Approximation by mixtures of multivariate Erlang distributions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915874335621120 |
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| author | Nguyen, Hien Duy |
| author_facet | Nguyen, Hien Duy |
| contents | 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 Szász--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 Hölder conditions of order $α\in(0,1]$, rates of order $n^{-α/2}$ 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 $K^{-α/(2d)}$; in global weighted sup norms we obtain the explicit algebraic component-count rate $K^{-α/[2d(2d+α)]}$; 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; Szász--Mirakjan--Kantorovich operator; density approximation; weighted $L^{p}$ approximation; approximation rates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_18506 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Approximation by mixtures of multivariate Erlang distributions Nguyen, Hien Duy Statistics Theory 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 Szász--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 Hölder conditions of order $α\in(0,1]$, rates of order $n^{-α/2}$ 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 $K^{-α/(2d)}$; in global weighted sup norms we obtain the explicit algebraic component-count rate $K^{-α/[2d(2d+α)]}$; 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; Szász--Mirakjan--Kantorovich operator; density approximation; weighted $L^{p}$ approximation; approximation rates. |
| title | Approximation by mixtures of multivariate Erlang distributions |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2603.18506 |