# pr.probability – Monotonic coupling between "double-sided Gumbel" distributions

I am interested in finding a monotonic coupling between two random variables. Let $$alpha_1> alpha_2$$, $$b . Define the following two densities (non-normalized):
begin {align *} f_ {12} (x) &: = exp left { alpha_2 (a-x) + alpha_1 (x-b) -e ^ {a-x} -e ^ {x-b} right }, \ f_ {21} (x) &: = exp left { alpha_1 (a-x) + alpha_2 (x-b) -e ^ {a-x} -e ^ {x-b} right }. end {align *}
trivially, $$int _ { mathbb {R}} f {{}} (x) dx = int _ { mathbb {R}} f {{21} (x) dx$$.
Moreover, we can check that
for everyone $$y$$, we have
$$begin {equation *} int_x ^ { infty} f_ {12} (y) dy ge int_x ^ { infty} f_ {21} (y) dy. end {equation *}$$

Then there should be a monotonic coupling
enter $$f_ {12}$$ and $$f_ {21}$$.
In other words, we should be able to find a
family of probability measures with densities $$p_x (y) ge0$$
such as:
$$begin {equation *} int _ {- infty} ^ x p_x (y) dy = 1, qquad int_y ^ infty f_ {12} (x) p_x (y) dx = f_ {21} (y) end {equation *}$$
for everyone $$x$$ and $$y$$, respectively.

When $$alpha_1- alpha_2 = 1$$, one of these families is given by
$$begin {equation *} p ^ {(1)} _ x (y) = exp left {a-y-e ^ {a-y} + e ^ {a-x} right } mathbf {1} _ {y le x}. end {equation *}$$

My question is: can we find a generalization $$p_x ^ {( alpha_1- alpha_2)} (y)$$ of $$p_x ^ {(1)}$$ that works for everyone $$alpha_1> alpha_2$$?

(Of course we can take measurements with atoms for $$p_x (y)$$also – there is no need to limit densities.)

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