Abstract:In this paper, we provide a sufficient condition for the absolute continuity of one-dimensional push-forwards of dependent random vectors. Suppose that $X$ has an absolutely continuous distribution and that the conditional distribution of an $\mathbb{R}^d$-valued random vector $Y$ given $X=x$ is nondecreasing in $x\in \mathbb{R}$ in the usual stochastic order. For Borel maps $g\colon \mathbb{R}\times\mathbb{R}^d\to\mathbb{R}$ satisfying a coordinatewise monotonicity condition in $Y$ and a uniform lower-increment condition in $X$, we prove that $g(X,Y)$ has an absolutely continuous distribution. The result requires neither independence nor a joint density, and allows the marginal law of $Y$ to be completely arbitrary. Moreover, the result remains valid if $\mathbb{R}^d$ is replaced by an arbitrary measurable space endowed with a reflexive binary relation. We discuss consequences for monotone risk aggregation and extensions of the familiar regularization by convolution beyond independent random variables.
From: Max Nendel [view email]
[v1]
Fri, 19 Jun 2026 02:16:37 UTC (10 KB)
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