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Suppose that $(\Omega,\Sigma,\mathbb P)$ is a probability space and let $Y$ and $Z$ be two real-valued, $\Sigma$-to-Borel measurable functions on this space.

Suppose that the distribution of $Y$ conditional on $Z$ satisfies $$\mathbb P(Y\leq y\lvert\rvert Z)=F(y-Z)\quad\text{almost surely, for each $y\in\mathbb R$},\tag{$\clubsuit$}$$ where $F:\mathbb R\to[0,1]$ is some non-decreasing (a fortiory Borel-measurable) distribution function.

The form of the conditional probabilities ($\clubsuit$) leads me to formulate the following two conjectures, which I have had a hard time proving:

Conjecture 1: The unconditional distribution of $Y-Z$ satisfies \begin{align*} \mathbb P(Y-Z\leq y)=F(y)\quad\text{for each $y\in\mathbb R$}; \end{align*}

and

Conjecture 2: The variables $Y-Z$ and $Z$ are independent.

Any hints would be greatly appreciated.

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(A) Let $G$ be the distribution function of $Z$. Then, $$P(Y-Z\leq y)=\int P(Y\leq z+y|Z=z)dG(z)=\int F(y-z+z)dG(z)=F(y)\int dG(z)=F(y)$$

(B) $P[Y-Z\leq y|Z]=P[Y\leq Z+y|Z]=F(Z+y-Z)=F(y)$ independent of $Z$. Hence independence follows.

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