PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 89(103) (2010), 19–36 DOI: 10.2298/PIM1103019O

THE DIFFERENCE BETWEEN THE PRODUCT AND THE CONVOLUTION PRODUCT OF DISTRIBUTION FUNCTIONS IN ℝ^{n}

E. Omey and R. Vesilo

Communicated by Slobodanka Janković

Abstract. Assume that X⃗ and Y⃗ are independent, nonnegative d-dimen-sional random vectors with distribution function (d.f.) F (x⃗) and G(x⃗), respectively. We are interested in estimates for the difference between the product and the convolution product of F and G, i.e.,

D(x⃗) = F (x⃗)G(x⃗) − F ∗ G(x⃗).

Related to D(x⃗) is the difference R(x⃗) between the tail of the convolution and the sum of the tails:

R(x⃗) = (1 − F ∗ G(x⃗)) − (1 − F (x⃗) + 1 − G(x⃗)).

We obtain asymptotic inequalities and asymptotic equalities for D(x⃗) and R(x⃗). The results are multivariate analogues of univariate results obtained by several authors before.

1. Introduction

Assume that X⃗ and Y⃗ are independent, nonnegative d-dimensional random vectors with distribution function (d.f.) F (x⃗) and G(x⃗), respectively. In this paper, we are interested in estimates for the difference between the product and the convolution product of F and G, i.e.,

D(x⃗) = F (x⃗) G(x⃗) − F ∗ G(x⃗).

Here, and throughout the paper, we set

F ∗ G(x⃗) = P (X⃗ + Y⃗ ⩽ x⃗) = ∫ x⃗ ; 0 F (x⃗ − y⃗) dG(y⃗).

2010 Mathematics Subject Classification: Primary 26A12, 26B99, 60E99, 60K99. Key words and phrases: Subexponential distribution, regular variation, O-regularly varying

functions, sums of random vectors.

Clearly, we have D(x⃗) ⩾ 0. Related to D(x⃗) is the difference between the tail of the convolution and the sum of the tails:

R(x⃗) = \overline{F ∗ G}(x⃗) − F̄(x⃗) + Ḡ(x⃗),

where, here and elsewhere, we use the notation F̄ = 1 − F for the tails of d.fs. Clearly, we have

R(x⃗) = D(x⃗) − F̄(x⃗) Ḡ(x⃗).

In the case of one single d.f. F, we are interested in the following differences:

D_{n}(x⃗) = F^{n}(x⃗) − F^{∗n}(x⃗), and R_{n}(x⃗) = \overline{F^{∗n}}(x⃗) − nF̄(x⃗).

Note that we have

|R_{n}(x⃗) − D_{n}(x⃗)| ⩽ (1 − F (x⃗))^{2} \frac{n(n − 1)}{2}.

The interest in D_{n} and R_{n} comes from the class S(ℝ^{d}) of subexponential distributions. In Omey (2006) and Baltrunas, Van Gulck and Omey (2006, section 4) we studied the class of d.f. satisfying

\frac{1 − F^{∗n}(x⃗)}{1 − F (x⃗)} → n,

as x^{0} = min(x_{1}, x_{2}, …, x_{d}) → ∞. See also Daley, Omey and Vesilo (2007, Section 5). In dimension d = 1, this is the usual subexponential class S. In Omey, Mallor and Santos (2006), we considered d.fs satisfying a relation of the form

lim_{t→∞} \frac{1 − F^{∗n}(tx⃗ + ā)}{1 − F (tx⃗)} = n, for each a⃗ and each x⃗ > 0, with x^{0} < ∞.

Using R_{n} and D_{n}, we are able to study the rate of convergence in these definitions.

We are interested in two types of results. In the first place, we are interested in general inequalities and upper bounds. Secondly, we are interested in asymptotic equalities. In the next section, we briefly discuss the one-dimensional case, which has been studied, among others, by Omey (1994), Daley et al. (2007) and Baltrunas and Omey (1998).

2. Results in the one-dimensional case

In the one-dimensional case, we start from nonnegative random variables X and Y with d.f. F and G, respectively. We are interested in the following quantities:

D(x) = F (x) G(x) − F ∗ G(x);  D_{n}(x) = F^{n}(x) − F^{∗n}(x);
R(x) = \overline{F ∗ G}(x) − F̄(x) − Ḡ(x); R_{n}(x) = \overline{F^{∗n}}(x) − nF̄(x).

To state some of the known results, we need the following classes of positive and measurable functions, cf. Omey (1994): (throughout, we assume limits as x → ∞.)

f ∈ L:  for all y we have f(x + y)/f(x) → 1;
f ∈ OD(m): for all y we have |f(x + y) − f(x)| = O(1) m(x);
f ∈ D(m, α): for all y we have (f(x + y) − f(x))/m(x) → αy;
f ∈ ORV:  for all y > 0 we have f(xy) = O(1)f(x);
f ∈ RV (α): for all y > 0 we have f(xy)/f(x) → y^{α};
f ∈ Π_{α}(m): for all y > 0 we have (f(xy) − f(x))/m(x) → α log(y).

Note that f ∈ D(m, α) with α ≠ 0 implies that f(log(x)) ∈ Π_{α}(g(x)), where g(x) = m(log(x)) ∈ RV (0). For a measurable function f, the upper and lower Matuszewska indices are defined as follows:

α(f) = lim_{x→∞} \frac{log lim sup_{t→∞} f(tx)/f(t)}{log(x)},

and

β(f) = lim_{x→∞} \frac{log lim inf f(tx)/f(t)}{log(x)}.

It can be proved that f ∈ ORV if and only if α(f) < ∞ and β(f) > −∞. Properties of these indices can be found in the books of Bingham et al. (1987) or Geluk and de Haan (1987).

We have the following representation theorem for the classes OD(m) and D(m, α).

Theorem 2.1 (Omey 1994, 1995). (i)(Representation theorem for OD(m) and D(m, 0)) Assume that f ∈ OD(m), resp.f ∈ D(m, 0) and suppose that m(log(x)) ∈ ORV. Then there exist constants C and x^{0} > 0 such that

f(x) = C + η(x) m(x) + ∫_{x^{0}}^{x} ϕ(z) m(z) dz, x ⩾ x^{0},

where the measurable functions η(x) and ϕ(x) are bounded, resp. η(x) = o(1) and ϕ(x) = o(1).

(ii) (Representation theorem for D(m, α), α ≠ 0) Assume that f ∈ D(m, α) and suppose that α ≠ 0. Then there exist constants C and x^{0} > 0 such that

f(x) = C + η(x) m(x) + ∫_{x^{0}}^{x} ϕ(z) m(z) dz, x ⩾ x^{0},

where η and ϕ are measurable functions satisfying η(x) = o(1) and ϕ(x) → α.

The representation theorems can be used to obtain upper bounds.

Proposition 2.1 (Omey 1994, 1995). Suppose that f ∈ OD(m) and assume that m(x) ∈ ORV. Then there exist constants C and x^{0} such that

|f(x + y) − f(x)| ⩽ C(1 + |y|) m(x), x ⩾ x^{0}, |y| ⩽ x/2.

If f ∈ D(m, 0), for each ε > 0 we can find x(ε) such that

|f(x + y) − f(x)| ⩽ ε(1 + |y|) m(x), x ⩾ x(ε), |y| ⩽ x/2.

If F (x) is a d.f. such that F ∈ OD(m), respectively F ∈ D(m, 0), with m ∈ ORV and α(m) < −1, then F̄(x) = O(1)xm(x), respectively F̄(x) = o(1)xm(x) (Omey, 1994, Prop. 3.1.1). In the proposition below, we use integrals of the form

F_{1}(x) = ∫_{0}^{x} y dF (y).

If m ∈ ORV, it is clear that F ∈ OD(m) if and only if F_{1} ∈ OD(M), where M(x) = xm(x). If m ∈ ORV, with β(m) > −2, one can prove that F_{1}(x) → ∞ and that F_{1}(x) = O(1)x^{2}m(x).

Using these propositions, Omey (1994) proved the following results.

Theorem 2.2. (a) Suppose that F ∈ OD(m), G ∈ OD(n), with m, n ∈ ORV.

(i) We have D(x) = O(1) m(x) G_{1}(x) + O(1) n(x)F_{1}(x).

(ii) If β(n) > −2, β(m) > −2, then E(X) = E(Y) = ∞ and D(x) =

O(1)x^{2}n(x) m(x).

(iii) ∀n ⩾ 2, D_{n}(x) = O(1) m(x)F_{1}(x). If β(m) > −2, then D_{n}(x) =

O(1)x^{2}m^{2}(x).

(b) If F ∈ D(m, 0), G ∈ D(n, 0), with m, n ∈ ORV, then the results of (a) hold with the O(1)-terms replaced by o(1)-terms.

(c) Suppose that F ∈ OD(m) with m ∈ ORV.

(i) We have R_{n}(x) = O(1) m(x)F_{1}(x) + O(1)F̄^{2}(x).

(ii) If β(m) > −2, then R_{n}(x) = O(1)x^{2}m^{2}(x) + O(1)F̄^{2}(x). (iii) If β(m) > −2 and α(m) < −1, then R_{n}(x) = O(1)x^{2}m^{2}(x).

Related results can be found in Baltrunas and Omey (1998) or Baltrunas et al. (2006).

In the next result, we reformulate some results related to asymptotic equalities.

Theorem 2.3 (Omey, 1994, 1995). Suppose that F ∈ D(m, a), G ∈ D(n, b) with m, n ∈ ORV ∩ L. Also, suppose that E(X) + E(Y) < ∞.

(i) We have D(x) = aE(Y) m(x) + bE(X) n(x) + o(1) m(x) + o(1) n(x).

(ii) For all n ⩾ 2, we have D_{n}(x) = 2a\binom{n}{2}E(X) m(x) + o(1) m(x).
(iii) If α(m) < −1, we have R_{n}(x) = 2a\binom{n}{2}E(X) m(x) + o(1) m(x).

3. Results in the multidimensional case

3.1. Inequalities. We denote by F_{i} and G_{i} (i = 1, …, d) the marginal distributions of F and G and let D_{i}(x) = F_{i}(x) G_{i}(x) − F_{i} ∗ G_{i}(x). The following lemma is easy to prove.

Lemma 3.1. We have 0 ⩽ D(x⃗) ⩽ ∑_{i=1}^{d} D_{i}(x_{i}).

Proof. In the bivariate case, let F (x, y) = P (X_{1} ⩽ x, X_{2} ⩽ y) and G(x, y) = P (Y_{1} ⩽ x, Y_{2} ⩽ y). We have D(x, y) = I + II, where

I = P ( max(X_{1}, Y_{1}) ⩽ x, max(X_{2}, Y_{2}) ⩽ y) − P (X_{1} + Y_{1} ⩽ x, max(X_{2}, Y_{2}) ⩽ y)

and

II = P (X_{1} + Y_{1} ⩽ x, max(X_{2}, Y_{2}) ⩽ y) − P (X_{1} + Y_{1} ⩽ x, X_{2} + Y_{2} ⩽ y).

First consider I. Define the following events A = {X_{1} + Y_{1} ⩽ x}, B = {max(X_{1}, Y_{1}) ⩽ x} and C = {max(X_{2}, Y_{2}) ⩽ y}. Since A ⊂ B, we have

I = P (B ∩ C) − P (A ∩ C) ⩽ P (B) − P (A) = D_{1}(x).

In a similar way, we have II ⩽ D_{2}(y). □

From Theorem 2.2, we have the following upper bounds for the marginals. Lemma 3.2. Suppose that m_{i}, n_{i} ∈ ORV.

(i) If F_{i} ∈ OD(m_{i}) and G_{i} ∈ OD(n_{i}), then

D_{i}(x) = O(1) (m_{i}(x) ∫_{0}^{x} y dG_{i}(y) + n_{i}(x) ∫_{0}^{x} y dF_{i}(y)) .

(ii) If F_{i} ∈ D(m_{i}, 0) and G_{i} ∈ D(n_{i}, 0), then

D_{i}(x) = o(1) (m_{i}(x) ∫_{0}^{x} y dG_{i}(y) + n_{i}(x) ∫_{0}^{x} y dF_{i}(y)) .

Combining Lemmas 3.1 and 3.2, we obtain our first new result. We consider limits as x^{0} → ∞.

Theorem 3.1. Suppose that for each i we have m_{i}, n_{i} ∈ ORV.

(i) Suppose that for each i, F_{i} ∈ OD(m_{i}) and G_{i} ∈ OD(n_{i}), then

D(x⃗) = O(1)(∑_{i=1}^{d} m_{i}(x_{i}) ∫_{0}^{x_{i}} y dG_{i}(y) + ∑_{i=1}^{d} n_{i}(x_{i}) ∫_{0}^{x_{i}} y dF_{i}(y)).

(ii) Suppose that for each i, F_{i} ∈ D(m_{i}, 0) and G_{i} ∈ D(n_{i}, 0), then

D(x⃗) = o(1)(∑_{i=1}^{d} m_{i}(x_{i}) ∫_{0}^{x_{i}} y dG_{i}(y) + ∑_{i=1}^{d} n_{i}(x_{i}) ∫_{0}^{x_{i}} y dF_{i}(y)).

To prove a result for D_{n}(x⃗), we need the following closure properties. Lemma 3.3 (Omey 1994, Corollary 3.3.3). Suppose that m_{i} ∈ ORV.

(i) If F_{i} ∈ OD(m_{i}) and G_{i} ∈ OD(m_{i}), then F_{i} ∗ G_{i} ∈ OD(m_{i}) and F_{i}^{∗n} ∈ OD(m_{i}).

(ii) If F_{i} ∈ D(m_{i}, 0) and G_{i} ∈ D(m_{i}, 0), then F_{i} ∗ G_{i} ∈ D(m_{i}, 0) and F_{i}^{∗n} ∈

D(m_{i}, 0). Now we can formulate a result for D_{n}(x⃗). Again, we consider statements as

x^{0} → ∞.

Theorem 3.2. Suppose that for each i we have m_{i} ∈ ORV.

(i) Suppose for each i that F_{i} ∈ OD(m_{i}); then for all n ⩾ 2,

D_{n}(x⃗) = O(1)(∑_{i=1}^{d} m_{i}(x_{i}) ∫_{0}^{x_{i}} y dF_{i}(y)).

(ii) Suppose for each i that F_{i} ∈ D(m_{i}, 0); then for all n ⩾ 2,

D_{n}(x⃗) = o(1)(∑_{i=1}^{d} m_{i}(x_{i}) ∫_{0}^{x_{i}} y dF_{i}(y)).

Proof. For n = 2, this is the content of Theorem 3.1. Now let G(x⃗) = F^{∗n−1}(x⃗) and X_{i,j} (j = 1, ⋯, n − 1) be i.i.d. random variables with d.f. F_{i}. From Lemma 3.3, we have that G_{i} ∈ OD(m_{i}). Also note that

∫_{0}^{x} y dG_{i}(y) = E(I_{{∑_{j=1}^{n−1} X_{i,j}⩽x}} ∑_{j=1}^{n−1} X_{i,j}) = (n − 1)E (X_{i,1}I_{{∑_{j=1}^{n−1} X_{i,j}⩽x}})

⩽ (n − 1)E(X_{i,1}I_{{X_{i,1}⩽x}}) = (n − 1) ∫_{0}^{x} y dF_{i}(y).

Using Theorem 3.1 we obtain that

G(x⃗)F (x⃗) − F ∗ G(x⃗) = O(1)(∑_{i=1}^{d} m_{i}(x_{i}) ∫_{0}^{x_{i}} y dF_{i}(y)).

Now note that D_{n}(x⃗) = G(x⃗)F (x⃗) − F ∗ G(x⃗) + F (x⃗)D_{n−1}(x⃗). The proof now easily follows by induction on n. □

If the means are finite, we obtain from Theorem 3.2(i) that

D_{n}(x⃗) = O(1)(∑_{i=1}^{d} m_{i}(x_{i})).

If m_{i}(x) ∈ ORV with −2 < β(m_{i}), then the means are infinite and then (cf. Theorem 2.2) it follows that D_{n,i}(x) = O(1) x_{i}^{2}m_{i}^{2}(x_{i}). In this case, Theorem 3.2(i) gives

D_{n}(x⃗) = O(1)(∑_{i=1}^{d} x_{i}^{2}m_{i}^{2}(x_{i})).

In order to formulate a result about R(x⃗), recall that R(x⃗) = D(x⃗)−F̄(x⃗) Ḡ(x⃗). We have to deal with F̄(x⃗) Ḡ(x⃗). First note that F̄(x⃗) ⩽ ∑_{i=1}^{d} F̄_{i}(x_{i}). Now suppose that F_{i} ∈ OD(m_{i}), G_{i} ∈ OD(n_{i}), where m_{i}, n_{i} ∈ ORV with α(m_{i}) < −1 and α(n_{i}) < −1. In this case, we have F̄_{i}(x_{i}) = O(1)x_{i}m_{i}(x_{i}) and G⃗_{i}(x_{i}) = O(1)x_{i}n_{i}(x_{i}), cf. (Omey 1994, Proposition 3.1.1(v)).

(a) In the finite means case, we find that x_{i}F̄_{i}(x_{i}) → 0, as x_{i} → ∞, and it follows that x^{0}F̄(x⃗) ⩽ x^{0}F̄(x^{0}1) ⩽ x^{0} ∑_{i=1}^{d} F̄_{i}(x^{0}) → 0, as x^{0} → ∞. Hence, as x^{0} → ∞, we obtain that

F̄(x⃗) Ḡ(x⃗) ⩽ F̄(x^{0}1) Ḡ(x^{0}1) ⩽ F̄(x^{0}1) ∑_{i=1}^{d} x^{0}n_{i}(x^{0})

= x^{0}F̄(x^{0}1) ∑_{i=1}^{d} n_{i}(x^{0}) = o(1) ∑_{i=1}^{d} n_{i}(x^{0}).

THE DIFFERENCE BETWEEN THE PRODUCT AND THE CONVOLUTION … 25

In a similar way, we have F̄(x⃗) Ḡ(x⃗) = o(1) ∑_{i=1}^{d} m_{i}(x^{0}).

(b) If −2 < β(m_{i}) and α(m_{i}) < −1, then E(X_{i}) = ∞, and we have

∫_{0}^{x_{i}} y dF_{i}(y) = O(1) x_{i}^{2}m_{i}(x_{i}).

In this case, it follows that

F̄(x⃗) Ḡ(x⃗) = O(1) ∑_{i=1}^{d} x^{0}m_{i}(x^{0}) ∑_{i=1}^{d} x^{0}n_{i}(x^{0}),
D(x⃗) = O(1)(∑_{i=1}^{d} x_{i}^{2}m_{i}(x_{i}) n_{i}(x_{i})).

We conclude Corollary 3.1. Suppose m_{i}, n_{i} ∈ ORV with α(m_{i}) < −1, α(n_{i}) < −1.

(i) If F_{i} ∈ OD(m_{i}), G_{i} ∈ OD(n_{i}) with finite means,, then

R(x⃗) = O(1)(∑_{i=1}^{d} m_{i}(x_{i}) + ∑_{i=1}^{d} n_{i}(x_{i})) + o(1)(∑_{i=1}^{d} m_{i}(x^{0}) + ∑_{i=1}^{d} n_{i}(x^{0})).

(ii) If also −2 < β(m_{i}), −2 < β(n_{i}), we have

R(x⃗) = O(1)(∑_{i=1}^{d} x_{i}^{2}m_{i}(x_{i}) n_{i}(x_{i})) + O(1)(∑_{i=1}^{d} x^{0}m_{i}(x^{0}))(∑_{i=1}^{d} x^{0}n_{i}(x^{0})).

(iii) Similar results hold if we start from F_{i} ∈ D(m_{i}, 0) and G_{i} ∈ D(n_{i}, 0). For R_{n}(x⃗), the analysis is similar. Now we obtain the following result. Corollary 3.2. Suppose that m_{i} ∈ ORV with α(m_{i}) < −1.

(i) If F_{i} ∈ OD(m_{i}) with finite means, then for all n ⩾ 2,

R_{n}(x⃗) = O(1) ∑_{i=1}^{d} m_{i}(x_{i}) + o(1) ∑_{i=1}^{d} m_{i}(x^{0}).

(ii) If also −2 < α(m_{i}), we have

R_{n}(x) = O(1) ∑_{i=1}^{d} x_{i}^{2}m_{i}^{2}(x_{i}) + O(1)(∑_{i=1}^{d} x^{0}m_{i}(x^{0}))^{2}.

(iii) Similar results hold if we start from F_{i} ∈ D(m_{i}, 0). 3.2. Asymptotic equalities. In this section, we are seeking an asymptotic

equality in the place of an upper bound. Our starting point is the following useful inequality (cf. Lemma 3.4). We suppose that X⃗ and Y⃗ are independent nonnegative random vectors with distribution function F (x⃗) and G(x⃗) respectively. We give an estimate for the difference D(x⃗) = F (x⃗) G(x⃗) − F ∗ G(x⃗).

Lemma 3.4. We have D(x⃗) = A(x⃗) + B(x⃗) + C(x⃗), where

A(x⃗) = ∫ x⃗/2 ; 0(F (x⃗) − F (x⃗ − y⃗)) dG(y⃗),
B(x⃗) = ∫ x⃗/2 ; 0(G(x⃗) − G(x⃗ − y⃗)) dF (y⃗),
|C(x⃗)| ⩽ 2(F (x⃗) − F (x⃗/2))(G(x⃗) − G(x⃗/2)).

Proof. We rewrite F ∗ G(x⃗) as follows. We have

F ∗ G(x⃗) = P (X⃗ + Y⃗ ⩽ x⃗, X⃗ ⩽ x⃗/2, Y⃗ ⩽ x⃗/2)
+ P (X⃗ + Y⃗ ⩽ x⃗, {X⃗ ⩽ x⃗/2}^{c}, Y⃗ ⩽ x⃗/2)
+ P (X⃗ + Y⃗ ⩽ x⃗, X⃗ ⩽ x⃗/2, {Y⃗ ⩽ x⃗/2}^{c})
+ P (X⃗ + Y⃗ ⩽ x⃗, {X⃗ ⩽ x⃗/2}^{c}, {Y⃗ ⩽ x⃗/2}^{c})
= I + II + III + IV.

For I, it is easy to see that I = F (x⃗/2) G(x⃗/2). For II, we have

II = ∫ x⃗/2 ; 0 P (X⃗ ⩽ x⃗ − y⃗, {X⃗ ⩽ x⃗/2}^{c}) dG(y⃗)

= ∫ x⃗/2 ; 0 (F (x⃗ − y⃗) − F (x⃗/2)) dG(y⃗)

= −A(x⃗) + ∫ x⃗/2 ; 0 (F (x⃗) − F (x⃗/2)) dG(y⃗)

= −A(x⃗) + (F (x⃗) − F (x⃗/2)) G(x⃗/2).

In a similar way, we obtain that III = −B(x⃗) + (G(x⃗) − G(x⃗/2))F (x⃗/2). Hence, we obtain that

D(x⃗) = F (x⃗) G(x⃗) − F (x⃗/2) G(x⃗/2) + A(x⃗) + B(x⃗)
− (F (x⃗) − F (x⃗/2)) G(x⃗/2) − (G(x⃗) − G(x⃗/2))F (x⃗/2) − IV
= (F (x⃗) − F (x⃗/2))(G(x⃗) − G(x⃗/2)) + A(x⃗) + B(x⃗) − IV.

Now note that in IV we have {X⃗ + Y⃗ ⩽ x⃗} ⊂ {X⃗ ⩽ x⃗} ∩ {Y⃗ ⩽ x⃗} and we find that

IV ⩽ P (X⃗ ⩽ x⃗, Y⃗ ⩽ x⃗/2, {X⃗ ⩽ x⃗/2}^{c}, {Y⃗ ⩽ x⃗/2}^{c})
⩽ P (X⃗ ⩽ x⃗, {X⃗ ⩽ x⃗/2}^{c}) P (Y⃗ ⩽ x⃗/2, {Y⃗ ⩽ x⃗/2}^{c})
= (F (x⃗) − F (x⃗/2))(G(x⃗) − G(x⃗/2)).

This proves the result. □

3.2.1. The class D_{d}(m, λ). Now we introduce a multivariate analogue of the class D(m, α).

Definition 1. We say that the d.f. F is in the class 𝔇_{d}(m, λ) if we have

\frac{F (tx⃗) − F (tx⃗ − a⃗)}{m(t)} → λ(x⃗, a⃗), t → ∞,

for all x⃗ > 0, with x^{0} < ∞ and a⃗ ∈ ℝ^{d}. The function λ(x⃗, a⃗) is called the limit function.

Note that in the definition, we assume that the defining property holds for each of the marginals of F. For the i-th marginal we have

\frac{F_{i}(tx) − F_{i}(tx − a)}{m(t)} → λ_{i}(x, a), for each x > 0, and each a ∈ ℝ.

By taking x = 1, we obtain that

\frac{F_{i}(t) − F_{i}(t − a)}{m(t)} → λ_{i}(1, a), for each x > 0, and each a ∈ ℝ.

If m ∈ L,we obtain that

\frac{F_{i}(t) − F_{i}(t − a − b)}{m(t)} = \frac{F_{i}(t) − F_{i}(t − a)}{m(t)} + \frac{F_{i}(t − a) − F_{i}(t − a − b)}{m(t − a)} \frac{m(t − a)}{m(t)}
→ λ_{i}(1, a) + λ_{i}(1, b).

It follows that λ_{i}(1, a + b) = λ_{i}(1, a) + λ_{i}(1, b), and then also that λ_{i}(1, a) = α_{i}a for some real constant α_{i}.

Now observe that

\frac{F_{i}(tx) − F_{i}(tx − a)}{m(t)} = \frac{F_{i}(tx) − F_{i}(tx − a)}{m(tx)} \frac{m(tx)}{m(t)}.

After taking limits, we obtain that

λ_{i}(x, a) = α_{i}a lim_{t→∞} \frac{m(tx)}{m(t)}.

If α_{i} ≠ 0 and a ≠ 0, we obtain that

lim_{t→∞} \frac{m(tx)}{m(t)} = \frac{λ_{i}(x, a)}{α_{i}a}.

In this case, it follows under minimal conditions, that m ∈ RV (δ) for some real number δ, and as a consequence, that λ_{i}(x, a) = α_{i}x^{δ}a.

In what follows, we will assume that in the definition of 𝔇_{d}(λ, m), m is a regularly varying function.

Definition 2. We say that the d.f. F is in the class D_{d}(m, λ) if we have m ∈ RV (δ) and if

\frac{F (tx⃗) − F (tx⃗ − a⃗)}{m(t)} → λ(x⃗, a⃗), t → ∞,

for all x⃗ > 0, with x^{0} < ∞ and a⃗ ∈ ℝ^{d}. The function λ(x⃗, a⃗) is called the limit function. In this case, the marginals F_{i} ∈ D(m, λ_{i}), where λ_{i}(x, a) = α_{i}ax^{δ}.

In the next section, it will be convenient to assume that F ∈ D_{d}(m, λ), and that the defining property holds locally uniformly in x⃗.

Definition 3. We say that the d.f. F is in the class DL_{d}(m, λ) if we have m ∈ RV (δ) and if

\frac{F (tx⃗) − F (tx⃗ − a⃗)}{m(t)} → λ(x⃗, a⃗), t → ∞,

for all x⃗ > 0, with x^{0} < ∞ and a⃗ ∈ ℝ^{d}, holds locally uniformly in x⃗.

In this case, we can show that the limit function is an additive function in a⃗. To show this, note that we have

\frac{F (tx⃗) − F (tx⃗ − a⃗ − b⃗)}{m(t)} → λ(x⃗, a⃗ + b⃗).

On the other hand, we also have that

\frac{F (tx⃗) − F (tx⃗ − a⃗ − b⃗)}{m(t)} = \frac{F (tx⃗) − F (tx⃗ − a⃗)}{m(t)} + \frac{F (tx⃗ − a⃗) − F (tx⃗ − a⃗ − b⃗)}{m(t)}

= \frac{F (tx⃗) − F (tx⃗ − a⃗)}{m(t)} + \frac{F (t(x⃗ − a⃗/t)) − F (t(x⃗ − a⃗/t) − b⃗)}{m(t)}

→ λ(x⃗, a⃗) + λ(x⃗, b⃗)

and it follows that λ(x⃗, a⃗ + b⃗) = λ(x⃗, a⃗) + λ(x⃗, b⃗).

3.2.2. Main result. This is the main result of this section. Recall that X⃗ has d.f. F and Y⃗ has d.f. G.

Theorem 3.3. Suppose that F ∈ D_{d}(m, λ), G ∈ D_{d}(n, θ) have finite means and that m, n ∈ RV. Then

D(tx⃗ − a⃗) = [Eλ(x⃗, a⃗ + Y⃗) − λ(x⃗, a⃗)]m(t) + [Eθ(x⃗, a⃗ + X⃗) − θ(x⃗, a⃗)]n(t)

+ o(1) n(t) + o(1) m(t).

Proof. We analyze the three terms in Lemma 3.4. First, take A(x⃗) and replace x⃗ by tx⃗ − a⃗. We have

A(tx⃗ − a⃗) = ∫ (tx⃗−a⃗)/2 ; 0 (F (tx⃗ − a⃗) − F (tx⃗ − a⃗ − y⃗)) dG(y⃗)

= (F (tx⃗ − a⃗) − F (tx⃗)) G((tx⃗ − a⃗)/2)

+ ∫ (tx⃗−a⃗)/2 ; 0 (F (tx⃗) − F (tx⃗ − a⃗ − y⃗)) dG(y⃗)

= I + II.

We first analyze II. Clearly, we have

\frac{F (tx⃗) − F (tx⃗ − a⃗ − y⃗)}{m(t)} → λ(x⃗, a⃗ + y⃗).

We want to apply Lebesgue’s theorem on dominated convergence. Clearly, we have

F (tx⃗) − F (tx⃗ − a⃗ − y⃗) ⩽ ∑_{i=1}^{d}(F_{i}(tx_{i}) − F_{i}(tx_{i} − a_{i} − y_{i}))

and 0 ⩽ y_{i} ⩽ (tx_{i} − a_{i})/2. Using Proposition 2.1, we get that

F_{i}(tx_{i}) − F_{i}(tx_{i} − a_{i} − y_{i}) ⩽ C_{i}(1 + |a_{i} + y_{i}|) m(tx_{i}), tx_{i} ⩾ x^{∗}.

Using m ∈ RV, and since x⃗ is fixed, we find that for y⃗ ⩽ (tx⃗−a⃗)/2 and t sufficiently large, we have

|F (tx⃗) − F (tx⃗ − a⃗ − y⃗)| ⩽ ∑_{i=1}^{d} C_{i}(1 + |a_{i} + y_{i}|) m(t).

Since, by assumption, the means E(Y_{i}) are finite, we can apply Lebesgue’s theorem on dominated convergence, and we conclude that

\frac{1}{m(t)}A(tx⃗ − a⃗) → −λ(x⃗, a⃗) + Eλ(x⃗, a⃗ + Y⃗).

In a similar way, we obtain that

\frac{1}{n(t)}B(tx⃗ − a⃗) → −θ(x⃗, a⃗) + Eθ(x⃗, a⃗ + X⃗).

For C(tx⃗ − a⃗), we proceed as follows. First, we have that, cf. Proposition 2.1,

F (tx⃗ − a⃗) − F ((tx⃗ − a⃗)/2) ⩽ ∑_{i=1}^{d} (F_{i}(tx_{i} − a_{i}) − F_{i}((tx_{i} − a_{i})/2))

= O(1) ∑_{i=1}^{d} tx_{i}m(tx_{i}) = O(1)tm(t).

Also, we have that

G(tx⃗ − a⃗) − G((tx⃗ − a⃗)/2) ⩽ ∑_{i=1}^{d} (G_{i}(tx_{i} − a_{i}) − G_{i}((tx_{i} − a_{i})/2)) = o(1)t^{−1},

since the means are assumed to be finite. It follows that C(tx⃗ − a⃗) = o(1) n(t). By changing the role of F and G, we can also deduce that C(tx⃗ − a⃗) = o(1) m(t). Now, we can combine the estimates. This proves the result. □

In the special case that m = n, we find that

\frac{D(tx⃗ − a⃗)}{m(t)} → E[λ(x⃗, a⃗ + Y⃗)] − λ(x⃗, a⃗) + E[θ(x⃗, a⃗ + X⃗)] − θ(x⃗, a⃗).

Now note that F ∗ G = D + F G. If m = n, it follows from Theorem 3.3 that

(3.1) \frac{F ∗ G(tx⃗ − a⃗) − F ∗ G(tx⃗)}{m(t)} = \frac{D(tx⃗ − a⃗) − D(tx⃗)}{m(t)}

+ \frac{(F (tx⃗ − a⃗) − F (tx⃗)) G(tx⃗ − a⃗)}{m(t)} + \frac{F (tx⃗)(G(tx⃗ − a⃗) − G(tx⃗))}{m(t)}

→ E[λ(x⃗, a⃗ + Y⃗) − λ(x⃗, Y⃗)] + E[θ(x⃗, a⃗ + X⃗) − θ(x⃗, X⃗)]

As a consequence, we have F ∗ G ∈ D(m, ξ) for some limit function ξ.

Taking F = G, we get that

\frac{F^{∗2}(tx⃗ − a⃗) − F^{∗2}(tx⃗)}{m(t)} = 2E[λ(x⃗, a⃗ + X⃗) − λ(x⃗, X⃗)].

For convenience, we set

λ_{1}(x⃗, a⃗) = λ(x⃗, a⃗), λ_{2}(x⃗, a⃗) = 2E[λ(x⃗, a⃗ + X⃗) − λ(x⃗, X⃗)].

Now let X⃗_{1}, X⃗_{2}, …, X⃗_{n} denote i.i.d. copies of X⃗ and for i ⩾ 1, let S⃗_{i} = X⃗_{1}+⋯+X⃗_{i}. Taking G = F^{∗2} in expression (1), we obtain that

\frac{F^{∗3}(tx⃗ − a⃗) − F^{∗3}(tx⃗)}{m(t)} = E[λ_{2}(x⃗, a⃗ + S⃗_{2}) − λ_{2}(x⃗, S⃗_{2})]
+ E[λ_{1}(x⃗, a⃗ + X⃗) − λ_{1}(x⃗, X⃗)]
≡ λ_{3}(x⃗, a⃗).

Proceeding in a similar way, we take G = F^{∗n−1} in (1) to obtain that

\frac{F^{∗n}(tx⃗ − a⃗) − F^{∗n}(tx⃗)}{m(t)} = E[λ_{n−1}(x⃗, a⃗ + S⃗_{n−1}) − λ_{n−1}(x⃗, S⃗_{n−1})]
+ E[λ_{1}(x⃗, a⃗ + X⃗) − λ_{1}(x⃗, X⃗)]
≡ λ_{n}(x⃗, a⃗).

We have proved the following result.

Corollary 3.3. Suppose that F ∈ D_{d}(m, λ) has a finite mean and that m ∈ RV. Then, for each n ⩾ 2, we have F^{∗n} ∈ D_{d}(m, λ_{n}), where λ_{n} is defined recursively above.

If the defining property of F ∈ D_{d}(m, λ) holds locally uniformly in x⃗ we have (cf. previous section) that λ(x⃗, a⃗ + b⃗) = λ(x⃗, a⃗) + λ(x⃗, b⃗) and we can simplify the expressions for λ_{n}. We clearly have

λ_{2}(x⃗, a⃗) = 2E[λ(x⃗, a⃗ + X⃗) − λ(x⃗, X⃗)] = 2λ(x⃗, a⃗),
λ_{3}(x⃗, a⃗) = E[λ_{2}(x⃗, a⃗ + X⃗_{1} + X⃗_{2}) − λ_{2}(x⃗, X⃗_{1} + X⃗_{2})]
+ E[λ_{1}(x⃗, a⃗ + X⃗) − λ_{1}(x⃗, X⃗)]
= 2E[λ(x⃗, a⃗ + X⃗_{1} + X⃗_{2}) − λ(x⃗, X⃗_{1} + X⃗_{2})]
+ λ(x⃗, a⃗) = 3λ(x⃗, a⃗).

The final result is that λ_{n}(x⃗, a⃗) = nλ(x⃗, a⃗).

3.2.3. A result for D_{n}. Now, we consider D_{n}(x⃗). Clearly, we have

D_{2}(x⃗) = F^{2}(x⃗) − F^{∗2}(x⃗),
D_{n+1}(x⃗) = F^{n+1}(x⃗) − F^{∗n+1}(x⃗) = F (x⃗)D_{n}(x⃗) + D(x⃗),

where D(x⃗) = F (x⃗) G(x⃗) − F ∗ G(x⃗) and G(x⃗) = F^{∗n}(x).

As before, let let X⃗_{1}, X⃗_{2}, …, X⃗_{n} denote i.i.d. copies of X⃗, and let S⃗_{j} = X⃗_{1} + X⃗_{2} + ⋯ + X⃗_{j}.

From Theorem 3.3 and Corollary 3.3, we have that G(x⃗) ∈ D_{d}(m, λ_{n}) and

\frac{1}{m(t)}D(tx⃗ − a⃗) → E[λ(x⃗, a⃗ + S⃗_{n})] − λ(x⃗, a⃗) + E[λ_{n}(x⃗, a⃗ + X⃗)] − λ_{n}(x⃗, a⃗).

From Theorem 3.3, we also have that

\frac{1}{m(t)}D_{2}(tx⃗ − a⃗) → 2E(λ(x⃗, a⃗ + X⃗) − λ(x⃗, a⃗)) = Ψ_{2}(x⃗, a⃗).

It follows that \frac{1}{m(t)}D_{n+1}(tx⃗ − a⃗) → Ψ_{n+1}(x⃗, a⃗), where

Ψ_{n+1}(x⃗, a⃗) = Ψ_{n}(x⃗, a⃗) + E[λ(x⃗, a⃗ + S⃗_{n})] − λ(x⃗, a⃗) + E[λ_{n}(x⃗, a⃗ + X⃗)] − λ_{n}(x⃗, a⃗).

In the case where λ(x⃗, a⃗ + b⃗) = λ(x⃗, a⃗) + λ(x⃗, b⃗), we have

Ψ_{2}(x⃗, a⃗) = 2Eλ(x⃗, X⃗),
Ψ_{n+1}(x⃗, a⃗) = Ψ_{n}(x⃗, a⃗) + Eλ(x⃗, S⃗_{n}) + Eλ_{n}(x⃗, X⃗) = Ψ_{n}(x⃗, a⃗) + 2nEλ(x⃗, X⃗).

It easily follows by induction that Ψ_{n}(x⃗, a⃗) = 2\binom{n}{2}Eλ(x⃗, a⃗). Summarizing, we have proved the following result

Corollary 3.4. Suppose that F ∈ D_{d}(m, λ) has a finite mean and that m ∈ RV. Then, for each n ⩾ 2, we have \frac{1}{m(t)}D_{n}(tx⃗−a⃗) → Ψ_{n}(x⃗, a⃗), where Ψ_{n} is defined recursively above. Moreover, if F ∈ DL_{d}(m, λ), we have Ψ_{n}(x⃗, a⃗) = 2\binom{n}{2}Eλ(x⃗, a⃗).

In the case of R_{n}, we have the following result.

Corollary 3.5. Suppose that F ∈ D_{d}(m, λ) has a finite mean and that m ∈ RV. Also, assume that

\frac{(1 − F (tx⃗ − a⃗))^{2}}{m(t)} → 0, for all x⃗ > 0 with x^{0} < ∞ and a⃗ ∈ ℝ^{d}.

Then, for each n ⩾ 2, we have \frac{1}{m(t)}R_{n}(tx⃗ − a⃗) → Ψ_{n}(x⃗, a⃗), where Ψ_{n} is defined recursively above.

3.2.4. Examples. Example 1. For simplicity, we consider d.fs in ℝ^{2}. We say that a positive and measurable function f is regularly varying in the sense of Yakymiv (1982) if there exists function h ∈ RV such that \frac{f(tx,ty)}{h(t)} → λ(x, y) locally uniformly in (x, y) > (0, 0). Notation: f ∈ RV_{Y} (h, λ). We say a function of several variables is monotone if it is increasing or decreasing in each variable. If f is monotone, then pointwise convergence automatically implies convergence locally uniformly.

In our first example, we assume that the d.f. F has first partial derivatives f_{1} and f_{2} that are both regularly varying in the sense of Yakymiv, with the same auxiliary function m ∈ RV. Now we write

F (tx, ty) − F (tx − a, ty − b)
= ∫_{tx−a}^{tx} f_{1}(u, ty) du + ∫_{ty−b}^{ty} f_{2}(tx − a, w) dw

= ∫_{−a}^{0} f_{1}(tx + u, ty) du + ∫_{−b}^{0} f_{2}(tx − a, ty + w) dw.

By our assumption, we have f_{i}(tx, ty)/m(t) → λ_{i}(x, y), locally uniformly in (x, y) > (0, 0). Considering the term with f_{1} and using

\frac{f_{1}(tx + u, ty)}{m(t)} = \frac{f_{1}(t(x + u/t), ty)}{m(t)} → λ_{1}(x, y),

and, similarly, for the term containing f_{2}, we find that

\frac{F (tx, ty) − F (tx − a, ty − b)}{m(t)} → aλ_{1}(x, y) + bλ_{2}(x, y).

If also the marginals are in some class D(m, α_{i}), we obtain that F ∈ D_{2}(m, λ), where λ(x, y) = aλ_{1}(x, y) + bλ_{2}(x, y).

Example 2. Assume F (x, y) = F_{1}(x)F_{2}(y)(1 + θF̄_{1}(x)F̄_{2}(y)). We have

F (x + a, y + b) − F (x, y)

= F_{1}(x + a)F_{2}(y + b)(1 + θF̄_{1}(x + a)F̄_{2}(y + b))

− F_{1}(x)F_{2}(y)(1 + θF̄_{1}(x)F̄_{2}(y))

= (F_{1}(x + a) − F_{1}(x))F_{2}(y + b)(1 + θF̄_{1}(x + a)F̄_{2}(y + b))

+ F_{1}(x)(F_{2}(y + b) − F_{2}(y))(1 + θF̄_{1}(x + a)F̄_{2}(y + b))

+ θF_{1}(x)F_{2}(y)(F̄_{1}(x + a)F̄_{2}(y + b) − F̄_{1}(x)F̄_{2}(y))

= (1 + o(1))(F_{1}(x + a) − F_{1}(x) + F_{2}(y + b) − F_{2}(y))

+ θ(1 + o(1))((F̄_{1}(x + a) − F̄_{1}(x))F̄_{2}(y + b) + F̄_{1}(x)(F̄_{2}(y + b) − F̄_{2}(y)))

= (1 + o(1))(F_{1}(x + a) − F_{1}(x) + F_{2}(y + b) − F_{2}(y))

+ θ(1 + o(1))((F_{1}(x + a) − F_{1}(x))F̄_{2}(y + b) + F̄_{1}(x)(F_{2}(y + b) − F_{2}(y))).

If F_{1} ∈ D(m, α) and F_{2} ∈ D(m, β), where m ∈ RV (λ), we obtain that

F (tx + a, ty + b) − F (tx, ty)
= (1 + o(1))(F_{1}(tx + a) − F_{1}(tx) + F_{2}(ty + b) − F_{2}(ty))
+ θ(1 + o(1))((F_{1}(tx + a) − F_{1}(tx))F̄_{2}(ty + b)
+ F̄_{1}(tx)(F_{2}(ty + b) − F_{2}(ty)))

and then

\frac{F (tx + a, ty + b) − F (tx, ty)}{m(t)} → αax^{λ} + βby^{λ}, min(x, y) < ∞.

Example 3. Now suppose that

F (x, y) = \frac{F_{1}(x)F_{2}(y)}{1 + θF̄_{1}(x)F̄_{2}(y)}.

We obtain a similar result to Example 2.

Example 4. Suppose the bivariate distribution function, F, of (X, Y) is given by the copula function Ψ defined by F (x, y) = Ψ(F_{1}(x), F_{2}(y)), where F_{1} and F_{2} are the marginal distribution functions of X and Y, respectively.

Suppose F_{1} ∈ D_{d}(m, λ_{1}), with λ_{1}(x, a) = αax^{−ρ} (α > 0), and F_{2} ∈ D_{d}(m, λ_{2}), with λ_{2}(x, a) = βax^{−ρ} (β > 0). We determine whether F ∈ D_{d}(m, λ_{3}) for some function λ_{3}.

Start by considering

F (tx + a, ty + b) − F (tx, ty) = Ψ(F_{1}(tx + a), F_{2}(ty + b)) − Ψ(F_{1}(tx), F_{2}(ty)).

Assume that ψ has continuous derivatives of order 2. Using Taylor’s Theorem with remainder, there exists z = (z_{1}, z_{2}) with 0 ⩽ z_{1} ⩽ x and 0 ⩽ z_{2} ⩽ y such that

Ψ(x + a, y + b) − Ψ(x, y)

= aΨ_{1}(x, y) + bΨ_{2}(x, y) + \frac{a^{2}}{2} Ψ_{1,1}(z_{1}, z_{2}) + \frac{b^{2}}{2} Ψ_{2,2}(z_{1}, z_{2}) + abΨ_{1,2}(z_{1}, z_{2}),

where Ψ_{i} and Ψ_{i,j} are the first and second partial derivatives of Ψ. If the derivatives of order 2 are bounded, then

Ψ(x + a, y + b) − Ψ(x, y) = aΨ_{1}(x, y) + bΨ_{2}(x, y) + O(1)(a^{2} + b^{2} + ab)

which gives

Ψ(F_{1}(tx + a), F_{2}(ty + b)) − Ψ(F_{1}(tx), F_{2}(ty))
= [F_{1}(tx + a) − F_{1}(tx)] Ψ_{1}(F_{1}(tx), F_{2}(ty))
+ [F_{2}(ty + b) − F_{2}(ty)] Ψ_{2}(F_{1}(tx), F_{2}(ty))
+ O(1) [F_{1}(tx + a) − F_{1}(tx)]^{2} + O(1) [F_{2}(ty + b) − F_{2}(ty)]^{2}
+ O(1) [F_{1}(tx + a) − F_{1}(tx)] [F_{2}(ty + b) − F_{2}(ty)] .

This gives

(3.2) \frac{Ψ(F_{1}(tx + a), F_{2}(ty + b)) − Ψ(F_{1}(tx), F_{2}(ty))}{m(t)}
∼ αax^{ρ}Ψ_{1}(F_{1}(tx), F_{2}(ty))βby^{ρ}Ψ_{2}(F_{1}(tx), F_{2}(ty)) + O(1) m(t),

for some α, β.

From Omey (1994, p. 113) we have that m(t) = O(F̄_{1}(t)) and m(t) = O(F̄_{2}(t)). Hence,

(3.3) \frac{Ψ(F_{1}(tx + a), F_{2}(ty + b)) − Ψ(F_{1}(tx), F_{2}(ty))}{m(t)}

∼ αax^{ρ}Ψ_{1}(F_{1}(tx), F_{2}(ty)) + βby^{ρ}Ψ_{2}(F_{1}(tx), F_{2}(ty)).

As a specific example, suppose we define Ψ(x, y) by an Archimedian copula (see, for example, Balakrishnan and Lai (2009), p. 37) that is defined by a continuous, decreasing generator function ϕ from [0, 1] to [0, ∞) such that ϕ(Ψ(x, y)) = ϕ(x) + ϕ(y).

Ψ is a copula if and only if its pseudoinverse, given by

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

is decreasing and convex, in which case, Ψ(x, y) = ϕ^{−1}(ϕ(x) + ϕ(y)).

The complementary copula is defined by

Ψ̂(x, y) = x + y − 1 + Ψ(1 − x, 1 − y),

so that

Ψ(x, y) = x + y − 1 + Ψ̂(1 − x, 1 − y).

A particular type of Archimedian copula is the bivariate Pareto copula for which ϕ(t) = t^{−1/δ} − 1, δ > 0. This gives Ψ̂(x, y) = (x^{−1/δ} + y^{−1/δ} − 1)^{−δ} and

Ψ(x, y) = x + y − 1 + ((1 − x)^{−1/δ} + (1 − y)^{−1/δ} − 1)^{−δ}.

Differentiating Ψ gives

Ψ_{1}(x, y) = 1 − [(1 − x)^{−1/δ} + (1 − y)^{−1/δ} − 1]^{−δ−1}(1 − x)^{−1−1/δ}

= 1 − [(1 − x)^{1/δ}((1 − x)^{−1/δ} + (1 − y)^{−1/δ} − 1)]^{−δ−1}

= 1 − [1 + (\frac{1 − x}{1 − y})^{1/δ} − (1 − x)^{1/δ}]^{−δ−1}.

Substituting F_{1} and F_{2} gives

Ψ_{1}(F_{1}(tx), F_{2}(ty)) = 1 − [1 + (\frac{F̄_{1}(tx)}{F̄_{2}(ty)})^{1/δ} − F̄_{1}(x)^{1/δ}]^{−δ−1}.

If α > β, then F̄_{1}(tx) = o(F̄_{2}(ty)) and Ψ_{1}(F_{1}(tx), F_{2}(ty)) → 0. If α < β, then, F̄_{1}(x)/F̄_{2}(x) ∈ RV_{−α+β} and

Ψ_{1}(F_{1}(tx), F_{2}(ty)) ∼ 1 − (x/y)^{(−α+β)(−δ−1)/δ} → 1.

If F̄_{1} ∼ F̄_{2} ∈ RV_{−α}, so that α = β, then

Ψ_{1}(F_{1}(tx), F_{2}(ty)) → 1 − [1 + (x/y)^{−α/δ}]^{−δ−1} .

After deriving similar results for Ψ_{2}(F_{1}(tx), F_{2}(ty)) we can obtain expressions for (3.3).

4. Concluding remarks

(1) To obtain inequalities, we only need detailed information about the marginal distributions. In order to obtain asymptotic equalities, we have to assume that in each of the tails we can use the same auxiliary regularly varying function m(t). We can relax this assumption by looking at the following class of d.f.. For simplicity, we only give the definition in ℝ^{2}.

Definition 4. We say that the d.f. F is in the class D_{d}(m, c⃗, λ) if we have m(t), c_{1}(t), c_{2}(t) ∈ RV and if

\frac{F (c_{1}(t)x_{1}, c_{2}(t)x_{2}) − F (c_{1}(t)x_{1} − a_{1}, c_{2}(t)x_{2} − a_{2})}{m(t)} → λ(x⃗, a⃗),

for all x⃗ > 0 with x^{0} < ∞ and a⃗ ∈ ℝ^{2}.

(2) It is not clear under what general conditions F ∈ D_{d}(m, λ) implies that the defining property holds locally uniformly in x⃗.

(3) In our future research, we will discuss properties of the classes defined below. In each case, we consider limits as x^{0} → ∞.

f ∈ OD(m) : ∀y⃗, |f(x⃗ + y⃗) − f(x⃗)| = O(1) m(x⃗);
f ∈ D(m, λ) : ∀y⃗, (f(x⃗ + y⃗) − f(x⃗))/m(x⃗) → λ(y⃗);
f ∈ L(λ) :  ∀y⃗, f(x⃗ + y⃗)/f(x⃗) → λ(y⃗);

and, so on.

Suppose, for example, that F is a d.f.with marginals F_{i} ∈ OD(m_{i}). Using the inequality |F (x⃗ + y⃗) − F (x⃗)| ⩽ ∑_{i=1}^{d} |F_{i}(x_{i} + y_{i}) − F_{i}(x_{i})| and F_{i} ∈ OD(m_{i}), we obtain that |F (x⃗ + y⃗) − F (x⃗)| = O(1) ∑_{i=1}^{d} m_{i}(x_{i}), as x^{0} → ∞. It follows that F ∈ OD(m) with m(x⃗) = ∑_{i=1}^{d} m_{i}(x_{i}). Note that if m_{i}(x) ∈ ORV, then as x^{0} → ∞,

m(x_{1}y_{1}, …, x_{d}y_{d}) = ∑_{i=1}^{d} m_{i}(x_{i}y_{i}) = O(1) ∑_{i=1}^{d} m_{i}(x_{i}) = O(1) m(x⃗).

(4) In Corollary 3.5, we provided conditions under which

\frac{1}{m(t)}(1 − F^{∗n}(tx⃗ − a⃗) − n(1 − F (tx⃗ − a⃗))) → Ψ_{n}(x⃗, a⃗).

Taking a⃗ = 0, we have the following probabilistic interpretation. We have

1 − F^{∗n}(tx⃗) = P (max_{i}(S_{n,i}/x_{i}) > t),
1 − F (tx⃗) = P (max_{i}(X_{i}/x_{i}) > t).

It follows that

P (max_{i}(S_{n,i}/x_{i}) > t) ∼ nP (max_{i}(X_{i}/x_{i}) > t)

and that

P (max_{i}(S_{n,i}/x_{i}) > t) − nP (max_{i}(X_{i}/x_{i}) > t) ∼ m(t)Ψ_{n}(x⃗,0).

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EHSAL-HUB (Received 11 03 2010) Stormstraat 2 (Revised 02 02 2011) 1000 Brussels, Belgium.

 edward.omey@hubrussel.be

Department of Electronic Engineering Macquarie University, NSW, 2109 Australia

 rein.vesilo@mq.edu.au