Approximation of probability density functions via location-scale finite mixtures in Lebesgue spaces

TrungTin Nguyen; Faicel Chamroukhi; Hien D. Nguyen; Geoffrey J. McLachlan

Abstract

The class of location-scale finite mixtures is of enduring interest both from applied and theoretical perspectives of probability and statistics. We establish and prove the following results: to an arbitrary degree of accuracy, (a) location-scale mixtures of a continuous probability density function (PDF) can approximate any continuous PDF, uniformly, on a compact set; and (b) for any finite $p \geq 1$ , location-scale mixtures of an essentially bounded PDF can approximate any PDF in $ℒ_{p}$ , in the $ℒ_{p}$ norm.

¹Normandie Univ, UNICAEN, CNRS, LMNO, 14000 Caen, France.
²School of Engineering and Mathematical Sciences. Department of Mathematics and Statistics, La Trobe University, Melbourne, Victoria, Australia.
³School of Mathematics and Physics, University of Queensland, St. Lucia, Brisbane, Australia.
^∗∗Corresponding author.

Keywords: Mixture models, approximation theory, uniform approximation, probability density functions.

1 Introduction

Define $(𝔼, ∥ \cdot ∥_{𝔼})$ to be a normed vector space (NVS), and let $x \in (ℝ^{n}, ∥ \cdot ∥_{2})$ , for some $n \in ℕ$ , where $∥ \cdot ∥_{2}$ is the Euclidean norm. Let $f : ℝ^{n} \to ℝ$ be a function satisfying $f \geq 0$ and $\int f d λ = 1$ , where $λ$ is the Lebesgue measure. We say that $f$ is a probability density function (PDF) on the domain $ℝ^{n}$ (which we will omit for brevity, from hereon in). Let $g : ℝ^{n} \to ℝ$ be another PDF and define the functional class $ℳ^{g} = ⋃_{m \in ℕ} ℳ_{m}^{g}$ , where

ℳ_{m}^{g} = {h_{m}^{g} : h_{m}^{g} ​ (\cdot) = \sum_{i = 1}^{m} \frac{c_{i}}{σ_{i}^{n}} ​ g ​ (\frac{\cdot - μ_{i}}{σ_{i}}), μ_{i} \in ℝ^{n}, σ_{i} \in ℝ_{+}, c \in 𝕊^{m - 1}, i \in [m]} ​,

$c^{⊤} = (c_{1}, \dots, c_{m})$ , $ℝ_{+} = (0, \infty)$ ,

𝕊^{m - 1} = {c \in ℝ^{m} : \sum_{i = 1}^{m} c_{i} = 1, c_{i} \geq 0, i \in [m]} ​,

$[m] = {1, \dots, m}$ , and ${(\cdot)}^{⊤}$ is the matrix transposition operator. We say that $h_{m}^{g} \in ℳ^{g}$ is an $m -component$ location-scale finite mixture of the PDF $g$ . The class $ℳ^{g}$ has enjoyed enduring practical and theoretical interest throughout the years, as reported in the volumes of Everitt and Hand, (1981), McLachlan and Basford, (1988), Lindsay, (1995), McLachlan and Peel, (2000), Frühwirth-Schnatter, (2006), Mengersen et al., (2011), Frühwirth-Schnatter et al., (2019), and Nguyen et al., (2021).

We say that $f$ is compactly supported on $𝕂 \subset ℝ^{n}$ , if $𝕂$ is compact and if $𝟏_{𝕂^{∁}} f = 0,$ where $𝟏_{𝕏}$ is the indicator function that takes value 1 when $x \in 𝕏$ , and $0$ elsewhere, and where ${(\cdot)}^{∁}$ is the set complement operator (i.e. $𝕏^{∁} = ℝ^{n} \ 𝕏$ ). Here, $𝕏$ is a generic subset of $ℝ^{n}$ . Further, say that $f \in ℒ_{p} (𝕏)$ for any $1 \leq p < \infty$ , if

{‖ f ‖}_{ℒ_{p} ​ (𝕏)} = {(\int {| 𝟏_{𝕏} ​ f |}^{p} ​ d ​ λ)}^{1 / p} < \infty ​,

and say that $f \in ℒ_{\infty} (𝕏)$ , the class of essentially bounded measurable functions, if

{‖ f ‖}_{ℒ_{\infty} ​ (𝕏)} = inf {a \geq 0 : λ ​ ({x \in 𝕏 : | f ​ (x) | > a}) = 0} < \infty ​,

where we call $∥ \cdot ∥_{ℒ_{p} (𝕏)}$ the $ℒ_{p} -norm$ on $𝕏$ . Denote the class of all bounded functions on $𝕏$ by

ℬ ​ (𝕏) = {f \in ℒ_{\infty} ​ (𝕏) : \exists a \in [0, \infty) ​, such that ​ | f ​ (x) | \leq a, \forall x \in 𝕏}

and write

{‖ f ‖}_{ℬ ​ (𝕏)} = sup_{x \in 𝕏} | f ​ (x) | ​ .

For brevity, we shall write $ℒ_{p} (ℝ^{n}) = ℒ_{p}$ , $ℬ (ℝ^{n}) = ℬ$ , ${‖ f ‖}_{ℒ_{p} (ℝ^{n})} = {‖ f ‖}_{ℒ_{p}}$ , and ${‖ f ‖}_{ℬ (ℝ^{n})} = {‖ f ‖}_{ℬ}$ .

Lastly, we denote the class of continuous functions and uniformly continuous functions by $𝒞$ and $𝒞^{u}$ , respectively. The classes of bounded continuous function shall be denoted by $𝒞_{b} = 𝒞 \cap ℬ$ . Note that the class of continuous functions that vanish at infinity, defined as

𝒞_{0} = {f \in 𝒞 : \forall ϵ > 0, \exists a compact ​ 𝕂 \subset ℝ^{n} ​, such that ​ {‖ f ‖}_{ℬ ​ (𝕂^{∁})} < ϵ} ​,

is a subset of $𝒞_{b}$ .

An important characteristic of the class $ℳ^{g}$ is its capability of approximating larger classes of PDFs in various ways. Motivated by the incomplete proofs of Xu et al., (1993, Lem 3.1) and Theorem 5 from Cheney and Light, (2000, Chapter 20), as well as the results of Nestoridis and Stefanopoulos, (2007), Bacharoglou, (2010), and Nestoridis et al., (2011), Nguyen et al., (2020) established and proved the following theorem regarding sequences of PDFs ${h_{m}^{g}}$ from $ℳ^{g}$ .

Theorem 1 (Theorem 5 from Nguyen et al., (2020)).

Let $h_{m}^{g} \in ℳ^{g}$ denote an $m -component$ location finite mixture PDF. If we assume that $f$ and $g$ are PDFs and that $g \in 𝒞_{0}$ , then the following statements are true.

(a)

For any $f \in 𝒞_{0}$ , there exists a sequence ${h_{m}^{g}}_{m = 1}^{\infty} \subset ℳ^{g}$ , such that

$lim_{m \to \infty} {‖ f - h_{m}^{g} ‖}_{ℒ_{\infty}} = 0 .$
(b)

For any $f \in 𝒞_{b}$ , and compact set $𝕂 \subset ℝ^{n}$ , there exists a sequence ${h_{m}^{g}}_{m = 1}^{\infty} \subset ℳ^{g}$ , such that

$lim_{m \to \infty} {‖ f - h_{m}^{g} ‖}_{ℒ_{\infty} (𝕂)} = 0 .$
(c)

For any $p \in (1, \infty)$ and $f \in ℒ_{p}$ , there exists a sequence ${h_{m}^{g}}_{m = 1}^{\infty} \subset ℳ^{g}$ , such that

$lim_{m \to \infty} {‖ f - h_{m}^{g} ‖}_{ℒ_{p}} = 0 .$
(d)

For any measurable $f$ , there exists a sequence ${h_{m}^{g}}_{m = 1}^{\infty} \subset ℳ^{g}$ , such that

$lim_{m \to \infty} h_{m}^{g} = f , almost everywhere.$
(e)

If $ν$ is a $σ -finite$ Borel measure on $ℝ^{n}$ , then for any $ν -measurable$ $f$ , there exists a sequence ${h_{m}^{g}} ℳ^{g}$ , such that

$lim_{m \to \infty} h_{m}^{g} = f , almost everywhere, with respect to ν .$

Further, if we assume that

g \in {g \in 𝒞_{0} : \forall x \in ℝ^{n} ​, ​ | g ​ (x) | \leq θ_{1} ​ {(1 + {‖ x ‖}_{2})}^{- n - θ_{2}} ​, ​ (θ_{1}, θ_{2}) \in ℝ_{+}^{2}} ​,

then the following is also true.

(f)

For any $f \in 𝒞$ , there exists a sequence ${h_{m}^{g}}_{m = 1}^{\infty} \subset ℳ^{g}$ , such that

$lim_{m \to \infty} {‖ f - h_{m}^{g} ‖}_{ℒ_{1}} = 0 .$

The goal of this work is to seek the weakest set of assumptions in order to establish approximation theoretical results over the widest class of probability density problems, possible. In this paper, we establish Theorem 2 which improves upon Theorem 1 in a number of ways. More specifically, while statements (a), (c), (d), and (e) still hold under the same assumptions as in Theorem 1; statement (b) from Theorem 1 is improved to apply to a larger class of target function $f \in 𝒞$ , see more in statement (a) of Theorem 2; and statement (f) from Theorem 1 is drastically improved to apply to any $f \in ℒ_{1}$ and $g \in ℒ_{\infty}$ , see more in statement (b) of Theorem 2. We note in particular that our improvement with respect to statement (b) from Theorem 1 yields exactly the result of Theorem 5 from Cheney and Light, (2000, Chapter 20), which was incorrectly proved (see also DasGupta, (2008, Theorem 33.2)).

The remainder of the article progresses as follows. The main result of this paper is stated in Section 2. Technical preliminaries to the proof of the main result are presented in Section 3. The proof is then established in Section 4. Additional technical results required throughout the paper are reported in the Appendix A.

2 Main result

Theorem 2.

Let $h_{m}^{g} \in ℳ^{g}$ denote an $m -component$ location finite mixture PDF. If we assume that $f$ and $g$ are PDFs, then the following statements are true.

(a)

If $f, g \in 𝒞$ and $𝕂 \subset ℝ^{n}$ is a compact set, then there exists a sequence ${h_{m}^{g}}_{m = 1}^{\infty} \subset ℳ^{g}$ , such that

$lim_{m \to \infty} {‖ f - h_{m}^{g} ‖}_{ℬ (𝕂)} = 0 .$
(b)

For $p \in [1, \infty)$ , if $f \in ℒ_{p}$ and $g \in ℒ_{\infty}$ , then there exists a sequence ${h_{m}^{g}}_{m = 1}^{\infty} \subset ℳ^{g}$ , such that

$lim_{m \to \infty} {‖ f - h_{m}^{g} ‖}_{ℒ_{p}} = 0 .$

3 Technical preliminaries

Let $f, g \in ℒ_{1}$ , and denote the convolution of $f$ and $g$ by $f ⋆ g = g ⋆ f$ . Further, we say that $g_{k} (\cdot) = k^{n} g (k \times \cdot)$ ( $k \in ℝ_{+}$ ) is a dilate of $g$ .

Notice that $ℳ_{m}^{g}$ can be parameterized via dilates. That is, we can write

ℳ_{m}^{g} = {h_{m}^{g} : h_{m}^{g} (\cdot) = \sum_{i = 1}^{m} c_{i} k_{i}^{n} g (k_{i} \times \cdot - k_{i} μ_{i}), μ_{i} \in ℝ^{n}, k_{i} \in ℝ_{+}, c \in 𝕊^{m - 1}, i \in [m]},

where $k_{i} = 1 / σ_{i}$ .

Let $𝔽$ be a subset of $𝔼$ , and denote the convex hull of $𝔽$ by $conv (𝔽)$ is the smallest convex subset in $𝔼$ that contains $𝔽$ (cf. Brezis, 2010, Chapter 1). By definition, we may write

conv ​ (𝔽) = {\sum_{i \in [m]} α_{i} ​ f_{i} : f_{i} \in 𝔽, α \in 𝕊^{m - 1}, i \in [m], m \in ℕ} ​,

where $α^{⊤} = (α_{1}, \dots, α_{m})$ .

Define the class of “basic” densities, which will serve as the approximation building blocks, as follows

𝒢^{g} = {k^{n} g (k \times \cdot - k μ), μ \in ℝ^{n}, k \in ℝ_{+}},

and suppose that we can choose a suitable NVS $(𝔼, ∥ \cdot ∥_{𝔼})$ , such that $𝒢^{g} \subset ℳ^{g} \subset 𝔼$ . Then, by definition, it holds that $ℳ^{g}$ is a convex hull of $𝒢^{g}$ .

For $u \in 𝔼$ and $r > 0$ , we define the open and closed balls of radius $r$ , centered around $u$ , by:

𝔹 ​ (u, r) = {v \in 𝔼 : {‖ u - v ‖}_{𝔼} < r} ​,

and

\bar{𝔹} ​ (u, r) = {v \in 𝔼 : {‖ u - v ‖}_{𝔼} \leq r} ​,

respectively. For brevity, we also write $𝔹_{r} = 𝔹 (0, r)$ and ${\bar{𝔹}}_{r} = \bar{𝔹} (0, r)$ . A set $𝔽 \subset 𝔼$ is open, if for every $u \in 𝔽$ , there exists an $r > 0$ , such that $𝔹 (u, r) \subset 𝔽$ . We say that $𝔽$ is closed if its complement is open, and by definition, we say that $𝔼$ and the empty set are both closed and open.

We call the smallest closed set containing $𝔽$ its closure, and we denote it by $\bar{𝔽}$ . A sequence ${u_{m}} \subset 𝔼$ converges to $u \in 𝔼$ , if ${lim}_{m \to \infty} {‖ u_{m} - u ‖}_{𝔼} = 0$ , and we denote it symbolically by ${lim}_{m \to \infty} u_{m} = u$ . That is, for every $ϵ > 0$ , there exists an $N (ϵ) \in ℕ$ , such that $m \geq N (ϵ)$ implies that ${‖ u_{m} - u ‖}_{𝔼} < ϵ$ .

By Lemma 6, we can write the closure of $𝔽$ as

\bar{𝔽} = {u \in 𝔼 : u = lim_{m \to \infty} u_{m}, u_{m} \in 𝔽}

and hence

\bar{ℳ^{g}} = {h \in 𝔼 : h = lim_{m \to \infty} h_{m}^{g}, h_{m}^{g} \in ℳ^{g}} ​ .

Thus, by definition, it holds that $\bar{ℳ^{g}}$ is a closed and convex subset of $𝔼$ .

If $f \in 𝒞$ is a PDF on $ℝ^{n}$ , we denote its support by

supp ​ f = {x \in ℝ^{n} : f ​ (x) \neq 0}

and furthermore, we denote the set of compactly supported continuous functions by

𝒞_{c} = {f \in 𝒞 : supp ​ f ​ is compact} ​ .

For open sets $𝕍 \subset ℝ^{n}$ , we will write $f ≺ 𝕍$ as shorthand for $f \in 𝒞_{c}$ , $0 \leq f \leq 1$ , and $supp f \subset 𝕍$ .

The following lemmas permit us to prove the primary technical mechanism that is used to prove our main result presented in Theorem 2.

Lemma 1.

Let $f \in 𝒞$ be a PDF. Then, for every compact $𝕂 \subset ℝ^{n}$ , we can choose $h \in 𝒞_{c}$ , such that $supp h \subset 𝔹_{r}$ , $0 \leq h \leq f$ , and $h = f$ on $𝕂$ , for some $r \in ℝ_{+}$ .

Proof.

Since $𝕂$ is bounded, there exists some $r \in ℝ_{+}$ , such that $𝕂 \subset 𝔹_{r}$ . Lemma 10 implies that there exists a function $u ≺ 𝔹_{r}$ , such that $u (x) = 1$ , for all $x \in 𝕂$ . We can then set $h = u f$ to obtain the desired result of Lemma 1. ∎

Lemma 2.

Let $h \in 𝒞_{c}$ , such that $supp h \subset 𝔹_{r}$ , $0 \leq h$ , and $\int h d λ \leq 1$ , and let $g \in 𝒞$ be a PDF. Then, for any $k \in ℝ_{+}$ , there exists a sequence ${h_{m}^{g}}_{m = 1}^{\infty} \subset ℳ^{g}$ , so that

lim_{m \to \infty} {‖ g_{k} ⋆ h - h_{m}^{g} ‖}_{ℬ ​ ({\bar{𝔹}}_{r})} = 0 ​ .

(1)

Furthermore, if $g \in 𝒞_{b}^{u}$ , we have the stronger result that

lim_{m \to \infty} {‖ g_{k} ⋆ h - h_{m}^{g} ‖}_{ℬ} = 0 ​ .

(2)

Proof.

It suffices to show that given any $r, k, ϵ \in ℝ_{+}$ , there exists a sufficiently large $m (ϵ, r, k) \in ℕ$ such that for all $m \geq m (ϵ, r, k)$ , there exists a $h_{m}^{g} \in ℳ_{m}^{g}$ satisfying

{‖ g_{k} ⋆ h - h_{m}^{g} ‖}_{ℬ ​ ({\bar{𝔹}}_{r})} < ϵ ​ .

(3)

First, write

	$(g_{k} ⋆ h) (x)$	$=$	$\int g_{k} (x - y) h (y) d λ (y) = \int 𝟏_{{y : y \in {\bar{𝔹}}_{r}}} g_{k} (x - y) h (y) d λ (y)$
		$=$	$\int 𝟏_{{y : y \in {\bar{𝔹}}_{r}}} k^{n} g (k x - k y) h (y) d λ (y) = \int 𝟏_{{z : z \in {\bar{𝔹}}_{r k}}} g (k x - z) h (\frac{z}{y}) d λ (z) ,$

where ${\bar{𝔹}}_{r k}$ is a continuous image of a compact set, and hence is also compact (cf. Rudin, 1976, Theorem 4.14). By Lemma 11, for any $δ > 0$ , there exist $κ_{i} \in ℝ^{n}$ ( $i \in [m - 1]$ , for some $m \in ℕ$ ), such that ${\bar{𝔹}}_{r k} \subset ⋃_{i = 1}^{m - 1} 𝔹 (κ_{i}, δ / 2)$ . Further, if $𝔹_{i}^{δ} = 𝔹_{r k}^{δ} = {\bar{𝔹}}_{r k} \cap 𝔹 (κ_{i}, δ / 2)$ , then ${\bar{𝔹}}_{r k} = ⋃_{i = 1}^{m - 1} 𝔹_{i}^{δ}$ . We can hence obtain a disjoint covering of ${\bar{𝔹}}_{r k}$ by taking $𝔸_{1}^{δ} = 𝔹_{1}^{δ}$ , and $𝔸_{i}^{δ} = 𝔹_{i}^{δ} \ ⋃_{j = 1}^{i - 1} 𝔹_{j}^{δ}$ ( $i \in [m - 1]$ ) (cf. Cheney and Light, 2000, Chapter 24). Notice that ${\bar{𝔹}}_{r k} = ⋃_{i = 1}^{m - 1} 𝔸_{i}^{δ}$ , each $𝔸_{i}^{δ}$ is a Borel set, and $diam (𝔸_{i}^{δ}) \leq δ$ , by construction.

We shall denote the disjoint cover of ${\bar{𝔹}}_{r k}$ by $Π_{m}^{δ} = {𝔸_{i}^{δ}}_{i = 1}^{m - 1}$ . We seek to show that there exists an $m \in ℕ$ and $Π_{m}^{δ}$ , such that

{‖ g_{k} ⋆ h - \sum_{i = 1}^{m} c_{i} ​ k_{i}^{n} ​ g ​ (k_{i} ​ x - z_{i}) ‖}_{ℬ ​ ({\bar{𝔹}}_{r})} < ϵ ​,

where $k_{i} = k$ , $c_{i} = k^{- n} \int 𝟏_{{z : z \in 𝔸_{i}^{δ}}} h (z / k) d λ (z)$ , and $z_{i} \in 𝔸_{i}^{δ}$ , for $i \in [m - 1]$ . We then set $z_{m} = 0$ and $c_{m} = 1 - \sum_{i = 1}^{m - 1} c_{i}$ . Here, $c_{m}$ depends only on $r$ and $ϵ$ . Suppose that $c_{m} > 0$ . Then, since $g \neq 0$ , there exists some $s \in ℝ_{+}$ such that $C_{s} = {sup}_{w \in {\bar{𝔹}}_{s}} g (w) > 0$ . We can choose

k_{m} = \min {\frac{s}{r}, {(\frac{ϵ}{2 ​ c_{m} ​ C_{s}})}^{1 / n}} ​,

so that $∥ g (k_{m} \times \cdot) ∥_{ℬ ({\bar{𝔹}}_{r})} \leq C_{s}$ and

∥ g (k_{m} \times \cdot) ∥_{ℬ ​ ({\bar{𝔹}}_{r})} \leq \frac{c_{m} ​ ϵ ​ C_{s}}{2 ​ c_{m} ​ C_{s}} = ϵ / 2 .

Moreover, if we assume that $g \in 𝒞_{b}^{u}$ , then there exists a constant $C \in (0, \infty)$ such that ${‖ g ‖}_{ℬ} \leq C$ . In this case, we can choose $k_{m}^{n} = ϵ / (2 c_{m} C)$ to obtain

∥ c_{m} k_{m}^{n} g (k_{m} \times \cdot - z_{m}) ∥_{ℬ} \leq ϵ / 2 .

Since $0 \leq h$ and $\int h d λ \in [0, 1]$ , the sum $\sum_{i = 1}^{m - 1} c_{i}$ satisfies the inequalities:

	$0 \leq \sum_{i = 1}^{m - 1} c_{i}$	$=$	$k^{- n} \sum_{i = 1}^{m - 1} \int 𝟏_{{z : z \in 𝔸_{i}^{δ}}} h (\frac{z}{k}) d λ (z)$
		$=$	$k^{- n} \int 𝟏_{{z : z \in k 𝕂}} h (\frac{z}{k}) d λ (z) = \int 𝟏_{{x : x \in 𝕂}} h d λ \leq 1 .$

Thus, $c_{m} \in [0, 1]$ , and our construction of $h_{m}^{g}$ implies that $h_{m}^{g} = \sum_{i = 1}^{m} c_{i} k_{i}^{n} g (k_{i} x - z_{i}) \in ℳ_{m}^{g}$ .

We can then bound the left-hand side of (3) as follows:

	${‖ g_{k} ⋆ h - h_{m}^{g} ‖}_{ℬ ({\bar{𝔹}}_{r})}$
$\leq$	$∥ g_{k} ⋆ h - \sum_{i = 1}^{m - 1} c_{i} k_{i}^{n} g (k_{i} \times \cdot - z_{i}) ∥_{ℬ ({\bar{𝔹}}_{r})} + ∥ c_{m} k_{m}^{n} g (k_{m} \times \cdot - z_{m}) ∥_{ℬ ({\bar{𝔹}}_{r})}$
$\leq$	$∥ g_{k} ⋆ h - \sum_{i = 1}^{m - 1} c_{i} k_{i}^{n} g (k_{i} \times \cdot - z_{i}) ∥_{ℬ ({\bar{𝔹}}_{r})} + \frac{ϵ}{2}$
$=$	${‖ \int 1_{{z : z \in {\bar{𝔹}}_{r k}}} g (k x - z) h (\frac{z}{k}) d λ (z) - \sum_{i = 1}^{m - 1} \int 1_{{z : z \in 𝔸_{i}^{δ}}} g (k x - z) h (\frac{z}{k}) d λ (z) ‖}_{ℬ ({\bar{𝔹}}_{r})}$
	$+ \frac{ϵ}{2}$
$\leq$	$\sum_{i = 1}^{m - 1} \int 1_{{z : z \in 𝔸_{i}^{δ}}} \| g (k x - z) - g (k x - z_{i}) \| h (\frac{z}{k}) d λ (z) + \frac{ϵ}{2} .$	(4)

Since $x \in {\bar{𝔹}}_{r}$ , $z \in 𝔸_{i}^{δ}$ , and $z_{i} \in {\bar{𝔹}}_{r k}$ , it holds that ${‖ k x - z_{i} ‖}_{2} = {‖ k x - z ‖}_{2} \leq 2 r k ,$ and

{‖ k ​ x - z - (k ​ x - z_{i}) ‖}_{2} = {‖ z - z_{i} ‖}_{2} \leq diam ​ (𝔸_{i}^{δ}) \leq δ ​ .

Note that $g \in 𝒞$ , and thus $g$ is uniformly continuous on the compact set ${\bar{𝔹}}_{2 r k}$ , implying that

| g ​ (k ​ x - z) - g ​ (k ​ x - z_{i}) | \leq w ​ (g, 2 ​ r ​ k, δ) ​,

for each $i \in [m - 1]$ , where

w (g, r, δ) = sup {| g (x) - g (y) | : ∥ x - y ∥_{2} \leq δ and x, y \in {\bar{𝔹}}_{r}}

denotes a modulus of continuity. Since ${lim}_{δ \to 0} w (g, 2 r k, δ) = 0$ (cf. Makarov and Podkorytov, 2013, Theorem 4.7.3), we may choose a $δ (ϵ, r, k) > 0$ , such that

w ​ (g, 2 ​ r ​ k, δ ​ (ϵ, r, k)) < \frac{ϵ}{2 ​ k^{n}} ​ .

We then proceed from (4) as follows:

${‖ g_{k} ⋆ h - h_{m}^{g} ‖}_{ℬ ({\bar{𝔹}}_{r})}$	$\leq w (g, 2 r k, δ (ϵ, r, k)) \int 𝟏_{{z : z \in {\bar{𝔹}}_{r k}}} h (\frac{z}{k}) d λ (z) + \frac{ϵ}{2}$
	$= w (g, 2 r k, δ (ϵ, r, k)) k^{n} \int h d λ + \frac{ϵ}{2}$
	$\leq w (g, 2 r k, δ (ϵ, r, k)) k^{n} + \frac{ϵ}{2} < \frac{ϵ}{2} + \frac{ϵ}{2} = ϵ .$	(5)

To conclude the proof of (1), it suffices to choose an appropriate sequence of partitions $Π_{m}^{δ (ϵ, r, k)}$ , such that $m \geq m (ϵ, r, k)$ , for some sufficiently large $m (ϵ, r, k)$ , so that (4) and (5) hold. This is possible via Lemma 11. When $g \in 𝒞_{b}^{u}$ , we notice that (4) and (5) both hold for all $x \in ℝ^{n}$ . Thus, we have the stronger result of (2). ∎

We present the primary tools for proving Theorem (2) in the following pair of lemma. The first one in Lemma 3 permits the approximation of convolutions of the form $g_{k} ⋆ f$ in the $ℒ_{1}$ functional space, and the second presented in Lemma 4 generalizes this first result to the spaces $ℒ_{p}$ , where $p \in [1, \infty)$ , under an essentially bounded assumption.

Lemma 3.

If $f$ and $g$ are PDFs in the NVS $(ℒ_{1}, ∥ \cdot ∥_{ℒ_{1}})$ , then $ℳ^{g} \subset ℒ_{1}$ and $g_{k} ⋆ f \in ℒ_{1}$ , for every $k \in ℝ_{+}$ . Furthermore, there exists a sequence ${h_{m}^{g}}_{m = 1}^{\infty} \subset ℳ^{g}$ , such that

lim_{m \to \infty} {‖ g_{k} ⋆ f - h_{m}^{g} ‖}_{ℒ_{1}} = 0 ​ .

Proof.

For any $k \in ℝ_{+}$ , we can show that $g_{k} \in ℒ_{1}$ , since

{‖ g_{k} ‖}_{ℒ_{1}} = \int g_{k} ​ d ​ λ = \int k^{n} ​ g ​ (k ​ x) ​ d ​ λ ​ (x) = \int g ​ d ​ λ = 1 ​ .

If $h_{m}^{g} \in ℳ_{m}^{g}$ , then $h_{m}^{g} \in ℒ_{1}$ , since it is a finite sum of functions in $ℒ_{1}$ , and thus, $ℳ^{g} \subset ℒ_{1}$ . Note that since $f$ is a PDF, we have $f \in ℒ_{1}$ , and by Lemma 13, we also have that $g_{k} ⋆ f \in ℒ_{1}$ . By Lemma 14, it then follows that

${‖ g_{k} ⋆ f ‖}_{ℒ_{1}}$	$=$	$\int g_{k} ⋆ f d λ$
	$=$	$\int [\int g_{k} (x - y) f (y) d λ (y)] d λ (x)$
	$=$	$\int [\int g_{k} (x - y) d λ (x)] f (y) d λ (y)$
	$=$	${‖ g_{k} ‖}_{ℒ_{1}} {‖ f ‖}_{ℒ_{1}} = 1 $

By definition of of the closure of $ℳ^{g}$ in $ℒ_{1}$ , it suffices to show that for any $k \in ℝ_{+}$ , $g_{k} ⋆ f \in \bar{ℳ^{g}}$ . We seek a contradiction by assuming that $g_{k} ⋆ f \notin \bar{ℳ^{g}}$ . Then, we can choose $𝔸 = \bar{ℳ^{g}}$ and $𝔹 = {g_{k} ⋆ f}$ so that $𝔸, 𝔹 \subset ℒ_{1}$ are nonempty convex subsets, such that $𝔸 \cap 𝔹 = \emptyset$ . Furthermore, $𝔸$ is closed and $𝔹$ is compact. By Lemma 7, there exists a continuous linear functional $ϕ \in ℒ_{1}^{*}$ , such that $ϕ (v) < α < ϕ (w)$ , for all $v \in 𝔸$ and $w \in 𝔹$ . By definition of $𝔹$ , for all $v \in \bar{ℳ^{g}} \subset ℒ_{1}$ we have

ϕ ​ (v) < α < ϕ ​ (g_{k} ⋆ f) ​ .

By Lemma 9, with $ϕ \in ℒ_{1}^{*}$ , there exists a unique function $u \in ℒ_{\infty}$ , such that, for all $v \in ℒ_{1}$ ,

ϕ ​ (v) = \int u ​ (x) ​ v ​ (x) ​ d ​ λ ​ (x) ​ .

If we let $v = g_{k} (\cdot - μ) \in \bar{ℳ^{g}} \subset ℒ_{1}$ , then we obtain the inequalities

sup_{μ \in ℝ^{n}} \int u ​ (x) ​ g_{k} ​ (x - μ) ​ d ​ λ ​ (x) < α < \int u ​ (x) ​ (g_{k} ⋆ f) ​ (x) ​ d ​ λ ​ (x) ​ .

The left-hand inequality can be reduced as follows:

	$α$	$< \int u (x) (g_{k} ⋆ f) (x) d λ (x)$
		$= \int u (x) [\int g_{k} (x - μ) f (μ) d λ (μ)] d λ (x)$
		$= \int f (μ) [\int u (x) g_{k} (x - μ) d λ (x)] d λ (μ)$
		$< α \int f (μ) d λ (μ) = α ,$

where the third line is due to Lemma 14 and the final equality is because $f$ is a PDF. This yields the sought contradiction. ∎

Lemma 4.

If $f, g \in ℒ_{\infty}$ are PDFs in the NVS $(ℒ_{\infty}, ∥ \cdot ∥_{ℒ_{p}})$ , for $p \in [1, \infty)$ , then, $ℳ^{g} \subset ℒ_{p}$ and $g_{k} ⋆ f \in ℒ_{p}$ , for any $k \in ℝ_{+}$ . Furthermore, there exists a sequence ${h_{m}^{g}}_{m = 1}^{\infty} \subset ℳ^{g}$ , such that

lim_{m \to \infty} {‖ g_{k} ⋆ f - h_{m}^{g} ‖}_{ℒ_{p}} = 0 ​ .

Proof.

We obtain the result for $p = 1$ via Lemma 3. Otherwise, since $g \in ℒ_{1} \cap ℒ_{\infty}$ , we know that $g \in ℒ_{p}$ and $g_{k} \in ℒ_{p}$ , for each $k \in ℝ_{+}$ , via Lemma 12. For any $h_{m}^{g} \in ℳ_{m}^{g}$ , we then have $h_{m}^{g} \in ℒ_{p}$ via finite summation, and hence $ℳ^{g} \in ℒ_{p}$ . Since $f \in ℒ_{1}$ , Lemma 13 implies that $g_{k} ⋆ f \in ℒ_{p}$ . By definition of the closure of $ℳ^{g},$ it suffices to show that $g_{k} ⋆ f \in \bar{ℳ^{g}}$ , for any $k \in ℝ_{+}$ . This can be achieved by seeking a contradiction under the assumption that $g_{k} ⋆ f \notin \bar{ℳ^{g}}$ and using Lemma 8 in the same manner as Lemma 9 is used in the proof of Lemma 3. ∎

4 Proof of main result

4.1 Proof of Theorem 2 (a)

To prove the statement (a) of Theorem 2, it suffices to show that there exists a sufficiently large $m (ϵ, 𝕂) \in ℕ$ , such that for all $m \geq m (ϵ, 𝕂)$ , there exists a $h_{m}^{g} \in ℳ_{m}^{g}$ , such that ${‖ f - h_{m}^{g} ‖}_{ℬ (𝕂)} < ϵ$ , for any $ϵ > 0$ and compact set $𝕂 \subset ℝ^{n}$ .

First, Lemma 1 implies that we can choose a $h \in 𝒞_{c}$ , such that $supp h \subset {\bar{𝔹}}_{r}$ , $0 \leq h \leq f$ , and $h = f$ on $𝕂$ , for some $r > 0$ , where $𝕂 \subset {\bar{𝔹}}_{r}$ . We then have ${‖ f - h ‖}_{ℬ (𝕂)} = 0$ .

Since $h \in 𝒞_{c} \subset 𝒞_{b}^{u}$ , Lemma 5 and Corollary 1 then imply that there exists a $k (ϵ) \in ℝ_{+}$ , such that for all $k \geq k (ϵ)$ , ${‖ h - g_{k} ⋆ h ‖}_{ℬ (𝕂)} < ϵ / 2$ . We shall assume that $k \geq k (ϵ)$ , from hereon in.

Lemma 2 then implies that there exists an $m (ϵ, r, k) \in ℕ$ , such that for any $m \geq m (ϵ, r, k)$ , there exists a $h_{m}^{g} \in ℳ_{m}^{g}$ , such that ${‖ g_{k} ⋆ h - h_{m}^{g} ‖}_{ℬ (𝕂)} < {‖ g_{k} ⋆ h - h_{m}^{g} ‖}_{ℬ ({\bar{𝔹}}_{r})} < ϵ / 2$ . The triangle inequality then completes the proof.

4.2 Proof of Theorem 2 (b)

To prove the statement (a) of Theorem 2, it suffices to show that there exists a sufficiently large $m (ϵ) \in ℕ$ , such that for all $m \geq m (ϵ)$ , there exists a $h_{m}^{g} \in ℳ_{m}^{g}$ , such that ${‖ f - h_{m}^{g} ‖}_{ℒ_{p}} < ϵ$ , for any $ϵ > 0$ .

First, Lemma 5 and Corollary 1 imply that there exists a $k (ϵ) \in ℝ_{+}$ , such that for any $k \geq k (ϵ)$ , it follows that ${‖ f - g_{k} ⋆ f ‖}_{ℒ_{p}} < ϵ / 2$ . We shall assume $k \geq k (ϵ)$ , from hereon in.

Lemmas 3 and 4 imply that there exists an $m (ϵ) \in ℕ$ , such that for all $m \geq m (ϵ)$ , there exists a $h_{m}^{g} \in ℳ_{m}^{g}$ , such that ${‖ g_{k} ⋆ f - h_{m}^{g} ‖}_{ℒ_{p}} < ϵ / 2$ . The triangle inequality then completes the proof.

Appendix A Technical results

We state a number of technical results that are used throughout the main text, in this Appendix. Sources for unproved results are provided at the end of the section.

Lemma 5.

Let ${g_{k}}$ be a sequence of PDFs in $ℒ_{1}$ , such that for every $δ > 0$ ,

lim_{k \to \infty} \int 𝟏_{{x : {‖ x ‖}_{2} > δ}} ​ g_{k} ​ d ​ λ = 0 ​ .

Then, for $f \in ℒ_{p}$ and $p \in [1, \infty)$ ,

lim_{k \to \infty} {‖ g_{k} ⋆ f - f ‖}_{ℒ_{p}} = 0 ​ .

Furthermore, for $f \in 𝒞_{b}$ and compact $𝕂 \subset ℝ^{n}$ ,

lim_{k \to \infty} {‖ g_{k} ⋆ f - f ‖}_{ℒ_{\infty} ​ (𝕂)} = 0 ​ .

The sequences ${g_{k}}$ of Lemma 5 are often referred to as approximate identities or approximations of identity (cf. Makarov and Podkorytov, 2013, Sec. 7.6). A typical construction of approximate identities is to consider the sequence of dilations, of the form: $g_{k} (\cdot) = k^{n} g (k \times \cdot)$ , which permits the following corollary.

Corollary 1.

Let $g$ be a PDF. Then, the sequence ${g_{k} : g_{k} (\cdot) = k^{n} g (k \times \cdot)}$ satisfies the hypothesis of Lemma 5 and hence permits its conclusion.

Lemma 6.

Let $(𝔼, ∥ \cdot ∥_{𝔼})$ be an NVS, and let $𝔽 \subset 𝔼$ and $u \in 𝔼$ . Then the following statements are equivalent: (a) $u \in \bar{𝔽}$ ; (b) $𝔹 (u, r) \cap 𝔽 \neq \emptyset$ , for all $r > 0$ ; and (c) there exists a sequence ${u_{m}} \subset 𝔽$ that converges to $u$ .

Let $𝔼$ be a locally convex linear topological space over $ℝ$ and recall that a functional is a function defined on $𝔼$ (or some subspace of $𝔼$ ), with values in $ℝ$ . We denote the due space of $𝔼$ (the space of all continuous linear functions on $𝔼$ ) by $𝔼^{*}$ .

Lemma 7 (Second geometric form of the Hahn-Banach theorem).

Let $𝔸, 𝔹 \subset 𝔼$ be two nonempty convex subsets, such that $𝔸 \cap 𝔹 \neq \emptyset$ . Assume that $𝔸$ is closed and that $𝔹$ is compact. Then, there exists a continuous linear functional $ϕ \in 𝔼^{*}$ , such that its corresponding hyperplane $H = {u \in 𝔼 : ϕ (u) = α}$ ( $α \in ℝ$ ) strictly separates $𝔸$ and $𝔹$ . That is, there exists some $ϵ > 0$ , such that $ϕ (u) \leq α - ϵ$ and $ϕ (v) \geq α + ϵ$ , for all $u \in 𝔸$ and $v \in 𝔹$ . Or, in other words, ${sup}_{u \in 𝔸} ϕ (u) < {inf}_{v \in 𝔹} ϕ (v)$ .

Lemma 8 (Riesz representation theorem for $ℒ_{p}$ , $p \in ℝ_{+}$ ).

If $p \in ℝ_{+}$ , and $ϕ \in {(ℒ_{p})}^{*}$ , then, there exists a unique function $u \in ℒ_{q}$ , such that for all $v \in ℒ_{q}$ ,

ϕ ​ (v) = \int u ​ (x) ​ v ​ (x) ​ d ​ λ ​ (x) ​,

where $1 / p + 1 / q = 1$ .

Lemma 9 (Riesz representation theorem for $ℒ_{1}$ ).

If $ϕ \in {(ℒ_{1})}^{*}$ , then there exists a unique $u \in ℒ_{\infty}$ , such that for all $v \in ℒ_{1}$ ,

ϕ ​ (v) = \int u ​ (x) ​ v ​ (x) ​ d ​ λ ​ (x) ​ .

Lemma 10.

Let $𝕍_{1}, \dots, 𝕍_{n}$ be open subsets of $ℝ^{n}$ , and let $𝕂$ be a compact set, such that $𝕂 \subset ⋃_{i = 1}^{n} 𝕍_{i}$ . Then, there exists functions $h_{i} ≺ 𝕍_{i}$ ( $i \in [n]$ ), such that $\sum_{i = 1}^{n} h_{i} (x) = 1$ , for all $x \in 𝕂$ . The set ${h_{i}}$ is referred to as the partition of unity on $𝕂$ , subordinated to the cover ${𝕍_{i}}$ .

Lemma 11.

If $𝕏 \subset ℝ^{n}$ is bounded, then for any $r > 0$ , $𝕏$ can be covered by $⋃_{i = 1}^{m} 𝔹 (x_{i}, r)$ , for some finite $m \in ℕ$ , where $x_{i} \in ℝ^{n}$ and $i \in [m]$ .

Lemma 12.

If $1 \leq p \leq q \leq r \leq \infty$ , then $ℒ_{p} \cap ℒ_{r} \subset ℒ_{q}$ .

Lemma 13.

If $f \in ℒ_{p}$ ( $1 \leq p \leq \infty$ ) and $g \in ℒ_{1}$ , then $f ⋆ g$ exists and we have ${‖ f ⋆ g ‖}_{ℒ_{p}} \leq {‖ f ‖}_{ℒ_{p}} {‖ f ‖}_{ℒ_{1}}$ . Furthermore, if $p$ and $q$ are such that $1 / p + 1 / q = 1$ , then $f \in ℒ_{p}$ and $g \in ℒ_{q}$ , then $f ⋆ g$ exists, is bounded and uniformly continuous, and ${‖ f ⋆ g ‖}_{ℒ_{\infty}} \leq {‖ f ‖}_{ℒ_{p}} {‖ f ‖}_{ℒ_{q}}$ . In particular, if $p \in ℝ_{+}$ , then $f ⋆ g \in 𝒞_{0}$ .

Lemma 14 (Fubini’s Theorem).

Let $(𝕏, 𝒳, ν_{1})$ and $(𝕐, 𝒴, ν_{2})$ be $σ -finite$ measure spaces, and assume that $f$ is a $(𝒳 \times 𝒴) -measurable$ function on $𝕏 \times 𝕐$ . If

\int_{𝕏} [\int_{𝕐} | f ​ (x, y) | ​ d ​ ν_{1} ​ (x)] ​ d ​ ν_{2} ​ (y) < \infty ​,

then

\int_{𝕏 \times 𝕐} | f | ​ d ​ (ν_{1} \times ν_{2})

= \int_{𝕏} [\int_{𝕐} | f ​ (x, y) | ​ d ​ ν_{1} ​ (x)] ​ d ​ ν_{2} ​ (y) = \int_{𝕐} [\int_{𝕏} | f ​ (x, y) | ​ d ​ ν_{2} ​ (y)] ​ d ​ ν_{1} ​ (x) < \infty ​ .

Sources for results

Lemma 5 appears in Makarov and Podkorytov, (2013, Thm. 9.3.3) and Cheney and Light, (2000, Ch. 20, Thm. 2). Corollary 1 is obtained from Cheney and Light, (2000, Ch. 20, Thm. 4). Lemmas 6, 12, and 13 are taken from Propositions 0.22, 6.10, and 8.8 Folland, (1999). Lemmas 7–9 appear in Brezis, (2010) as Theorems 1.7, 4.11, and 4.14, respectively. Lemmas 10 and 14 can be found in Rudin, (1987) as Theorems 2.13 and Theorem 8.8, respectively. Lemma 11 is obtained from Conway, (2012, Thm. 1.2.2).

Appendix B Acknowledgements

The authors would like to very much thank Pr. Eric Ricard for the interesting discussions with him and for his suggestions. TTN is supported by “Contrat doctoral” from the French Ministry of Higher Education and Research and by the French National Research Agency (ANR) grant SMILES ANR-18-CE40-0014. HDN and GJM are funded by Australian Research Council grant number DP180101192.

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Approximation of probability density functions via location-scale finite mixtures in Lebesgue spaces

Abstract

1 Introduction

Theorem 1 (Theorem 5 from Nguyen et al., (2020)).

2 Main result

Theorem 2.

3 Technical preliminaries

Lemma 1.

Proof.

Lemma 2.

Proof.

Lemma 3.

Proof.

Lemma 4.

Proof.

4 Proof of main result

4.1 Proof of Theorem 2 (a)

4.2 Proof of Theorem 2 (b)

Appendix A Technical results

Lemma 5.

Corollary 1.

Lemma 6.

Lemma 7 (Second geometric form of the Hahn-Banach theorem).

Lemma 8 (Riesz representation theorem for ℒp, p∈ℝ+).

Lemma 9 (Riesz representation theorem for ℒ1).

Lemma 10.

Lemma 11.

Lemma 12.

Lemma 13.

Lemma 14 (Fubini’s Theorem).

Sources for results

Appendix B Acknowledgements

References

Cite this paper

Related work

Lemma 8 (Riesz representation theorem for $ℒ_{p}$ , $p \in ℝ_{+}$ ).

Lemma 9 (Riesz representation theorem for $ℒ_{1}$ ).