Weak Dependence Notions and Their Mutual Relationships

: New weak notions of positive dependence between the components X and Y of a random pair ( X , Y ) have been considered in recent papers that deal with the effects of dependence on conditional residual lifetimes and conditional inactivity times. The purpose of this paper is to provide a structured framework for the deﬁnition and description of these notions, and other new ones, and to describe their mutual relationships. An exhaustive review of some well-know notions of dependence, with a complete description of the equivalent deﬁnitions and reciprocal relationships, some of them expressed in terms of the properties of the copula or survival copula of ( X , Y ) , is also provided.


Introduction
In the past few decades, a number of different concepts and notions of dependence between the components X and Y of a random pair (X, Y) have been defined and applied in a variety of fields like reliability theory or actuarial sciences. Since the main motivation for these notions is in the understanding of the effects of the dependence between X and Y on the reliability of one of them given the behavior of the other, most of these notions have been described and defined by using classical stochastic comparisons, comparing in some stochastic sense the distribution of one variable under different conditions for the other. For example, a well-known notion is the stochastically increasing property: given (X, Y), the variable X is said to be stochastically increasing in Y if, and only if, Pr(X > x|Y = y) increases in y for all fixed x (respectively in the corresponding supports of Y and X), or equivalently, if, and only if, (X|Y = y 1 ) ≤ ST (X|Y = y 2 ) for all y 1 ≤ y 2 , where ≤ ST denotes the usual stochastic order (whose definition is recalled next). Similarly, other notions have been defined (and studied in detail) by replacing the usual stochastic order with other common stochastic comparisons and considering other conditional events, such as, for example, {Y > y}.
The purpose of this paper is twofold: on the one hand, it is our intention to provide a complete review of the main notions introduced so far, clearly describing equivalent definitions and reciprocal relationships; on the other hand, it is to enlarge the set of dependence notions with two properties recently considered in [1,2] and with other new ones, all defined as above, but considering stochastic orders for which the corresponding notions have not been characterized in the previous literature. The first point is motivated by the fact that characterizations of the previously defined notions can be found in many different articles and books, where the equivalent definitions and mutual connections are separately and partially analyzed, so that it is difficult to find them in a few exhaustive statements. For the second point, the aim is also to describe in an systematic manner some notions introduced in recent literature and to define a unified framework for dependence notions and their mutual relationships.
The paper also contains a result of general interest in the context of total positivity theory, for which we have not found the statement in the literature. This result, stated and proven in Section 2, deals with the preservation properties of total positivity and is used to provide sufficient conditions for the new notions of dependence.
The paper is structured as follows. The definitions of the stochastic orders used to describe the old and new dependence properties are recalled in Section 2, together with other useful properties and characterizations. Section 3 contains the survey on the notions introduced so far in the literature; all of them imply the Positively Quadrant Dependent (PQD) property, defined later, which is a necessary condition for a property of a vector (X, Y) to be considered a dependence property according to the classical definition described in [3]. The new dependence notions, which are called here "weak dependence notions" since they imply the non-negativity of the linear correlation index, but not the PQD property, are described in Sections 4 and 5. Finally, Section 6 contains counterexamples showing that equivalences among some of these notions are not satisfied.
Throughout the paper, we will consider bivariate vectors (X, Y) having an absolutely continuous joint distribution, unless otherwise stated, and the supports of X and Y will be denoted as S X and S Y . For any random variable or random vector X and an event A, the notation (X|A) describes the random variable whose distribution is the conditional distribution of X given A (tacitly assuming that it exists). The terms "increasing" and "decreasing" are used in place of "nondecreasing" and "nonincreasing", respectively. Furthermore, for a function h(x, y) : R 2 → R, the notations ∂ 1 h(x, y) and ∂ 2 h(x, y) are used for the partial derivatives with respect to the first or the second component, respectively, while ∂ 2 12 h(x, y) is used for the mixed second partial derivative.

Preliminaries
First, we briefly recall the definitions of the stochastic orders that will be used throughout this paper. Most of them are mainly used to compare random lifetimes or inactivity times. To this aim, let X and Y be two absolutely continuous random variables having supports S X and S Y , distribution functions F X and F Y , reliability (survival) functions F X = 1 − F X andF Y = 1 − F Y , and density functions f X and f Y , respectively. Furthermore, for t ∈ R, let X t = (X − t|X > t) and Y t = (Y − t|Y > t) be the corresponding residual lifetimes (RL), and let t X = (t − X|X < t) and t Y = (t − Y|Y < t) denote the corresponding inactivity times (IT). Then, we say that X is smaller than Y: increasing and convex functions φ for which the expectations exist; • in the increasing concave order (denoted by X ≤ ICV Y) if E(φ(X)) ≤ E(φ(Y)) for all increasing and concave functions φ for which the expectations exist.
We address the reader to [4] for a detailed description of these stochastic orders (except for the MIT order, for which we address the reader to [5]) and to [6] or [7] for a list of examples of applications in reliability theory and in actuarial sciences. Among these stochastic orders, there exist the well-known relationships shown in Table 1. The proof of all these relationships may be found in [4], except for those involving the MIT order, which can be found in [5].
We recall here some immediate properties and equivalent conditions of these stochastic orders. The proofs of the following statements, unless stated, can be found in [4] or are straightforward. Proposition 1. The following conditions are equivalent: Note that the third and the fourth conditions in the proposition above can be used to define the order also for variables that are not absolutely continuous.

Proposition 2.
The following conditions are equivalent: Proposition 3. The following conditions are equivalent: The proofs of the propositions above may be found in [4] or easily follow from statements there. For example, to prove the last one, we note that if m X (t) = E(X t ) = E(X − t|X > t) is the MRL of X, then the MRL of X t is m X t (x) = m X (x + t). Therefore, we get (i) ⇒ (ii). Furthermore, (ii) ⇒ (iii) holds because the MRL order implies the ICX order. Finally, (iii) ⇒ (i) holds because the ICX order implies the order in expectations.
Similar results can be stated for inactivity times. The proofs of the following results, again, can be found in [4] or are straightforward. Proposition 4. The following conditions are equivalent: Proposition 5. The following conditions are equivalent: Finally, we include the equivalences for the MIT order. Proposition 6. The following conditions are equivalent: The proof of the equivalence between Conditions (i) and (iii) in the previous proposition may be found in [8], while the equivalence with (ii) can be proven in a similar manner.
For the proof of some of the results stated in the next sections, we will use a few preliminary results whose proofs are given here. These results, for which we have not found the statements in the literature, are of general interest in the context of total positivity theory, regardless of their applications in the following sections.
For them, we recall the notion of total positivity of order two (see, e.g., [9]).

Definition 1.
A bivariate function l : R 2 → R + is said to be totally positive of order two (TP 2 ) if: We also recall the following useful property, the proof of which is given in Lemma 7.1(a) in Chapter 4 of [6] (see also the remark after the lemma).

Lemma 1.
Let W be a Lebesgue-Stieltjes measure, not necessarily positive, defined on the interval (a, b), and let φ be a non-negative function defined on (a, b).
We can now state and prove the first one of the preliminary results. Proof. We assume that h(x, y) is TP 2 in (x, y), which means that: Now observe that (1) holds if, and only if: From (1) and (2), we can rewrite the TP 2 property of h(x, y) as: Now, given y 1 ≤ y 2 , this is the same as: where: Now, consider the function: where: Then, by noting that: Since φ is increasing and non-negative, by using Lemma 1, we have: By repeating the argument in the opposite direction, we see that the last inequality is the same as Therefore, we have proven that (1) Rewriting this inequality as: , for x ≤ t, y 1 ≤ y 2 and taking x = t, we see that This means that ∞ x f (s, y)φ(s)ds is TP 2 in (x, y). The following is an immediate consequence of Theorem 1.

Proof.
Since h(x, y) is TP 2 in (x, y) and ψ is increasing, it follows from Theorem 1.1 of [9], p. 99, that h(ψ(x), y) is TP 2 in (x, y). This is equivalent to stating that: which is the same as: Equivalently, we can say that: is TP 2 in (x, y). Since ψ is strictly increasing and concave, the result follows from (5) by applying Theorem 1 with φ(s) = 1 ψ (s) .
Using the same procedure as above, but applying Lemma 7.1(b) in Chapter 4 of [6] instead of Lemma 7.1(a), one can also prove the following statements.
is also TP 2 in (x, y) for all non-negative and decreasing real functions φ.

Strong Dependence Notions
Starting from the seminal paper by Kimeldorf and Sampson [3], where a unified framework to deal with dependence concepts was proposed, many alternative notions of dependence (either positive or negative) have been investigated and applied in the literature with the intent of mathematically describing the different properties and aspects of this intuitive concept. According to [3], a bivariate property can be considered as a positive dependence notion if it satisfies a number of conditions, the first of which is the PQD property, whose definition and relevance in this context is recalled later.
Since all the monotone dependence properties based on the level of concordance between the components of a random vector are entirely described by its copula, whenever it is unique, one can rewrite the conditions described in [3] in terms of families of copulas, without taking care of the marginal distributions. See, e.g., [10] or [11] for the relationships between copulas and positive dependence notions or indexes. For this reason, we briefly recall here the definition of the copula and the survival copula of a random vector (X, Y).
Given a random vector (X, Y) with joint distribution function F and marginal distributions F X and F Y , the function C : In this case, it also holds, if F X and F Y are continuous, that: for all u, v ∈ [0, 1], where F −1 X denotes the quasi-inverse of F X (and similarly for F −1 Y ). Actually, the copula of a vector (X, Y), which is unique whenever F X and F Y are continuous, is the joint distribution function of a vector having uniformly distributed margins on [0, 1] ⊂ R, which entirely describes the dependence between the components of the vector. Moreover, since in some applicative fields, like reliability theory, the dependence among components of a vector is analyzed through the survival copula instead of through the copula, then we recall its definition as well: given (X, Y), and denotingF,F X , andF Y as its joint survival function and the two marginal survival functions, the survival copula C is defined as: If C exists, then one also has: for all x and y. For the basic properties of copulas and survival copulas, we refer the reader to [11,12].
The definitions of the dependence properties considered throughout the paper, and previously defined in the literature, are recalled here.
is increasing in s and t for all (x, y) ∈ R 2 (see [11], p. 198) or, equivalently, iff (X|Y ≤ t) is RHR-decreasing in t.
Note that some of these notions, like the PQD, are "symmetric", in the sense that they describe both the properties of X given Y and the properties of Y given X, while other notions, like RTI(X|Y), describe only the properties of X given Y, while the opposite notions (e.g., RTI(Y|X)) describe the properties of Y given X. For these cases, the symmetric notions can be defined similarly and will be denoted as P(Y|X), where P is the specific property.
As pointed out before, for all the above-mentioned notions, there exist equivalent definitions given in terms of the survival copula C (or connecting copula C) of the vector (X, Y). These alternative definitions, proven in [11], [13], or [14], will be pointed out in the following statements, which also describe more interesting equivalences. Here, and throughout the rest of the paper, the notations X t,s and Y t,s are used to describe double conditioning, i.e., X t,

Proposition 7.
For continuousF X andF Y , the following conditions are equivalent: The equivalences between the first three conditions can be obtained from the equivalences given in the preceding section. The equivalences between (i) and (iv) can be obtained both from Proposition 3.1 in [2] or from Proposition 3.3 (iii) in [13]. The equivalence between (iv) and (v) can be found in [11] (Theorem 5.2.5). It is interesting to observe that, even if RTI(Y|X) refers to a property describing the behavior of the law of Y given X, Conditions (i), (ii), and (iii) refer to the behavior of X given Y. A discussion of and the formal motivation for this fact were clearly provided in [14].
Note that a similar proposition could be stated just for a fixed s. Note that if (i) holds for a fixed s > 0 and allF X ,F Y , then (iv) holds for all u =F X (t) > 0 and v =F Y (s) > 0; by changingF Y , we cover the interval (0, 1). However, if we fixF X andF Y , then the equivalence is not true when we also fix s (the RTI(Y|X) property is no longer satisfied). In this case, (v) does not hold, but one could write (iv) for a fixed value v =F Y (s).
The following statement deals with the SI notion.
Proposition 8. The following conditions are equivalent: The equivalences between the first three conditions can be obtained from the equivalences given in the preceding section. The equivalences with (iv) can be obtained both from Proposition 3.2 in [2] or from Proposition 3.3 (v), in [13]. Again, it is interesting to observe that SI(Y|X) refers to a property describing the behavior of the law of Y given X, while Conditions (i), (ii), and (iii) refer to the behavior of X given Y (see [14]).
The following statement describes equivalent definitions for the RCSI notion.
Proposition 9. The following conditions are equivalent: for all t 1 ≤ t 2 and allF X ,F Y ; (iii) X t,s 1 ≤ HR X t,s 2 for all t, all s 1 ≤ s 2 and allF X ,F Y ; (iv) X t,s 1 ≤ ST X t,s 2 for all t, all s 1 ≤ s 2 and allF X ,F Y ; (v) Y t 1 ,s ≤ HR Y t 2 ,s for all s, all t 1 ≤ t 2 and allF X , The proofs of the equivalences in the above proposition can be found in [1,14] or [13]. The preceding propositions can be completed with the following result (extracted from Proposition 3.3 in [13]) that does not have an equivalent result based on residual lifetimes. For the equivalence between Conditions (iv) and (v), see [15]. Proposition 10. The following conditions are equivalent: ≥ 0 for all increasing real functions τ and φ for which the covariance exists; The following two propositions hold as well.
Proposition 11. The following conditions are equivalent: The proof of equivalence among (i), (iv), and (v) can be found in [13] or [14]. The proof of equivalence among (ii), (iii), and (v) can be found in [1]. Once more, it is interesting to observe that SIRL(Y|X) refers to a property describing the behavior of the law of Y given X, while Conditions (i), (ii), and (iii) refer to the behavior of X given Y (see [14]). Proposition 12. The following conditions are equivalent: (i) (Y|X = t 1 ) ≤ LR (Y|X = t 2 ) for all t 1 < t 2 , t 1 , t 2 ∈ S X and allF X ,F Y ; (ii) (X|Y = s 1 ) ≤ LR (X|Y = s 2 ) for all s 1 < s 2 , s 1 , s 2 ∈ S Y and allF X ,F Y ; (iii) (X t |Y = s 1 ) ≤ LR (X t |Y = s 2 ) for all t, s 1 < s 2 , s 1 , s 2 ∈ S Y and allF X ,F Y ; (iv) (X t |Y = s 1 ) ≤ RHR (X t |Y = s 2 ) for all t, s 1 < s 2 , s 1 , s 2 ∈ S Y and allF X ,F Y ; The proof of equivalence between (vii) and (viii) can be found in [11]. The proof of equivalence among (i), (ii), and (viii) can be found in [13] or [14]. The proof of equivalence among (iii), (iv), (v), and (vi) follows from Proposition 2.1.
In order to synthesize all these properties with a uniform notation and to introduce similarly defined new properties, the definitions that follow were proposed in [13,14]. The first one of them deals with the SI notion. Definition 3. We say that (X, Y) is stochastically increasing (decreasing) in the order ORD, denoted SI ORD (X|Y) (SD ORD (X|Y)), if: holds for all s 1 < s 2 , with s 1 , s 2 ∈ S Y , for a given stochastic order ORD.
The classes for (Y|X) are defined in a similar manner. With this definition, the class SI(X|Y) can also be written as SI ST (X|Y); the class SIRL(X|Y) can also be written as SI HR (X|Y); and the condition (i) in Proposition 12 as SI LR (Y|X). Note that in this last case, SI LR (Y|X) is equivalent to SI LR (X|Y), and so, we can just write SI LR .
One can do the same with the RTI and LTD classes by proposing the following definitions (see [14]).

Definition 4.
We say that (X, Y) is right tail increasing (decreasing) in the order ORD, denoted RTI ORD (X|Y) (RTD ORD (X|Y)), if: holds for all s 1 < s 2 for a given stochastic order ORD.

Definition 5.
We say that (X, Y) is left tail decreasing (increasing) in the order ORD, denoted LTD ORD (X|Y) (LTI ORD (X|Y)), if: holds for all s 1 < s 2 for a given stochastic order ORD.
The classes for (Y|X) are defined in a similar manner. With this definition, the class RTI(X|Y) can also be written as RTI ST (X|Y); the conditions (i) and (ii) in Proposition 9 can also be written as RTI HR (X|Y) and RTI HR (Y|X); and the condition (i) in Proposition 11 as RTI LR (X|Y).
We can add more notions by considering the following definitions.

Definition 6.
We say that (X, Y) is right tail increasing (decreasing) at zero in the order ORD, denoted RTI 0 ORD (X|Y) (RTD 0 ORD (X|Y)), if: holds for all s for a given stochastic order ORD.

Definition 7.
We say that (X, Y) is left tail decreasing (increasing) at infinity in the order ORD, holds for all s for a given stochastic order ORD.
With these definitions, PQD can also be written as RTI 0 ST (Y|X) or RTI 0 ST (X|Y), but also as LTD ∞ ST (Y|X) or LTD ∞ ST (X|Y). Analogously, RTI(Y|X) and LTD(Y|X) can be written as RTI 0 HR (X|Y) and LTD ∞ RHR (X|Y), respectively. Furthermore, SI(Y|X) can be written as RTI 0 LR (X|Y) or as LTD ∞ LR (X|Y). The proofs of these equivalences can be found in [2,13]. Note also that some of these notions are actually equivalent, like, e.g., RTI 0 LR (Y|X) and SI ST (X|Y) or RTI 0 HR (Y|X) and RTI ST (X|Y) (see [14] for details). The preceding propositions can be connected by using the relationships among the stochastic orders shown in Table 1. They are summarized in Table 2. For example, to prove SIRL(X|Y) ⇒ RCSI(X, Y), we just note that RCSI(X, Y) = RTI HR (X|Y), that SIRL(X|Y) = RTI LR (X|Y), and that the LR order implies the HR order. These relationships prove that they are positive dependence properties, since all of them imply the PQD property. Analogously, the corresponding negative dependence properties imply the NQD property. Some of these relationships were given in [13], p. 166, and in [14]. Connections with the ordering properties of coherent systems and extremes values were given in [16,17]. Table 2. Relationships among positive dependence properties.
The relationships for the reversed properties (i.e., based on cumulative distributions rather than survival functions) are given in Table 3. For them, observe, e.g., that SI RHR (Y|X) iff ∂ 1 C(u, v) is TP 2 and that SI RHR (X|Y) iff ∂ 2 C(u, v) is TP 2 (see [13], where more equivalences were also provided). More equivalences can be seen in [13]. Note that two of the notions described in this table (SI RHR (Y|X) and SI RHR (X|Y)) were previously considered in [13]. Table 3. Relationships among reversed positive dependence properties.
Negative dependence properties can be considered as well. For them, one has to consider the RTD and LTI properties, obtaining relationships similar to those described in Tables 2 and 3.

Weak Dependence Notions
In this section, new dependence notions are introduced and discussed. These notions, which are defined as those satisfying RTI ORD (X|Y) and RTI 0 ORD (X|Y) where ORD is one of the orders ICX, or MRL, or simply in Expectation (E), are "weak" in the sense that they do not imply the PQD property, which is a necessary condition for a property to be considered a "dependence notion" according to the classical definition given in [3]. However, they imply the non-negativity of the linear correlation coefficient r X,Y , as mentioned after Proposition 15, and this is the reason to consider them as "weak dependence notions". Note that some of these notions have been already considered and applied in the literature; this is the case, for example, of RTI 0 ICX (X|Y), which was named positive stop-loss dependence in [7].
For RTI 0 ICX (X|Y), we have the following properties.
Proposition 13. Let (X, Y) be a random vector with a continuous marginal distribution functions F X and F Y . Then, the following conditions are equivalent: (i) X ≤ ICX (X|Y > y) for all y ∈ R (i.e., RTI 0 ICX (X|Y)); (ii) The survival copula C satisfies: Proof. It is clear that X ≤ ICX (X|Y > y) for all y, if, and only if, This is equivalent to: Since F X is continuous, this is the same as: Corollary 3. Under the above assumptions, ifF X is convex and: then X ≤ ICX (X|Y > y) for all y ∈ R and allF Y .
Proof. SinceF X is decreasing and convex, thenF −1 X (u) is decreasing and convex and −F −1 X (u) is increasing and concave. Then, by applying Lemma 7.1(b), of [6], to equation (8) above, we obtain: which is the same as (7).
In particular, we have that the PQD property implies both RTI 0 ICX (X|Y) for allF Y and all convexF X and RTI 0 ICX (Y|X) for allF X and all convexF Y . Note also that if RTI 0 ICX (X|Y) holds for a continuous survival functionF Y , then it holds for all continuous survival functionsF Y (since (6) holds for any v =F Y (y) and all y).
For RTI 0 MRL (X|Y), we have the following properties.

Proposition 14.
Let (X, Y) be a random vector with continuous marginal distribution functions F X and F Y . Then, the following conditions are equivalent: (i) X ≤ MRL (X|Y > y) for all y ∈ R (i.e., RTI 0 MRL (X|Y)); (ii) X t ≤ MRL X t,s for all (t, s) ∈ R 2 ; (iii) X t ≤ ICX X t,s for all (t, s) ∈ R 2 ; (iv) The survival copula C satisfies: Proof. The equivalence between the first three items follows from Proposition 3. For the equivalence with (iv), observe that X ≤ MRL (X|Y > y) for all y ∈ S Y if, and only if, decreases in t ∀y, i.e., if, and only if, This is equivalent to: for all v ∈ [0, 1] and t. The latter can be written as: which is the same as: i.e., (10).

Corollary 4.
Under the above assumptions, ifF X is strictly monotone and convex and: then X ≤ MRL (X|Y > y) for all y ∈ S Y and allF Y .
Proof. Observe that (11) is equivalent to: Defining: then (11) can be restated as TP 2 in the (z, i) property of 1 z K(1 − u, i)du. Thus, by applying Corollary 1 one gets that ∞ t K(1 − F X (u), i)du is also TP 2 in (t, i), i.e., that: i.e., X ≤ MRL (X|Y > y) for all y. Note that the strict monotonicity of F X is required in this proof.
Note that the sufficient conditions for RTI 0 MRL (X|Y) described in Corollary 4 were proven already in [2] with a different and longer proof. However, the strict monotonicity of F X is not required in the proof provided there. Moreover, note that (11) is also a necessary condition since equation (3.5) appearing in [2] is what one gets by choosing uniform distributions. However, we think that the convexity ofF X is not a necessary condition.
Alternative conditions for RTI 0 MRL (X|Y) are described also in Proposition 3.3 (vi), of [13] (obtained from [18]), and are: E(X) ≤ E(X|Y > y) ∀y ∈ R (13) and: Clearly, (13) is a necessary condition (because they are the values of the respective MRL functions at zero).
It is useful to point out that the counterexamples provided in Section 6 show that there exist no relationships between PQD (i.e., C(u, v) ≥ uv for all u, v ∈ [0, 1]) and both (11) and (14).
The Positive Quadrant Dependence in Expectation property (PQDE), defined in [19], is considered in the following statement. According to the definition provided in [19], and the notation used here, we say that the vector (X, Y) satisfies PQDE(X|Y) if, and only if, E(X) ≤ E(X|Y > y) for all y ∈ R (and similarly for PQDE(Y|X)). Note that, actually, a study on a negative dependence notion analog of the property PQDE(Y|X) goes back to [20].

Proposition 15.
Let (X, Y) be a vector of non-negative variables having continuous marginal distribution functions F X and F Y , respectively. Assuming that E(X|Y > y) and E(X|Y ≤ y) exist for all y, then the following conditions are equivalent: (i) E(X) ≤ E(X|Y > y) for all y ∈ R (i.e., PQDE(X|Y)); (ii) E(X) ≥ E(X|Y ≤ y) for all y ∈ R; (iii) Cov(X, φ(Y)) ≥ 0 for all increasing real functions φ for which the covariance exists; (iv) The survival copula C satisfies: Proof. For the equivalence between (i) and (ii), note that, for fixed y ∈ R, one has: which implies: Thus, E(X) − E(X|Y > y) and E(X|Y ≤ y) − E(X) must have the same sign, and the equivalence follows. In a similar manner, again by using Inequality (16), one can prove the equivalence between (i) and (iii) (see [20], Theorem 3.1, for details). For the relationship between (i) and (iv), observe that: and similarly for E(X|Y > y). If the marginal expectations are finite, then: We can thus verify that E(X) ≤ E(X|Y > y) for all y if, and only if, Note that, as an immediate consequence of the preceding proposition, we have that the PQD property implies E(X) ≤ E(X|Y > y) for all y and allF X ,F Y , if the expectations exist. Moreover, it is easy to verify that r X,Y ≥ 0 if E(X) ≤ E(X|Y > y) for all y, where: is the Pearson's correlation coefficient (here, σ X and σ Y are the standard deviations of X and Y, respectively). For the formula of r X,Y , see, e.g., [21]. For RTI MRL (X|Y), we have the following property.
Proposition 16. Let (X, Y) be a random vector with continuous marginal distribution functions F X and F Y . Then, the following conditions are equivalent: (i) (X|Y > y 1 ) ≤ MRL (X|Y > y 2 ) for all y 1 ≤ y 2 , y 1 , y 2 ∈ R (i.e., RTI MRL (X|Y)); (ii) (X t |Y > y 1 ) ≤ MRL (X t |Y > y 2 ) for all y 1 ≤ y 2 , y 1 , y 2 ∈ R, and for all t ∈ R; (iii) (X t |Y > y 1 ) ≤ ICX (X t |Y > y 2 ) for all y 1 ≤ y 2 , y 1 , y 2 ∈ R, and for all t ∈ R; (iv) The survival copula C satisfies: Proof. The equivalence between the first three items follows from Proposition 3. For the equivalence with (iv), observe that (X|Y > y 1 ) ≤ MRL (X|Y > y 2 ) for all y 1 ≤ y 2 if, and only if, i.e., if, and only if, This is equivalent to: The latter can be written as: for all z ∈ [0, 1] and 0 ≤ v 2 ≤ v 1 ≤ 1, which is equivalent to (17).
Note that, as an immediate consequence of the preceding proposition, we have that the RCSI property (i.e.,Ĉ is TP 2 ) implies (X|Y > y 1 ) ≤ MRL (X|Y > y 2 ) for all y 1 ≤ y 2 and allF X ,F Y .

Corollary 5.
Under the above assumptions, ifF X is strictly monotone and convex and: then (X|Y > y 1 ) ≤ MRL (X|Y > y 2 ) for all y 1 ≤ y 2 , y 1 , y 2 ∈ R and allF Y .
Proof. From (18) and the convexity ofF X , by applying Corollary 1, one gets that 1 t C(1 − F X (u), 1 − v)du is also TP 2 in (t, v). This implies that: Note that the sufficient conditions for RTI MRL (X|Y) described in Corollary 5 were proven already in [1] with a different and less immediate proof. However, the strict monotonicity ofF X is not required in the proof provided there.
The relationships between these new dependence notions and the ones described in the previous section are summarized in the following Table 4. Note that the fact that PQD(X, Y) is not implied by RTI MRL (X|Y), mentioned in the table, is because the MRL order does not imply the ST order. Table 4. Relationships among weak positive dependence properties.
To avoid false implications in Table 4, one can try to replace the PQD property with the RTI ICX (X|Y) property, for which we have the following statements. Proposition 17. Let (X, Y) be a random vector with continuous distribution functions F X and F Y . Then, the following conditions are equivalent: (i) (X|Y > y 1 ) ≤ ICX (X|Y > y 2 ) for all y 1 ≤ y 2 , y 1 , y 2 ∈ R (i.e., RTI ICX (X|Y)); (ii) The survival copula C satisfies: Proof. Observe that (X|Y > y 1 ) ≤ ICX (X|Y > y 2 ) for y 1 ≤ y 2 , if, and only if, i.e., if, and only if, The latter is the same as: for v 1 ≥ v 2 and t ∈ R, which can be also restated as: for v 1 ≥ v 2 and t ∈ R. This is the same as (19).
Thus, the relationships between the new dependence notions (and some of the strong ones) are summarized in the following Table 5, where only the dependencies of X given Y are considered (due to the fact that the RTI ICX (X|Y) is not symmetric, i.e., that RTI ICX (X|Y) is different than RTI ICX (Y|X)). However, a similar table can be obtained for (Y|X). Table 5. Relationships among weak positive dependence properties.
All the properties mentioned above are not independent of the marginal distributions of (X, Y), and the convexity of the marginal survival functions is required. However, interesting properties of the survival copula of the vector are introduced. In the following, these particular properties are studied in detail, and their mutual relationships are pointed out.
To this aim, first observe that considering the property of the copula rather than the property of the vector, one gets a different final result in terms of dependence indexes. In fact, letting P denote the property: and letting P denote the property: X ≤ ICX (X|Y > y) ∀y ∈ R and for fixed continuous survival functionsF X ,F Y , one immediately observes that both are positive dependence properties weaker than PQD, but with different implications. Property P verifies: is the Spearman's rho coefficient for X and Y (for the formula of ρ X,Y ,, see equation (5.1.15c) in [11]), and the first implication follows from (6). Property P verifies: Because of this fact, one can define weak dependence properties as has been done for P, by letting the margins be uniformly distributed on (0, 1) in the definitions above. Doing this, one gets the weak positive dependence properties described in the statement that follows (easy to prove). Note that property P described above is denoted here as P 2 .
Remark 1. The relationships between these properties can be better understood by considering their equivalent formulations described as follows.
For (23), observe that it is the same as: i.e., In a similar manner, one has that (24) holds if, and only if, Clearly, (29) implies (28), i.e., property P 2 implies P 1 . For (25), observe that it is the same as:

Reversed Weak Dependence Notions
By considering relationships between inactivity times rather than residual times, one can define other dependence notions, which we call "reversed weak dependence notions". In particular, the notions considered here are based on comparisons in the MIT and the ICV orders and on the LTD property. Thus, the notions considered here are LTD ∞ ICV (X|Y), LTD ∞ MIT (X|Y), LTD ICV (X|Y), LTD MIT (X|Y), and the one based on inequality E(X) ≥ E(X|Y ≤ y) for all y ≥ 0 (which is actually equivalent to PQDE(X|Y), as seen in Proposition 15).
All the proofs of the statements described in this section follow the same lines of those described in Section 4 and are therefore omitted, except for the first one (given as an example).

Proposition 19.
Let (X, Y) be a random vector with continuous marginal distribution functions F X and F Y . Then, X ≥ ICV (X|Y ≤ y) for all y ∈ R (i.e., LTD ∞ ICV (X|Y)) if, and only if: Proof. It is clear that X ≥ ICV (X|Y ≤ y) for all y, if, and only if, This is equivalent to: Since F X is continuous, this is the same as (33).

Corollary 7.
Under the above assumptions, if F X is convex and: then X ≥ ICV (X|Y ≤ y) for all y ∈ R (i.e., LTD ∞ ICV (X|Y)) for all F Y .
From Proposition 6, we obtain the following result.
Proposition 20. Let (X, Y) be a random vector with continuous marginal distribution functions F X and F Y . Then, the following conditions are equivalent: (i) X ≥ MIT (X|Y ≤ y) for all y ∈ R (i.e., LTD ∞ MIT (X|Y)); (ii) t X ≤ MIT ( t X|Y ≤ y) for all y, t ∈ R; (iii) t X ≤ ICX ( t X|Y ≤ y) for all y, t ∈ R;

Corollary 8.
Under the above assumptions, if F X is convex and: then X ≥ MIT (X|Y ≤ y) for all y ∈ R (i.e., LTD ∞ MIT (X|Y)) for all F Y .
The following theorem can also be obtained from Proposition 6.

Proposition 23.
Let (X, Y) be a random vector with continuous marginal distribution functions F X and F Y . Then, E(X) ≥ E(X|Y ≤ y) for all y ∈ R, whenever the expectations exist, if, and only if: Note that the following chain of implications holds.
Thus, all the notions described above are positive dependence notions, depending on the marginal distributions of (X, Y), whose relationships are described in Table 7.
As has been done for the positive dependence notions in the previous section, one can again define weak dependence properties that are independent of the margins by considering only the properties of the connecting copula C, thus letting the margins be uniformly distributed on (0, 1) in the definitions above. Doing so, one gets the weak positive dependence properties described in the statement that follows. Table 7. Relationships among reversed weak dependence properties.  [v 2 C(u, v 1 ) − v 1 C(u, v 2 )]du ≥ 0 for all z ∈ [0, 1] and 0 ≤ v 1 ≤ v 2 ≤ 1; (P R 5 ) (U|V ≤ v 2 ) ≥ MIT (U|V ≤ v 1 ) for all v 1 ≤ v 2 (i.e., LTD MIT (U|V)) if, and only if, The relationships among these notions can be proven as described for the weak positive dependence properties and are listed in Table 8. Table 8. Relationships among reversed weak dependence properties.

Counterexamples
Comments on the relationships among the above-described properties of copulas are given here, together with other useful counterexamples, like the first one that follows, which shows that the convexity ofF X is not a necessary condition for RTI 0 MRL (X|Y).

Example 1.
Let us consider an FGM survival copula, that is, for θ ∈ [−1, 1]. Then: is decreasing in u when θ ≥ 0. Hence, from Proposition 8, we get X ≤ LR (X|Y > s) for all s and all F X ,F Y (i.e., RTI 0 LR (X|Y)). Therefore, X ≤ MRL (X|Y > s) for all s and allF X ,F Y . Therefore, we do not need the condition "F X is convex" for RTI 0 MRL (X|Y) to hold. A straightforward calculation shows that (11) holds for this copula when θ ≥ 0.
In the next example, we discuss the relationships between C(u, v) ≥ uv (PQD property) and the conditions on C for RTI 0 MRL (X|Y) to hold, i.e., (11) (which is the same as RTI 0 MRL (U|V)) and (14). In particular, it proves that the PQD property does not imply (11) or (14). Moreover, it also proves that RTI 0 ICX (X|Y) does not imply RTI 0 MRL (X|Y).
The main disadvantage of the new dependence notions proposed here is that they depend on the marginal distributions (as the Pearson's correlation coefficient). This problem can be solved by replacing them with the respective copula properties obtained by assuming uniform marginals. In this case, they imply a positive Spearman's rho coefficient. Moreover, we must say that there are other weak dependence notions that come through in papers devoted to more specific areas, which are not studied here for the sake of brevity. This is the case, for example, of the Gini correlation introduced in economics in [23] and further studied in [24,25]. Given two random variables X and Y, the Gini correlation is a nonsymmetric measure given by: Cov(X, F X (X)) whose properties are a mixture of Pearson's and Spearman's correlations. It follows from Proposition 15(iii) that PQDE(X|Y) implies ρ X•Y ≥ 0. Furthermore, for simplicity, we have just studied the bivariate case. The study of other notions and the extensions of these dependence properties to n-dimensional random vectors are not straightforward and will be studied in future research projects.