged.

The rest of the paper is divided into sections as follows. In section 2, totally increasing functions are introduced. Then a characterization of AM(Δ, n) functions in terms of n-dimensional totally increasing functions is obtained. In section 3, we consider AM(Δ, n) functions in connection with Archimedean n-copulas and the transformed n-copulas considered in [9]. Finally, section 4 contains the proofs.

2. A characterization of AM(Δ′, n) functions

A complete discussion of AM functions can be found in Chapter IV of [16]. For our purpose we only sate the following result.

Theorem 2.1. [16, Chapter IV, Theorem 6] Let ϕ : [c, d] → R be AM(Δ, n) with n ⩾ 1 and let s, t ∈ [c, d] be such that s ⩽ t. Then Δ_{h}^{k}ϕ(s) ⩽ Δ_{h}^{k}ϕ(t) for all k = 0, …, n − 1 and all h > 0 such that y + kh ⩽ d.

In the sequel we suppose n ⩾ 1. In this section we do not use the non-negative condition for AM(Δ, n) functions. To simplify exposition, we say that if Δ_{h}^{k}ϕ(t) ⩾ 0 for all t ∈ [c, d], h > 0 such that t + nh ⩽ d and k = 1, …, n, then ϕ is absolutely monotonic (Δ′) of order n (abbreviated: ϕ is AM(Δ′, n)).

We have that a function ϕ is AM(Δ′, 2) if and only if ϕ satisfies

(2.1)  ϕ(t + h) − ϕ(t) ⩾ 0

for all t ∈ [c, d] and h > 0 such that t + h ∈ [c, d], and

(2.2)  ϕ(t + 2h) − 2ϕ(t + h) + ϕ(t) ⩾ 0

for all t ∈ [c, d] and h > 0 such that t + 2h ∈ [c, d]. Indeed AM(Δ′, 2) functions can be characterized in the following alternative manner,

Theorem 2.2. The function ϕ : [c, d] → R is AM(Δ′, 2) if and only if ϕ satisfies

(2.3)  ϕ(t_{4}) − ϕ(t_{3}) − ϕ(t_{2}) + ϕ(t_{1}) ⩾ 0

for all t_{1}, t_{2}, t_{3}, t_{4} ∈ [c, d] such that t_{1} ⩽ t_{2} ⩽ t_{4}, t_{1} ⩽ t_{3} ⩽ t_{4} and t_{4}−t_{3}−t_{2}+t_{1} ⩾ 0.

Proof. Suppose first that ϕ is AM(Δ′, 2) and set t_{1}, t_{2}, t_{3}, t_{4} ∈ [c, d], such that t_{1} ⩽ t_{2} ⩽ t_{4}, t_{1} ⩽ t_{3} ⩽ t_{4} and t_{4} − t_{3} − t_{2} + t_{1} ⩾ 0. Since t_{1} ⩽ t_{3}, then t_{1} + t_{4} − t_{3} ⩽ t_{4}. Since ϕ is AM(Δ′, 2), from Theorem 2.1 we obtain

ϕ(t_{1} + t_{4} − t_{3}) − ϕ(t_{1}) ⩽ ϕ(t_{4}) − ϕ(t_{3}).

We have t_{4} − t_{3} − t_{2} + t_{1} ⩾ 0 and ϕ is increasing, so

ϕ(t_{2}) ⩽ ϕ(t_{1} + t_{4} − t_{3}).

From these two inequalities we obtain (2.3).

Suppose now that ϕ satisfies (2.3) for all t_{1}, t_{2}, t_{3}, t_{4} ∈ [c, d] such that t_{1} ⩽ t_{2} ⩽ t_{4}, t_{1} ⩽ t_{3} ⩽ t_{4} and t_{4} − t_{3} − t_{2} + t_{1} ⩾ 0. Let first t ∈ [c, d] and h > 0 be such that t + h ∈ [c, d]. Setting t_{1} = t_{2} = t and t_{3} = t_{4} = t + h in (2.3), we obtain (2.1). Let now t ∈ [c, d] and h > 0 be such that t + 2h ∈ [c, d]. Setting t_{1} = t, t_{2} = t_{3} = t + h and t_{4} = t + 2h in (2.3), we obtain (2.2). Thus ϕ is AM(Δ′, 2). □

The above theorem shows that for AM(Δ′, 2) functions we can replace inequalities (2.1) and (2.2) (that involve two and three points in [c, d], respectively, related in a simple manner), by the only inequality (2.3) (that involves four points in [c, d] related in a more complicated manner). The aim of this section is to extend Theorem 2.2 for AM(Δ′, n) functions, with n ⩾ 2. To obtain this extension, we introduce a suitable expression for the needed points in [c, d]. This is achieved introducing totally increasing functions.

Let S_{1},…,S_{n} be partially ordered sets and, for simplicity, we denote with ⩽ the distinct orders. We consider in S_{1} × ⋯ × S_{n} the partial order of the product, i. e., if (a_{1}, …, a_{n}), (b_{1}, …, b_{n}) ∈ S_{1} × ⋯ × S_{n} then (a_{1}, …, a_{n}) ⩽ (b_{1}, …, b_{n}) if a_{i} ⩽ b_{i} for all i ∈ {1, …, n}; and we write (a_{1}, …, a_{n}) < (b_{1}, …, b_{n}) when a_{i} < b_{i} for all i ∈ {1, …, n}. If A and B are sets, then |A| denotes the cardinality of A and A ∖ B = {x ∈ A : x ∉ B}.

Let H : S_{1} × ⋯ × S_{n} → R and let (a_{1}, …, a_{n}), (b_{1}, …, b_{n}) ∈ S_{1} × ⋯ × S_{n} be such that (a_{1}, …, a_{n}) ⩽ (b_{1}, …, b_{n}). If (a_{1}, …, a_{n}) < (b_{1}, …, b_{n}), then the H-volume of {a_{1}, b_{1}} × ⋯ × {a_{n}, b_{n}} is given by

(2.4)  V_{H}({a_{1}, b_{1}} × ⋯ × {a_{n}, b_{n}}) = ∑ sign(c_{1}, …, c_{n})H(c_{1}, …, c_{n})
where the sum is taken over all (c_{1}, …, c_{n}) ∈ {a_{1}, b_{1}} × ⋯ × {a_{n}, b_{n}} and
sign(c_{1}, …, c_{n}) = (−1)^{k} with k = |{i ∈ {1, …, n} : c_{i} = a_{i}}|.

If there exists i ∈ {1, …, n} such that a_{i} = b_{i}, then V_{H}({a_{1}, b_{1}}×⋯×{a_{n}, b_{n}}) = 0.

Equivalently, the H-volume of {a_{1}, b_{1}}×⋯×{a_{n}, b_{n}} is the nth order difference of H on {a_{1}, b_{1}} × ⋯ × {a_{n}, b_{n}}, i.e.,

(2.5)  V_{H}({a_{1}, b_{1}} × ⋯ × {a_{n}, b_{n}}) = Δ_{(a_{1},...,a_{n}),(b_{1},...,b_{n})}H(x_{1}, …, x_{n})
= Δ_{a_{1},b_{1}} … Δ_{a_{n},b_{n}}H(x_{1}, …, x_{n}),

where we define the n first order differences of H as

Δ_{a_{k},b_{k}}H(x_{1}, …, x_{n}) = H(x_{1}, …, x_{k−1}, b_{k}, x_{k+1}, …, x_{n})
− H(x_{1}, …, x_{k−1}, a_{k}, x_{k+1}, …, x_{n}).

Also, by convention, if k = n we set

Δ_{(a_{1},...,a_{n−k}),(b_{1},...,b_{n−k})}H(x_{1}, …, x_{n}) = H(x_{1}, …, x_{n}).

Definition 2.1. A function H : S_{1} × ⋯ × S_{n} → R is said to be n-increasing if V_{H}({a_{1}, b_{1}} × ⋯ × {a_{n}, b_{n}}) ⩾ 0 for all (a_{1}, …, a_{n}), (b_{1}, …, b_{n}) ∈ S_{1} × ⋯ × S_{n} such that (a_{1}, …, a_{n}) ⩽ (b_{1}, …, b_{n}).

Definition 2.2. Let m ∈ {1, …, n} and let F : S_{1} × ⋯ × S_{n} → R. F is said to be m-increasing, if the functions obtained fixing any n − m variables are mincreasing, i.e., if for all m integers k_{1}, …, k_{m} such that 1 ⩽ k_{1} < ⋯ < k_{m} ⩽ n, and all y_{i} ∈ S_{i} with i ∈ {1, …, n} ∖ {k_{1}, …, k_{m}}, the function G : S_{k_{1}} × ⋯ × S_{k_{m}} → R given by

G(x_{k_{1}}, …, x_{k_{m}}) = F (y_{1}, …, y_{k_{1}−1}, x_{k_{1}}, y_{k_{1}+1}, …, y_{k_{m}−1}, x_{k_{m}}, y_{k_{m}+1}, …, y_{n})

is m-increasing.

Now we introduce n-dimensional totally increasing functions.

Definition 2.3. If F : S_{1}×⋯×S_{n} → R is m-increasing for all m ∈ {1, …, n}, then F is said to be totally increasing (abbreviated: F is T I).

As an immediate consequence of the previous definitions we have,

Lemma 2.1. Let F : S_{1}×⋯×S_{n} →R be n-increasing and let m ∈ {1, …, n−1}. Then the m-th differences of F are (n − m)-increasing functions.

Lemma 2.2. Let F : S_{1}×⋯×S_{n} → R be n-increasing. Suppose that S_{i} contains a least element a_{i}, i ∈ {1, …, n}. If F (x_{1}, …, x_{n}) = 0 for all (x_{1}, …, x_{n}) ∈ S_{1} × ⋯ × S_{n} such that x_{i} = a_{i} for at least some i ∈ {1, …, n}, then F is T I.

Using Lemma 2.2, we see that a simple and well known example of T I functions, from statistics and probability theory, are n-dimensional distribution functions. It is important to note that Rüschendorf in [11] considered T I increasing functions with S_{1} = ⋯ = S_{n} = [0, 1] and he called them Δ-monotone functions.

AM(Δ′, n) functions with n ⩾ 1, can be characterized in terms of totally increasing n-dimensional functions. As a first result in this direction we have,

Lemma 2.3. Let n ⩾ 2 and let ϕ : [c, d] → R. Let t_{0} ∈ [c, d] and let h > 0 be such that t_{0} + nh ⩽ d. Let S_{i} = {a_{i}, b_{i}} be ordered sets with a_{i} < b_{i}, i ∈ {1, …, n}. Let F : S_{1} × ⋯ × S_{n} → [c, d] be given by

F (x_{1}, …, x_{n}) = t_{0} + |{i ∈ {1, …, n} : x_{i} = b_{i}}|h.

Then, for all m ∈ {1, …, n} and all m integers k_{1}, …, k_{m} such that 1 ⩽ k_{1} < ⋯ < k_{m} ⩽ n it holds

(2.6) Δ_{(a_{k_{1}} ,...,a_{km}),(b_{k_{1}} ,...,b_{km})}ϕ ∘ F (x_{1}, …, x_{n})

= Δ_{h}^{m}ϕ(t_{0} + |{i ∈ {1, …, n} ∖ {k_{1}, …, k_{m}} : x_{i} = b_{i}}|h)

In particular, ϕ is AM(Δ′, n) if and only if ϕ ∘ F is T I for all such functions F.

Next appears the principal result of this article.

Theorem 2.3. Let ϕ : [c, d] → R be AM(Δ′, n). If F : S_{1} × ⋯ × S_{n} → [c, d] is T I, then ϕ ∘ F is T I.

The proof of Theorem 2.3 (see Section 4) is of elemental algebraic nature but it is very involved.

From Theorem 2.3 and Lemma 2.3 it results the following characterization for AM(Δ′, n) functions in terms of T I functions.

Theorem 2.4. A function ϕ : [c, d] → R is AM(Δ′, n) if and only if for all T I functions F : S_{1} × ⋯ × S_{n} → [c, d], the function ϕ ∘ F is T I.

Theorem 2.4 extends Theorem 2.2. Indeed, for n = 2, Theorem 2.4 is a reformulation of Theorem 2.2 in terms of T I functions.

3. Applications to copulas

An n-copula is the restriction to the unit n-cube [0, 1]^{n} of a multivariate distribution function whose marginals are uniform on [0, 1], more precisely,

Definition 3.1. An n-copula is a function C : [0, 1]^{n} → 0, 1] with the following properties:

1. C(x_{1}, …, x_{n}) = 0 if x_{i} = 0 for any i = 1, …, n.

2. C(1, …, 1, x_{i}, 1, …, 1) = x_{i} for each i = 1, …, n and all x_{i} ∈ [0, 1].

3. C is n-increasing.

Excellent references about copulas are [6], [10], [12] and [13]. In the sequel, we suppose n ⩾ 2 when we talk about n-copulas. The notion of copula has been introduced by A. Sklar in response to a question posed by M. Fréchet (see [14]). Sklar proved that if H is the joint distribution function of n random variables, X_{1}, …, X_{n}, and F_{1}, …, F_{n} are the distribution functions of X_{1}, …, X_{n}, respectively, then there exists an n-copula C such that

(3.1)  H(x_{1}, …, x_{n}) = C(F_{1}(x_{1}), …, F_{n}(x_{n})).

The n-copula C is uniquely determined on range(F_{1}) × ⋯ × range(F_{n}), so that C can be thought of as a description of the way in which a joint distribution function is related to its 1-dimensional marginals.

Because of (3.1),if we have a collection of n-copulas then we automatically have a collection of n-dimensional distributions with whatever 1-dimensional marginal distributions we desire. This is useful in modelling and simulation. Moreover, n-copulas are invariant under strictly increasing transformations of the random variables (see e.g., Theorem 6.5.6 in [13]), so they can be used in nonparametric statistic. Most of the dependence structure properties of an n-dimensional distribution function H are in a associated copula C, which does not depend on the marginals, and is often easier to handle than the original H. For these facts, it is very important in statistics to have a great variety of n-copulas. In general, we are interest in models that can lead to parametric families of multivariate distribution functions or copulas with close-form cumulative distribution functions, flexible dependence structure and partial closure under the taking the margins (see [6] for more details).

Different methods of constructing n-copulas have been proposed (see e.g. Chapter 3 and Chapter 4 in [10]). One of them yields an important class of n-copulas called Archimedean n-copulas. These n-copulas are of the form

(3.2)  C(x_{1}, …, x_{n}) = λ^{[−1]}(λ(x_{1}) + ⋯ + λ(x_{n})),

where λ : [0, 1] → [0, ∞] is a continuous strictly decreasing function such that λ(1) = 0 and λ^{[−1]} is the pseudo-inverse of λ given by

(3.3)  λ^{[−1]}(t) = \begin{cases}λ^{−1}(t), & if 0 ⩽ t ⩽ λ(0); \\ 0, & if λ(0) ⩽ t ⩽ ∞. \end{cases}

The function λ that appears in (3.2) is known as an additive generator of C. Setting φ(t) = exp(−λ(t)) and φ^{[−1]}(t) = λ^{[−1]}(−ln(t)), (3.2) can be written as

(3.4)  C(x_{1}, …, x_{n}) = φ^{[−1]}(φ(x_{1}) ⋯ φ(x_{n})).

The function φ that appears in (3.4) is known as a multiplicative generator of C. The study of Archimedean n-copulas is fundamentally done using (3.2). It is easy to see that C given by (3.2) satisfies 1 and 2 of Definition 3.1. Condition 3 of Definition 3.1 requires additional properties of λ.

Theorem 3.1. C defined by (3.2) with n = 2 is 2-increasing if and only if λ is convex.

A proof of the above theorem can be found in Chapter 4 of [10]. Kimberling in [7] gives necessary and sufficient conditions for a strict generator λ to guarantee that C given by (3.2) to be n-increasing for all n ⩾ 2. Indeed the arguments in [7] can be used to obtain the following results.

Theorem 3.2. If C given by (3.2) is n-increasing, then λ^{[−1]}(−t) is AM(Δ, n).

Theorem 3.3. If λ^{[−1]}(−t) is AM(D, n), then C given by (3.2) is n-increasing.

Note that in Theorem 3.2 and Theorem 3.3 conditions on λ^{[−1]} differ. This fact and Theorem 3.1 suggest the possibility of replace in Theorem 3.3, the condition on λ^{[−1]}(−t) of being AM(D, n) by the weaker condition of being AM(Δ, n). In fact, using Theorem 2.3, we can prove,

Theorem 3.4. If λ^{[−1]}(−t) is AM(Δ, n), then C given by (3.2) is n-increasing.

From Theorem 3.2 and Theorem 3.4 we obtain the following characterization of AM(Δ, n) functions in terms of Archimedean n-copulas.

Theorem 3.5. λ^{[−1]}(−t) is AM(Δ, n) if and only if C given by (3.2) is nincreasing.

Now we present the transformed copulas C_{φ}. These copulas are given by

(3.5)  C_{φ}(x_{1}, …, x_{n}) = φ^{[−1]}(C(φ(x_{1}), …, φ(x_{n})),

where C is an arbitrary n-copula, φ : [0, 1] → [0, 1] is a continuous strictly increasing function such that φ(1) = 1, and φ^{[−1]} is the pseudo-inverse of φ defined by

(3.6)  φ^{[−1]}(t) = \begin{cases}0, & if 0 ⩽ t ⩽ φ(0); \\ φ^{−1}(t), & if φ(0) ⩽ t ⩽ 1. \end{cases}

If C(x_{1}, …, x_{n}) = x_{1} ⋯ x_{n}, then (3.5) is reduced to (3.4), i.e., in this case (3.5) is the expression for an Archimedean n-copula with multiplicative generator φ.

Copulas C_{φ} were first considered by Genest and Rivest in [3]. In [9] conditions that φ must satisfy for C_{φ} to be an n-copula for every n-copula C are studied and some properties of n-copulas C_{φ} are explored. It is easy to see that C_{φ} satisfies conditions 1 and 2 of Definition 3.1. To guarantee that C_{φ} is n-increasing φ must satisfy additional properties. If n = 2 and if φ is concave then C_{φ} is 2-increasing (see Theorem 2.6 in [9]). In general, if n ⩾ 2 and if φ is AM(D, n) then C_{φ} is n-increasing (see Theorem 4.7 in [9]).

For copulas C_{φ} we have a situation similar to that for Archimedean n-copulas. We note that to guarantee that C_{φ} is n-increasing, the conditions impose to φ for the bivariate case are weaker than for the general case. It is natural to think that we can consider AM(Δ, n) functions instead of AM(D, n) functions in the general case. Indeed, from Theorem 2.3 we obtain,

Theorem 3.6. Let C be an n-copula. Let φ : [0, 1] → [0, 1] be a continuous strictly increasing function such that φ(1) = 1. If φ^{[−1]} is AM(Δ, n), then C_{φ} given by (3.5) is n-increasing.

In the bivariate case, non differentiable generators yield interest copulas. For example, if the additive generator of an Archimedean 2-copula C is piecewise linear, then C is singular. In the general case, in this article it has been shown that AM(Δ, n) functions can be used instead of AM(D, n) functions to obtain Archimedean n-copulas and n-copulas C_{φ}. Hereafter, all depend on the number of interest examples of AM(Δ, n) functions that can be obtained with the aim to have multivariate models with nice properties. This will be the subject of future investigations.

4. Proofs

Proof of Lemma 2.3. The proof proceeds by finite induction. Suppose m = 1 and consider k ∈ {1, …, n}. By definition of F we have

Δ_{a_{k},b_{k}}ϕ ∘ F (x_{1}, …, x_{n}) = ϕ ∘ F (x_{1}, …, x_{k−1}, b_{k}, x_{k+1}, …, x_{n})
− ϕ ∘ F (x_{1}, …, x_{k−1}, a_{k}, x_{k+1}, …, x_{n})
= ϕ(t_{0} + (|{i ∈ {1, …, n} ∖ {k} : x_{i} = b_{i}}| + 1) h)
− ϕ(t_{0} + |{i ∈ {1, …, n} ∖ {k} : x_{i} = b_{i}}|h)
= Δ_{h}^{1}ϕ(t_{0} + |{i ∈ {1, …, n} ∖ {k} : x_{i} = b_{i}}|h)

Thus (2.6) holds for m = 1. Suppose (2.6) is valid for 1 ⩽ m < n. Let now m ∈ {1, …, n}, and consider m integers k_{1}, …, k_{m} such that 1 ⩽ k_{1} < ⋯ < k_{m} ⩽ n.

We have

Δ_{(a_{k_{1}} ,...,a_{km}),(b_{k_{1}} ,...,b_{km})}ϕ ∘ F (x_{1}, …, x_{n})

= Δ_{(a_{k_{1}} ,...,a_{k_{m−1}} ),(b_{k_{1}} ,...,b_{k_{m−1}} )}Δ_{a_{km} ,b_{km}} ϕ ∘ F (x_{1}, …, x_{n})

= Δ_{(a_{k_{1}} ,...,a_{k_{m−1}} ),(b_{k_{1}} ,...,b_{k_{m−1}} )}ϕ ∘ F (x_{k_{1}}, …, x_{k_{m−1}}, b_{k_{m}}, x_{k_{m+1}}, …, x_{n})

− Δ_{(a_{k_{1}} ,...,a_{k_{m−1}} ),(b_{k_{1}} ,...,b_{k_{m−1}} )}ϕ ∘ F (x_{k_{1}}, …, x_{k_{m−1}}, a_{k_{m}}, x_{k_{m+1}}, …, x_{n})

= Δ_{h}^{m−1}ϕ(t_{0} + (|{k ∈ {1, …, n} ∖ {k_{1}, …, k_{m}} : x_{k} = b_{k}}| + 1)h)

− Δ_{h}^{m−1}ϕ(t_{0} + |{k ∈ {1, …, n} ∖ {k_{1}, …, k_{m}} : x_{k} = b_{k}}|h)

= Δ_{h}^{m−1}Δ_{h}^{1}ϕ(t_{0} + |{k ∈ {1, …, n} ∖ {k_{1}, …, k_{m}} : x_{k} = b_{k}}|h)

= Δ_{h}^{m}ϕ(t_{0} + |{k ∈ {1, …, n} ∖ {k_{1}, …, k_{m}} : x_{k} = b_{k}}|h)

This completes the induction procedure. □

To prove Theorem 2.3 we need the following two auxiliary statements. First of them follows easily from Theorem 2.1.

Lemma 4.1. Let ϕ : [c, d] → R be AM(Δ, n) or AM(Δ′, n), and let h be such that 0 < h < d − c. Then Δ_{h}^{1}ϕ : [c, d − h] → R is AM(Δ, n − 1).

Lemma 4.2. Let n ⩾ 2. Let S_{i} = {a_{i}, b_{i}} be ordered sets with a_{i} ⩽ b_{i}, i ∈ {1, …, n}. If F : S_{1} × ⋯ × S_{n} → R is k-increasing for k = 1, 2 and there exists j ∈ {1, …, n} such that F (b_{1}, …, b_{n}) = F (b_{1}, …, b_{j−1}, a_{j}, b_{j+1}, …, b_{n}), then

F (x_{1}, …, x_{j−1}, b_{j}, x_{j+1}, …, x_{n}) = F (x_{1}, …, x_{j−1}, a_{j}, x_{j+1}, …, x_{n}),

for all x_{i} ∈ S_{i}, i ∈ {1, …, n} ∖ {j}.

Proof. The proof proceeds by induction on n. Let n = 2. Suppose that j = 2 (the case j = 1 is similar). Thus, F (b_{1}, b_{2}) = F (b_{1}, a_{2}). Since F is k-increasing for k = 1, 2, we have 0 ⩽ F (a_{1}, b_{2}) − F (a_{1}, a_{2}) ⩽ F (b_{1}, b_{2}) − F (b_{1}, a_{2}) Therefore, F (a_{1}, b_{2}) = F (a_{1}, a_{2}). This verifies the lemma for n = 2.

Suppose n ⩾ 3 and the lemma is valid for n − 1. Suppose that j = n (the other cases are similar). Thus, F (b_{1}, …, b_{n}) = F (b_{1}, …, b_{n−1}, a_{n}). Clearly, the function F′ : S_{1} × ⋯ × S_{n−2} × S_{n} → R given by

F′(x_{1}, …, x_{n−2}, x_{n}) = F (x_{1}, …, x_{n−2}, b_{n−1}, x_{n})

satisfies the hypothesis of the lemma (with j = n). Then, by our supposition

(4.1)  F (x_{1}, …, x_{n−2}, b_{n−1}, b_{n}) = F (x_{1}, …, x_{n−2}, b_{n−1}, a_{n})

for all x_{i} ∈ S_{i}, i ∈ {1, …, n − 2}. Since F is k-increasing for k = 1, 2,

(4.2)  0 ⩽ F (x_{1}, …, x_{n−2}, a_{n−1}, b_{n}) − F (x_{1}, …, x_{n−2}, a_{n−1}, a_{n})
⩽ F (x_{1}, …, x_{n−2}, b_{n−1}, b_{n}) − F (x_{1}, …, x_{n−2}, b_{n−1}, a_{n})

for all x_{i} ∈ S_{i}, i ∈ {1, …, n − 2}. By (4.1) and (4.2),

(4.3)  F (x_{1}, …, x_{n−2}, a_{n−1}, b_{n}) = F (x_{1}, …, x_{n−2}, a_{n−1}, a_{n})

A CHARACTERIZATION OF ABSOLUTELY MONOTONIC (Δ) FUNCTIONS 101

for all x_{i} ∈ S_{i}, i ∈ {1, …, n − 2}. By (4.1) and (4.3),

F (x_{1}, …, x_{n−1}, b_{n}) = F (x_{1}, …, x_{n−1}, a_{n})

for all x_{i} ∈ S_{i}, i ∈ {1, …, n − 1}. This conclude the induction procedure. □

Now we are in position to prove Theorem 2.3.

Proof of Theorem 2.3. The proof proceeds by induction on n. For n = 1 the theorem is trivially valid. Suppose n ⩾ 2 and the theorem valid for n − 1.

Since ϕ is AM(Δ′, n), then it is AM(Δ′, n − 1). Thus we have,

Assertion 1. If T_{1}, …, T_{n−1} are ordered sets and G : T_{1} × ⋯ × T_{n−1} → [c, d] is T I, then ϕ ∘ G is T I.

By Assertion 1, since F is m-increasing for m ∈ {1, …, n − 1}, then ϕ ∘ F is mincreasing for m ∈ {1, …, n−1}. So, it only remains to see that ϕ∘F is n-increasing. Let (a_{1}, …, a_{n}), (b_{1}, …, b_{n}) ∈ S_{1} ×⋯×S_{n} be such that (a_{1}, …, a_{n}) ⩽ (b_{1}, …, b_{n}). Consider the function G_{a_{n}}^{b_{n}} defined on S_{1} × ⋯ × S_{n−1} by

G_{a_{n}}^{b_{n}}(x_{1}, …, x_{n−1}) = Δ_{a_{n},b_{n}}F (x_{1}, …, x_{n}).

Since F is T I, then G_{a_{n}}^{b_{n}} is T I. We have,

Δ_{(a_{1},...,a_{n}),(b_{1},...,b_{n})}ϕ ∘ F (x_{1}, …, x_{n}) (4.4)

= Δ_{(a_{1},...,a_{n−1}),(b_{1},...,b_{n−1})}[ϕ ∘ F (x_{1}, …, x_{n−1}, b_{n})−ϕ ∘ F (x_{1}, …, x_{n−1}, a_{n})]

= Δ_{(a_{1},...,a_{n−1}),(b_{1},...,b_{n−1})}[ϕ(F (x_{1}, …, x_{n−1}, a_{n}) + G_{a_{n}}^{b_{n}}(x_{1}, …, x_{n−1}))

− ϕ ∘ F (x_{1}, …, x_{n−1}, a_{n})]

There are three cases:

Case 1. G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−1}) = 0. By Lemma 4.2, G_{a_{n}}^{b_{n}}(x_{1}, …, x_{n−1}) = 0 for all x_{i} ∈ S_{i}, i ∈ {1, …, n − 1}. Therefore, by (4.4),

Δ_{(a_{1},...,a_{n}),(b_{1},...,b_{n})}ϕ ∘ F (x_{1}, …, x_{n}) = 0.

Case 2. G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−1}) = d − c. Since F (x_{1}, …, x_{n}) ∈ [c, d] and F is 1increasing, then F (x_{1}, …, x_{n−1}, a_{n}) = c for all x_{i} ∈ S_{i}, i ∈ {1, …, n − 1}. Then, by (4.4),

Δ_{(a_{1},...,a_{n}),(b_{1},...,b_{n})}ϕ ∘ F (x_{1}, …, x_{n})
= Δ_{(a_{1},...,a_{n−1}),(b_{1},...,b_{n−1})}[ϕ(F (x_{1}, …, x_{n−1}, b_{n})) − ϕ(c)]
= Δ_{(a_{1},...,a_{n−1}),(b_{1},...,b_{n−1})}ϕ(F (x_{1}, …, x_{n−1}, b_{n}))

By Assertion 1 and the previous equality, since F_{b_{n}} : S_{1} × ⋯ × S_{n−1} → [c, d] given by F_{b_{n}}(x_{1}, …, x_{n−1}) = F (x_{1}, …, x_{n−1}, b_{n}) is T I, then

Δ_{(a_{1},...,a_{n}),(b_{1},...,b_{n})}ϕ ∘ F (x_{1}, …, x_{n}) ⩾ 0.

Case 3. 0 < G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−1}) < d − c. Set h = G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−1}). By Lemma 4.1, since ϕ : [c, d] → R is AM(Δ′, n), then Δ_{h}^{1}ϕ : [c, d − h] → R is AM(Δ, n−1). Since F_{a_{n}} : S_{1} ×⋯×S_{n−1} → [c, d−h] given by F_{a_{n}}(x_{1}, …, x_{n−1}) = F (x_{1}, …, x_{n−1}, a_{n}) is T I, then (Δ_{h}^{1}ϕ) ∘ F_{a_{n}} is T I. Therefore,

Δ_{(a_{1},...,a_{n−2}),(b_{1},...,b_{n−2})}[ϕ(F (x_{1}, …, x_{n−2}, b_{n−1}, a_{n}) + G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−1})) (4.5)

− ϕ ∘ F (x_{1}, …, x_{n−2}, b_{n−1}, a_{n})]

⩾ Δ_{(a_{1},...,a_{n−2}),(b_{1},...,b_{n−2})}[ϕ(F (x_{1}, …, x_{n−2}, a_{n−1}, a_{n}) + G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−1}))

− ϕ ∘ F (x_{1}, …, x_{n−2}, a_{n−1}, a_{n})]

Since F and G_{a_{n}}^{b_{n}} are T I, then for all k such that 2 ⩽ k ⩽ n, the function H : S_{1} × ⋯ × S_{n−1} → R given by

(4.6)  H(x_{1}, …, x_{n−1}) = F (x_{1}, …, x_{n−k}, a_{n−k+1}, x_{n−k+2}, …, x_{n−1}, a_{n})
+ G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−k}, x_{n−k+1}, …, x_{n−1}),

is T I. Thus, from Assertion 1 and (4.5), we obtain

Assertion 2. For 2 ⩽ k ⩽ n we have

Δ_{(a_{1},...,a_{n−k},a_{n−k+2},...,a_{n−1}),(b_{1},...,b_{n−k},b_{n−k+2},...,b_{n−1})} (4.7)

[ϕ(F (x_{1}, …, x_{n−k}, b_{n−k+1}, x_{n−k+2}, …, x_{n−1}, a_{n})
+ G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−k+1}, x_{n−k+2}, …, x_{n−1}))
− ϕ ∘ F (x_{1}, …, x_{n−k}, b_{n−k+1}, x_{n−k+2}, …, x_{n−1}, a_{n})]
⩾ Δ_{(a_{1},...,a_{n−k},a_{n−k+2},...,a_{n−1}),(b_{1},...,b_{n−k},b_{n−k+2},...,b_{n−1})}
[ϕ(F (x_{1}, …, x_{n−k}, a_{n−k+1}, x_{n−k+2}, …, x_{n−1}, a_{n})
+ G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−k}, a_{n−k+1}, x_{n−k+2}, …, x_{n−1}))
− ϕ ∘ F (x_{1}, …, x_{n−k}, a_{n−k+1}, x_{n−k+2}, …, x_{n−1}, a_{n})]

To see Assertion 2 we proceed by finite induction. For k = 2, the function H : S_{1} × ⋯ × S_{n−1} → [c, d] given by (4.6) takes the form

H(x_{1}, …, x_{n−1}) = F (x_{1}, …, x_{n−2}, a_{n−1}, a_{n}) + G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−2}, x_{n−1})

By Assertion 1, since H is T I, then ϕ ∘ H is T I. Thus,

Δ_{(a_{1},...,a_{n−2}),(b_{1},...,b_{n−2})}[ϕ(F (x_{1}, …, x_{n−2}, a_{n−1}, a_{n}) + G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−1})) (4.8)

−ϕ ∘ F (x_{1}, …, x_{n−2}, a_{n−1}, a_{n})]

⩾ Δ_{(a_{1},...,a_{n−2}),(b_{1},...,b_{n−2})}[ϕ(F (x_{1}, …, x_{n−2}, a_{n−1}, a_{n}) + G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−2}, a_{n−1}))

−ϕ ∘ F (x_{1}, …, x_{n−2}, a_{n−1}, a_{n})]

By (4.5) and (4.8),

Δ_{(a_{1},...,a_{n−2}),(b_{1},...,b_{n−2})}[ϕ(F (x_{1}, …, x_{n−2}, b_{n−1}, a_{n}) + G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−1}))
−ϕ ∘ F (x_{1}, …, x_{n−2}, b_{n−1}, a_{n})]
⩾ Δ_{(a_{1},...,a_{n−2}),(b_{1},...,b_{n−2})}[ϕ(F (x_{1}, …, x_{n−2}, a_{n−1}, a_{n}) + G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−2}, a_{n−1}))
−ϕ ∘ F (x_{1}, …, x_{n−2}, a_{n−1}, a_{n})]

A CHARACTERIZATION OF ABSOLUTELY MONOTONIC (Δ) FUNCTIONS 103

This verifies (4.7) for k = 2. Suppose (4.7) is valid for 2 ⩽ k < n and rewrite it to obtain Δ_{(a_{1},...,a_{n−(k+1)},a_{n−(k+1)+2},...,a_{n−1}),(b_{1},...,b_{n−(k+1)},b_{n−(k+1)+2},...,b_{n−1})} (4.9)

[ϕ(F (x_{1}, …, x_{n−(k+1)}, b_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}, a_{n})
+ G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}))
− ϕ ∘ F (x_{1}, …, x_{n−(k+1)}, b_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}, a_{n})]
⩾ Δ_{(a_{1},...,a_{n−(k+1)},a_{n−(k+1)+2},...,a_{n−1}),(b_{1},...,b_{n−(k+1)},b_{n−(k+1)+2},...,b_{n−1})}
[ϕ(F (x_{1}, …, x_{n−(k+1)}, a_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}, a_{n})
+ G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}))
− ϕ ∘ F (x_{1}, …, x_{n−(k+1)}, a_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}, a_{n})]

For k + 1, the function H : S_{1} × ⋯ × S_{n−1} → [c, d] given by (4.6) takes the form

H(x_{1}, …, x_{n−1}) = F (x_{1}, …, x_{n−(k+1)}, a_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}, a_{n})
+ G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−(k+1)}, x_{n−(k+1)+1}, …, x_{n−1})

By Assertion 1, since H is T I then ϕ ∘ H is T I. Thus, Δ_{(a_{1},...,a_{n−(k+1)},a_{n−(k+1)+2},...,a_{n−1}),(b_{1},...,b_{n−(k+1)},b_{n−(k+1)+2},...,b_{n−1})} (4.10)

[ϕ(F (x_{1}, …, x_{n−(k+1)}, a_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}, a_{n})
+ G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}))
− ϕ ∘ F (x_{1}, …, x_{n−(k+1)}, a_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}, a_{n})]
⩾ Δ_{(a_{1},...,a_{n−(k+1)},a_{n−(k+1)+2},...,a_{n−1}),(b_{1},...,b_{n−(k+1)},b_{n−(k+1)+2},...,b_{n−1})}
[ϕ(F (x_{1}, …, x_{n−(k+1)}, a_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}, a_{n})
+ G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−(k+1)}, a_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}))
− ϕ ∘ F (x_{1}, …, x_{n−(k+1)}, a_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}, a_{n})]

From (4.9) and (4.10) we obtain

Δ_{(a_{1},...,a_{n−(k+1)},a_{n−(k+1)+2},...,a_{n−1}),(b_{1},...,b_{n−(k+1)},b_{n−(k+1)+2},...,b_{n−1})}

[ϕ(F (x_{1}, …, x_{n−(k+1)}, b_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}, a_{n})
+ G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}))
− ϕ ∘ F (x_{1}, …, x_{n−(k+1)}, b_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}, a_{n})]
⩾ Δ_{(a_{1},...,a_{n−(k+1)},a_{n−(k+1)+2},...,a_{n−1}),(b_{1},...,b_{n−(k+1)},b_{n−(k+1)+2},...,b_{n−1})}
[ϕ(F (x_{1}, …, x_{n−(k+1)}, a_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}, a_{n})
+ G_{a_{n}}^{b_{n}}(b_{1}, …, b_{n−(k+1)}, a_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}))
− ϕ ∘ F (x_{1}, …, x_{n−(k+1)}, a_{n−(k+1)+1}, x_{n−(k+1)+2}, …, x_{n−1}, a_{n})]

This shows Assertion 2. Now, by Assertion 2 with k = n, we have

Δ_{(a_{2},...,a_{n}),(b_{2},...b_{n})}[ϕ(F (b_{1}, x_{2}, …, x_{n−1}, a_{n}) + G_{a_{n}}^{b_{n}}(b_{1}, x_{2}, …, x_{n−1}))
− ϕ ∘ F (b_{1}, x_{2}, …, x_{n−1}, a_{n})]
⩾ Δ_{(a_{2},...,a_{n}),(b_{2},...b_{n})}[ϕ(F (a_{1}, x_{2}, …, x_{n−1}, a_{n}) + G_{a_{n}}^{b_{n}}(a_{1}, x_{2}, …, x_{n−1}))
− ϕ ∘ F (a_{1}, x_{2}, …, x_{n−1}, a_{n})]

or equivalently,

Δ_{(a_{1},...,a_{n−1}),(b_{1},...,b_{n−1})}[ϕ(F (x_{1}, …, x_{n−1}, a_{n}) + G_{a_{n}}^{b_{n}}(x_{1}, …, x_{n−1}))
−ϕ ∘ F (x_{1}, …, x_{n−1}, a_{n})] ⩾ 0.

Thus, by (4.4),

Δ_{(a_{1},...,a_{n}),(b_{1},...,b_{n})}ϕ ∘ F (x_{1}, …, x_{n}) ⩾ 0.

This shows that F is n-increasing. □

Proof of Theorem 3.4. Let (a_{1}, …, a_{n}), (b_{1}, …, b_{n}) ∈ [0, 1]^{n} be such that (a_{1}, …, a_{n}) < (b_{1}, …, b_{n}). Set B = {a_{1}, b_{1}} × ⋯ × {a_{n}, b_{n}} and consider the following two cases:

Case 1. λ(0) < ∞; or λ(0) = ∞ and a_{i} ≠ 0 for all i ∈ {1, …, n}. Let c = max{λ(x_{1}) + ⋯ + λ(x_{n}) : (x_{1}, …, x_{n}) ∈ B}. Clearly c < ∞ and since λ is decreasing, then the function F : B → [−c, 0] given by

F (x_{1}, …, x_{n}) = −[λ(x_{1}) + ⋯ + λ(x_{n})]

is T I. By Theorem 2.3, since λ^{[−1]}(−t) is AM(Δ, n) then λ^{[−1]}(−F) is T I. In particular, λ^{[−1]}(−F) is n-increasing, and consequently V_{C}(B) ⩾ 0.

Case 2. λ(0) = ∞ and a_{i} = 0 for some i ∈ {1, …, n}. For simplicity, suppose that a_{1} = ⋯ = a_{m} = 0 and a_{m+1} > 0, …, a_{n} > 0 for some m ∈ {1, …, n}. Consider the set

B_{ε} = {ε, b_{1}} × ⋯ × {ε, b_{m}} × {a_{m+1}, b_{m+1}} × ⋯ × {a_{n}, b_{n}}

where 0 < ε < min{b_{1}, …, b_{n}}. By case 1, V_{C}(B_{ε}) ⩾ 0. Since C is continuous, V_{C}(B_{ε}) tends to V_{C}(B) when ε tends to 0. Thus, V_{C}(B) ⩾ 0.

From the previous analysis it results that C is n-increasing. □

Proof of Theorem 3.6. Consider the function F : [0, 1]^{n} → [0, 1] given by F (x_{1}, …, x_{n}) = C(φ(x_{1}), …, φ(x_{n})). Let k ∈ {1, …, n} and let K ⊆ {1, …, n} with |K| = k. For simplicity, suppose K = {1, …, k}. Consider the set

B = {a_{1}, b_{1}} × ⋯ × {a_{k}, b_{k}}

where 0 ⩽ a_{i} < b_{i} ⩽ 1, i ∈ {1, …, k}. Since φ is strictly increasing, then (φ(a_{1}), …, φ(a_{k})) < (φ(b_{1}), …, φ(b_{k})). Consider now the set

B′ = {φ(a_{1}), φ(b_{1})} × ⋯ × {φ(a_{k}), φ(b_{k})}

Clearly,

Δ_{(a_{1},...,a_{k}),(b_{1},...,b_{k})}F (x_{1}, …, x_{n}) = Δ_{(φ(a_{1}),...,φ(a_{k})),(φ(b_{1}),...,φ(b_{k}))}C(x_{1}, …, x_{n})

By Lemma 2.2, C is T I. Then Δ_{(a_{1},...,a_{k}),(b_{1},...,b_{k})}F (x_{1}, …, x_{n}) ⩾ 0.

This shows that F is T I. Now, by Theorem 2.3, C_{φ} = φ^{[−1]} ∘ F is T I and, in particular, is n-increasing. □

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Instituto de Matemática Aplicada San Luis (Received 21 03 2005) UNSL-CONICET (Revised 29 12 2005) Ejército de los Andes 950 5700 San Luis, Argentina

    morillas@unsl.edu.ar