Approximation by mixtures of multivariate Erlang distributions

TL;DR

This paper proves that finite mixtures of multivariate Erlang distributions with a common rate parameter are dense in certain probability density classes on positive real spaces, providing constructive approximation methods and explicit convergence rates.

Contribution

It establishes the density of multivariate Erlang mixture densities with a common rate parameter in $L^{p}$ spaces and derives explicit approximation rates and bounds.

Findings

Erlang mixture densities are dense in $L^{p}$ densities on $ plus^d$.

Constructive approximation using tensor products of Szász--Mirakjan--Kantorovich operators.

Explicit convergence rates depending on the number of mixture components and domain properties.

Abstract

We prove that finite multivariate Erlang mixture densities with a common rate parameter are dense in the class of probability densities on $\mathbb{R}_{+}^{d}$ that belong to $L^{p}$, for every dimension $d\in\mathbb{N}$ and every $1\le p<\infty$. The argument is constructive: the one-dimensional Sz\'asz--Mirakjan--Kantorovich operator yields Erlang mixture approximations, and its tensor product yields multivariate approximants with a common scale. We then obtain several quantitative consequences. These include compact-set uniform approximation bounds and, under local H\"older conditions of order $\alpha\in(0,1]$, rates of order $n^{-\alpha/2}$ as the common scale $1/n$ tends to zero, whole-domain convergence in weighted sup norms, weighted and unweighted $L^{p}$ rates, and explicit rates for finite mixtures indexed by the number of mixture components. In particular, if the approximating density is required to have at most $K$ mixture components, then on fixed compact cubes we obtain an algebraic rate of order $K^{-\alpha/(2d)}$; in global weighted sup norms we obtain the explicit algebraic component-count rate $K^{-\alpha/[2d(2d+\alpha)]}$; and for $1<p<\infty$ we obtain corresponding weighted $L^{p}$ component-count rates. The results strengthen the weak-approximation theory for multivariate Erlang mixture distributions and yield immediate corollaries for broader classes such as product-gamma mixtures. \noindent\textbf{Keywords:} multivariate Erlang mixtures; Erlang distributions; Sz\'asz--Mirakjan--Kantorovich operator; density approximation; weighted $L^{p}$ approximation; approximation rates.