Absolute Continuity of Monotone Aggregations under Positive Regression Dependence

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 $g:\mathbb{R}\times {\mathbb{R}}^d\to \mathbb{R}$ 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.