🤖 AI Summary
This study addresses the absolute continuity of the distribution of outputs generated by monotone aggregation functions under positively regression-dependent structures. By introducing stochastic order monotonicity of conditional distributions and coordinate-wise monotonicity of mappings, the authors establish sufficient conditions that do not require independence or joint density assumptions, allowing for arbitrary marginal distributions. The results are extended to general measurable spaces equipped with reflexive binary relations. Leveraging tools from stochastic orders, conditional distributions, measure theory, and monotone functions, the paper proves the absolute continuity of the distribution of the aggregated variable \(g(X,Y)\). This extends the applicability of convolution regularization techniques and provides theoretical foundations for applications such as risk aggregation.
📝 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.