Comonotonicity of subsets of Rⁿ

A subset S of Rⁿ is called comonotonic^[5] (sometimes also nondecreasing^[6]) if, for all (x₁, x₂, . . . , x_n) and (y₁, y₂, . . . , y_n) in S with x_i < y_i for some i ∈ {1, 2, . . . , n}, it follows that x_j ≤ y_j for all j ∈ {1, 2, . . . , n}.

This means that S is a totally ordered set.

Comonotonicity of probability measures on Rⁿ

Let μ be a probability measure on the n-dimensional Euclidean space Rⁿ and let F denote its multivariate cumulative distribution function, that is

${\displaystyle F(x_{1},\ldots ,x_{n}):=\mu {\bigl (}\{(y_{1},\ldots ,y_{n})\in {\mathbb {R} }^{n}\mid y_{1}\leq x_{1},\ldots ,y_{n}\leq x_{n}\}{\bigr )},\qquad (x_{1},\ldots ,x_{n})\in {\mathbb {R} }^{n}.}$

Furthermore, let F₁, . . . , F_n denote the cumulative distribution functions of the n one-dimensional marginal distributions of μ, that means

${\displaystyle F_{i}(x):=\mu {\bigl (}\{(y_{1},\ldots ,y_{n})\in {\mathbb {R} }^{n}\mid y_{i}\leq x\}{\bigr )},\qquad x\in {\mathbb {R} }}$

for every i ∈ {1, 2, . . . , n}. Then μ is called comonotonic, if

${\displaystyle F(x_{1},\ldots ,x_{n})=\min _{i\in \{1,\ldots ,n\}}F_{i}(x_{i}),\qquad (x_{1},\ldots ,x_{n})\in {\mathbb {R} }^{n}.}$

Note that the probability measure μ is comonotonic if and only if its support S is comonotonic according to the above definition.^[7]

Comonotonicity of Rⁿ-valued random vectors

An Rⁿ-valued random vector X = (X₁, . . . , X_n) is called comonotonic, if its multivariate distribution (the pushforward measure) is comonotonic, this means

${\displaystyle \Pr(X_{1}\leq x_{1},\ldots ,X_{n}\leq x_{n})=\min _{i\in \{1,\ldots ,n\}}\Pr(X_{i}\leq x_{i}),\qquad (x_{1},\ldots ,x_{n})\in {\mathbb {R} }^{n}.}$

Upper Fréchet–Hoeffding bound for cumulative distribution functions

Let X = (X₁, . . . , X_n) be an Rⁿ-valued random vector. Then, for every i ∈ {1, 2, . . . , n},

${\displaystyle \Pr(X_{1}\leq x_{1},\ldots ,X_{n}\leq x_{n})\leq \Pr(X_{i}\leq x_{i}),\qquad (x_{1},\ldots ,x_{n})\in {\mathbb {R} }^{n},}$

hence

${\displaystyle \Pr(X_{1}\leq x_{1},\ldots ,X_{n}\leq x_{n})\leq \min _{i\in \{1,\ldots ,n\}}\Pr(X_{i}\leq x_{i}),\qquad (x_{1},\ldots ,x_{n})\in {\mathbb {R} }^{n},}$

with equality everywhere if and only if (X₁, . . . , X_n) is comonotonic.

Upper bound for the covariance

Let (X, Y) be a bivariate random vector such that the expected values of X, Y and the product XY exist. Let (X^*, Y^*) be a comonotonic bivariate random vector with the same one-dimensional marginal distributions as (X, Y).^{[note 1]} Then it follows from Höffding's formula for the covariance^[10] and the upper Fréchet–Hoeffding bound that

${\displaystyle {\text{Cov}}(X,Y)\leq {\text{Cov}}(X^{*},Y^{*})}$

and, correspondingly,

${\displaystyle \operatorname {E} [XY]\leq \operatorname {E} [X^{*}Y^{*}]}$

with equality if and only if (X, Y) is comonotonic.^[11]

Note that this result generalizes the rearrangement inequality and Chebyshev's sum inequality.