I'm currently reading a paper which mentions supremal and infimal convolutions. As I understood, they are upper and lower bounds for a joint distribution.

One of the formulas in that paper is as follows:

$J^{v}(z) = sup\{max(F(x) + G(z-x) - 1,0) \}$ where $z = x + y$

where $F$ and $G$ are defined as the CDFs of the $X$ and $Y$ distributions.

However for example if I have the following discrete distributions:

$X = \{1,2,3,4\}$

$Y = \{3,5,7,9\}$

For $J^{v}(11)$ we have two possible candidates as $F(4),G(7)$ and $F(2),G(9)$

$max(F(4) + G(7) - 1,0) = 0.75$ and $max(F(2) + G(9) - 1,0) = 0.5$, but how am i supposed to implement $sup$ operation ? If i need to sum the result, it would be meaningless since $0.75 = 0.5 = 1.25$.

How can i numerically calculate $J^{v}(11)$ ?


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