Chebyshev type inequalities by means of copulas

A copula is a function which joins (or ‘couples’) a bivariate distribution function to its marginal (one-dimensional) distribution functions. In this paper, we obtain Chebyshev type inequalities by utilising copulas.


Introduction
A copula is a function which joins (or 'couples') a bivariate distribution function to its marginal (one-dimensional) distribution functions. Mathematically defined, a copula C is a function C : [, ]  → [, ] with the following properties: (C) C(u, ) = C(, u) = , C(u, ) = u, and C(, u) = u for all u ∈ [, ], such that u  ≤ u  and v  ≤ v  . Property (C) is referred to as the -increasing property, or moderate growth []. The -increasing property implies the following properties for any copula C: (C) C is nondecreasing in each variable; (C) C satisfies the Lipschitz condition: for all u  , u  , v  , v  ∈ [, ], For further reading on copulas, we refer the readers to [] and []. While copulas join probability distributions, t-norms join membership functions of fuzzy sets, and hence combining probabilistic information and combining fuzzy information are not so different []. Mathematically defined, a t-norm T is a function T : A copula is a t-norm if and only if it is associative; conversely, a t-norm is a copula if and only if it is -Lipschitz []. The three main continuous t-norms, namely the minimum operator (M(x, y) = min{x, y}), the algebraic product (P(x, y) = xy), and the Lukasiewicz tnorm (W (x, y) = max{x + y -, }), are copulas.
The first importance of these copulas is given by the following: Let C be a copula, then The above inequality is referred to as the Fréchet-Hoeffding bounds for copulas and provides a basic inequality for copulas. Inequality (.) also holds in the contexts of probability theory and fuzzy probability calculus Let C be a copula, x, y ∈ [, ], and set u  = u  = x and v  = v  = y in property (C) (increasing property) above to obtain C(x, x) -C(x, y) -C(y, x) + C(y, y) ≥ , or equivalently, The relationship between the two notions is given in the following result.

Proposition  (Dragomir and Crstici
Consequently, the following Chebyshev type inequalities can be stated (see also are synchronous, then by Proposition , the product copula given by Motivated by this observation, we aim to obtain other types of Chebyshev inequalities by utilising (the general definition of ) copulas instead of the product copula as demonstrated above. Specifically, we provide inequalities for the dispersion of a function f defined on a measure space ( , , μ), with respect to a positive weight ω on with ω(t) dμ(t) = , that is,

Chebyshev type inequalities
The -increasing property of copulas gives us the following result.
Proof follows by (.) and Proposition  part . Now we state a more general form of this inequality. We start with the following lemma.
Thus, u  ≤ u  and v  ≤ v  since f and g are nondecreasing. Therefore, the -increasing property of C gives .
We show that F is Chebyshevian in both cases.
Lemma  and Proposition  part  give us the following.
The first inequality follows from Theorem  (by choosing the W copula) and the rest follows from the Fréchet-Hoeffding bound (.). Similarly, we have The last inequality follows from Theorem  (by choosing the M copula), and the rest follows from the Fréchet-Hoeffding bounds (.).
In what follows, we generalise Theorem  and Example . is Chebyshevian on  . We also have, for a nonnegative integrable function p : → R, Proof The Chebyshevian property of F follows from the -increasing property of copulas. Therefore, we have F(x, x) + F(y, y) ≥ F(x, y) + F(y, x) for all x, y ∈ , or equivalently Multiplying both sides by p(x) and p(y) and taking double integrals over  , we have This completes the proof.
Example  In this example, we obtain some Chebyshev type inequalities by choosing some examples of copulas. Let ( , , μ) be a measure space, f : → [, ] be a measurable function, and p : → R be a nonnegative integrable function. We have the following inequalities: We also have the following result.
Proof The -increasing property of copulas gives us Multiplying with ω(t) ≥  and integrating over give the desired result.
In the next section, we provide further inequalities of this type.

More inequalities
We denote the following: E ω (f ) := ωf dμ, where ω : We denote by D ω (f ) the dispersion of a function f defined on a measure space ( , , μ), with respect to a positive weight ω on with ω(t) dμ(t) = , that is, We have the following inequalities: Proof Firstly, we have From the Lipschitz property of copulas, we have Multiplying with ω(x)ω(y) ≥  and integrating twice over give Finally, Schwarz's inequality gives that is, This completes the proof.
then we have the inequalities The proof follows from Theorem  and a Grüss type inequality We omit the details.

Recall the notation
E ω (f ) := ωf dμ, , g(y) dμ(x) dμ(y), and introduce the following notation: In particular, we have We also have In particular, Proof We know that for any μ-ω-integrable functions k and l, we have for all x, y ∈ . If we multiply (.) by w(x)w(y) ≥  and integrate twice over , then we get By (.) and (.), we get This proves (.). We obtain (.) by setting f ≡ g in (.). From (.), we also have If we multiply (.) by w ≥  and integrate over , then we get Since Proof Using the Fréchet-Hoeffding bounds (.) and the fact that . This inequality is equivalent to Applying the reverse triangle inequality, we have for all u, v ∈ [, ].
By Schwarz's inequality, we also have D ω (f ).
We have the following result.