On uniformly consistent tests

TL;DR

This paper investigates the conditions under which uniformly consistent hypothesis tests exist across various statistical models, emphasizing the role of convexity and compactness in the sets of alternatives.

Contribution

It provides necessary and sufficient conditions for uniform consistency of tests in nonparametric and Gaussian models, linking convexity and compactness to test feasibility.

Findings

Uniformly consistent tests exist if and only if the convex set of alternatives is relatively compact.

Results apply to density hypothesis testing, signal detection in Gaussian noise, and linear ill-posed problems.

Small balls around hypotheses shrink as sample size increases, affecting test consistency.

Abstract

Necessary and sufficient conditions of uniform consistency are explored. A hypothesis is simple. Nonparametric sets of alternatives are bounded convex sets in $\mathbb{L}_p$, $p >1$ with "small" balls deleted. The "small" balls have the center at the point of hypothesis and radii of balls tend to zero as sample size increases. For problem of hypothesis testing on a density, we show that, for the sets of alternatives, there are uniformly consistent tests for some sequence of radii of the balls, if and only if, convex set is relatively compact. The results are established for problem of hypothesis testing on a density, for signal detection in Gaussian white noise, for linear ill-posed problems with random Gaussian noise and so on.

Equations

$$\Vert g{\Vert }_p^p={\int }_S|g{|}^{p}d{\mathbf{P}}_0<\infty .$$

$${\mathbb{H}}_0:f\doteq \frac{d\mathbb{P}}{d{\mathbb{P}}_0}-1=f_0\doteq 0$$

$${\mathbb{H}}_n:f\in V({\rho }_n)=U\setminus B_{{\rho }_n}(f_0)$$

$$\beta (K_n,V({\rho }_n))\doteq \underset{f\in V({\rho }_n)}{\sup }\beta (K_n,f)<1-\alpha -\delta$$

$${\mathfrak{E}}^{loc}(\theta ,\epsilon )=\log M(\epsilon /c(\theta ,\epsilon )\cap K)$$

$${\mathfrak{E}}^{loc}(\theta ,c_1\epsilon )\le {\mathfrak{E}}^{loc}(\theta ,\epsilon )\le {\mathfrak{E}}^{loc}(\theta ,c_2\epsilon ).$$

$${\mathbb{H}}_0:f\in W_0\subset {\mathbb{L}}_p$$

$${\mathbb{H}}_1:f\in W_1\subset {\mathbb{L}}_p$$

$$\lim \underset{n\to \infty }{\sup }(\underset{f_0\in W_0}{\sup }\alpha (K_n,f_0)+\underset{f\in W_1}{\sup }\beta (K_n,f))<1$$

$$d{Y}_{\epsilon }(t)=S(t)dt+\epsilon dw(t),\epsilon >0,$$

$${\mathbb{H}}_0:S(t)=S_0(t)\equiv 0,t\in [0,1],$$

$${\mathbb{H}}_{\epsilon }:S\in V({\rho }_{\epsilon })=U\setminus B({\rho }_{\epsilon }){\rho }_{\epsilon }>0.$$

$$\beta (K_{\epsilon },V({\rho }_{\epsilon }))=\underset{S\in V({\rho }_{\epsilon })}{\sup }\beta (K_{\epsilon },S)<1-\alpha -\delta .$$

$$Y_{\epsilon }=AS+\epsilon \xi ,\epsilon >0.$$

$${\mathbb{H}}_0:S=S_0=0,$$

$${\mathbb{H}}_{\epsilon }:S\in V_{(}{\rho }_{\epsilon })=U\setminus B_{{\rho }_{\epsilon }}(S_0){\rho }_{\epsilon }>0.$$

$$Y_{7\epsilon ,S}=AS+ϵ{\xi }_S$$

$$V({\epsilon }_i)\subset {\cup }_{l=1}^i{\cup }_{j=1}^{k_l}\Pi (g_{lj},c_{lj})$$

$$0\notin {\cup }_{l=1}^i{\cup }_{j=1}^{k_l}\Pi (g_{lj},c_{lj}),$$

$$\Pi(g_{lj},c_{lj})=\Bigl{\{}\,f\,:\,\int_{S}g_{lj}\,f\,d\,\mathbf{P}_{0}>c_{lj},f\in\mathbb{L}_{p}\Bigr{\}}.$$

$$\Vert f{\Vert }_p<\frac{{\epsilon }_{i-1}}{1-\delta }.$$

$$\begin{array}{rl}& \underset{f}{\sup }\underset{f_1}{\inf }\{\Vert f-f_1{\Vert }_p:f\in {{\rm Y}}_{i,j}\cap W_i\cap L_i,f_1\in W_{i-1}\cap L_{i-1}\}\\ & \le |\lambda {|}^{-1}R\underset{f}{\sup }\{\Vert f-f_1{\Vert }_p:f\in {{\rm Y}}_{i,j}\cap W_i\cap L_i\cap {\Gamma }_0,f_1\in W_{i-1}\cap L_{i-1}\cap {\Gamma }_0\}\\ & \le d_i|\lambda {|}^{-1}R.\end{array}$$

Full text

Institute of Problems of Mechanical Engineering RAS, Bolshoy pr., 61, VO, 1991178 St. Petersburg and St. Petersburg State University, Universitetsky pr., 28, Petrodvoretz, 198504 St. Petersburg, RUSSIA

AMS subject classification: 62F03, 62G10, 62G2

keywords : consistency, uniform consistency, goodness of fit tests, signal detection.

1 Introduction

Compactness assumption is necessary condition for existence of uniformly consistent estimator [9]. This assumption is also naturally arises if we explore ill-posed problems with deterministic noise [4]. In some sense compactness is rather severe assumption allowing to find uniformly consistent estimator. Now we see that the same severe assumption is necessary in natural setups of nonparametric hypothesis testing.

We say that family of sets of alternatives are uniformly consistent if, for these sets of alternatives, there is uniformly consistent family of tests.

Uniform consistency of nonparametric sets of alternatives has been explored from different viewpoints and different setups. A rather complete bibliography one can find in [5].

We explore the problem, if hypothesis is simple and nonparametric sets of alternatives are approaching with hypothesis. The research are based on Le Cam -Schwartz [17] results. Le Cam and Schwartz [17] have found necessary and sufficient conditions for existence of uniformly consistent tests and estimators. These conditions are rather cumbersome and are provided in terms of $n$-fold products of probability measures. A simple form of this conditions arises if probability measures of statistical experiment are compact in the topology of set-wise weak convergence, other words, if densities of probability measures are uniformly integrable.

By Lemma 2.3 in Gaensler [8], if set is compact in weak topology of set-wise convergence and there is another weak topology on this set such that the set is Hausdorff space for this topology then these two topologies coincide on the set. This allows to point out necessary and sufficient conditions of uniform consistency in terms of simple natural weak topologies for widespread statistical models: hypothesis testing on a density, signal detection in Gaussian white noise, hypothesis testing on intensity of Poisson process, hypothesis testing on Gaussian covariance operator and so on (see [5]). We can consider this research as natural complement to previous interesting numerous results on consistency of nonparametric tests [1, 3, 7, 10, 9, 13, 14, 15, 18].

In [5], sets of alternatives did not depend on sample size (power of signal and so on). For a long time the necessary and sufficient conditions of uniform consistency for approaching sets of alternatives remained unclear. Special examples of indistinquishability of hypotheses have been proposed. The first result on indistinguishability of hypotheses and alternatives has been established Le Cam [16]. For hypothesis testing on a density Le Cam [16] has shown that the center of the ball and the interior of the ball in $L_{1}$ are indistinguishable. For the problem of signal detection in Gaussian white noise Ibragimov and Hasminski [9] have proved that the center of a ball and the interior of a ball in $L_{2}$ are indistinguishable. For the same problem Burnashev [2] established the indistinguishability of the interior of a ball in $L_{p}$- spaces, $p>1$. For other setups we mention Ingster [11] and Ingster, Kutoyants [12] works. Ingster [11] and Ingster and Kutoyants [12] showed that we could not distinguish the center of a ball and the interior of a ball in $L_{2}$ in the problems of hypothesis testing on a density and on an intensity function of Poisson process respectively.

In [7], we showed that, if sets of alternatives are bounded center-symmetric convex set with deleted "small"balls in $\mathbb{L}_{2}$ having center at the zero point of hypothesis, then there are uniformly consistent tests for some sequence of radii of balls tending to zero as sample size increases, if and only if, the center-symmetric convex set is relatively compact.

Paper goal is to prove similar results without assumption of center-symmetry of convex set and to extend this result for deleted balls in $\mathbb{L}_{p}$, $p>1$. The technique allows to explorer balls from another Banach spaces. However we does not touch these setups.

2 Uniformly consistent sets

2.1 Hypothesis testing on a density

Let $X_{1},\ldots,X_{n}$ be i.i.d.r.v.’s having probability measure $\mathbf{P}$ defined on Borel sets of $\sigma$-field $\mathfrak{B}$ assigned on a set $S$. Denote $\mathcal{P}$ the set of all probability measures on $(S,\mathfrak{B})$.

For probability measure $\mathbf{P}_{0}\in\mathcal{P}$ denote $\mathbb{L}_{p}$, $1<p<\infty$, the set of all real measurable functions $g$ such that

[TABLE]

For each $\rho>0$ and for each $u\in\mathbb{L}_{p}$, define the ball $B_{\rho}(u)=\{\,g\,:\,\|g-u\|_{p}\leq\rho,\,g\in\mathbb{L}_{p}\}$.

Fix bounded convex set $U\subset\mathbb{L}_{p}$. For any $u\in U$ and any $\rho>0$, assume that there is bounded function $g\in B_{\rho}(u)\cap U$.

We say that $u\in U$ is internal point of convex set $U$ if for any bounded function $v\in U$ there is $\lambda<0$ such that $u+\lambda(v-u)\in U$. Note that traditional topological definition of internal point of set is different (p.410 [dun]).

Suppose that $f_{0}\equiv 0$ is internal point of $U$.

We explore problem of testing hypothesis

[TABLE]

versus alternatives

[TABLE]

with $\rho_{n}>0$, $\rho_{n}\to 0$ as $n\to\infty$.

For any test $K_{n}$ denote $\alpha(K_{n})=\mathbf{E}_{0}[K_{n}]$ its type I error probability and denote $\beta(K_{n},f)$ its type II error probability for alternative $f\in V(\rho_{n})$.

We say that sequence of tests $K_{n}$, $0<\alpha(K_{n},f_{0})\leq\alpha<1$, is uniformly consistent, if there is $n_{0}>0$ and $\delta>0$ such that, for all $n>n_{0}$, we have

[TABLE]

Otherwise, we say that sequence of tests $K_{n}$, $0<\alpha(K_{n},f_{0})\leq\alpha<1$, is uniformly inconsistent.

Theorem 2.1

There is a sequence $\rho_{n}\to 0$ as $n\to\infty$ such that sets of alternatives $V(\rho_{n})$ is uniformly consistent, if and only if, set $U$ are relatively compact in $\mathbb{L}_{p}$.

We say that $r_{n}$ is rates of distinguishability if sets of alternatives $V(\rho_{1n})$ is uniformly inconsistent for any sequence $\rho_{1n}>0$, $\rho_{1n}/r_{n}\to 0$ as $n\to\infty$, and sets of alternatives $V(\rho_{2n})$ is uniformly consistent for any sequence $\rho_{2n}>0$, $\rho_{2n}/r_{n}\to\infty$ as $n\to\infty$.

In proof of Theorem 2.1, for point zero, we construct set $W$ such that $U\subset W$ and show that Kolmogorov $n$-width $d_{i}$, $1\leq i<\infty$ of set $W$ tends to zero. The same reasoning we can conduct for any internal point $u\in U$ and construct set $W_{u}$, $U\subset W_{u}$ with Kolmogorov $n$-width $d_{ui}$, $1\leq i<\infty$. If we take $h=u$ in proof of Theorem 2.1 and, by the same reasoning, implementing similarity inequality (3.4), we get $cd_{i}\leq d_{ui}\leq Cd_{i}$, $1\leq i<\infty$, where coefficients $c,C$ are proportional to similarity coefficient.

For $\varepsilon>0$ and $\theta\in K$, denote $M(\varepsilon/c(\theta,\varepsilon)\cap K)$, the largest cardinality of an $\varepsilon/c$ packing set for $B(\theta,\varepsilon)\cap K$ (see definition 5.4 in [19]). We refer to

[TABLE]

as local entropy of set $K$ at the point $\theta$. The same similarity consideration allow to show that there are $c_{1}$ and $c_{2}$ such that, for any interior points $\theta_{1}\in K$ and $\theta_{2}\in K$ we have

[TABLE]

This is the base for the proof that we have the same rates of distinguishability in the problems of hypothesis testing $\mathbb{H}_{0}\,:\,f=0$ versus $\mathbb{H}_{n}\,:\,f\in U\setminus B_{\rho_{n}}(0)$ and $\mathbb{H}_{0u}\,:\,f=u$ versus $\mathbb{H}_{nu}\,:\,f\in U\setminus B_{\rho_{n}}(u)$. If point $u$ depends on $n$, one can get different rates of distinguishability [20].

Corollary 2.1

Let a priori information be provided that $f\in U$ where $U$ is bounded convex set in $\mathbb{L}_{p}$, $p>1$. Then there is uniformly consistent estimator of $\|f\|_{p}$, if and only if, the set $U$ is relatively compact.

Proof of Theorem 2.1 is based on Theorem 2.2 on uniform consistency of set of alternatives that does not depend on $n$.

For $p>1$, we explore the problem of testing hypothesis

[TABLE]

versus alternatives

[TABLE]

We say that sequence of tests $K_{n}$ is uniformly consistent if

[TABLE]

Theorem 2.2

i.* Let $W_{0}$ be bounded set in $L_{p}$, $p>1$. Then there are uniformly consistent tests if the closures of $W_{0}$ and $W_{1}$ are disjoint in weak topology.*

ii*. The converse holds if $W_{1}$ is also bounded sets in $L_{p}$, $p>1$.*

Theorem 2.2 follows from Le Cam, Schwartz (p.141, [17]) and Lemma 2.3 in Ganssler [8] (see Theorem 4.1 in [5] as well).

If closures in weak topology of sets $W_{0}$ and $W_{1}$ are disjoint then there is a finite number of functions $g_{1},\ldots,g_{k}\in\mathbb{L}_{q}$, $\frac{1}{p}+\frac{1}{q}=1$, and constants $c_{1},\ldots,c_{k}$ such that, for any $u\in W_{0}$ there holds $c_{i}\,>\,\int_{S}\,u\,g_{i}\,d\,\mathbf{P}_{0}$ for all $i$, $1\leq i\leq k$, and, for any $v\in W_{1}$, there is index $i$, $1\leq i\leq k$, such that $\int_{S}\,v\,g_{i}\,d\,\mathbf{P}_{0}\,>\,c_{i}$.

2.2 Hypothesis testing on intensity function of Poisson random process

Let we be given $n$ independent realizations $\kappa_{1},\ldots,\kappa_{n}$ of Poisson random process with mean measure $\mathbf{P}$ defined on Borel sets of $\sigma$-field $\mathfrak{B}$ assigned on set $S$. Suppose mean measure $\mathbf{P}$ has intensity function $h(t)=\frac{d\mathbf{P}}{d\nu}(t)$, $t\in S$. Here $\nu$ is a measure defined on Borel sets of $\sigma$-field $\mathfrak{B}$ as well and $\nu(S)<\infty$, $\mathbf{P}(S)<\infty$.

Problem is to test a hypothesis $H_{0}\,:\,f(t)\doteq f_{0}=h(t)-1\equiv 0$ versus alternatives $H_{n}\,:\,f\in V(\rho_{n})\subset\mathbb{L}_{p}$, $p>1$. The definition of sets $V(\rho_{n})$, $\rho_{n}>0$, is the same as in previous subsection with the unique difference that the densities are replaced with intensity functions. The definition of internal point of convex set $U$ is also the same as in the previous subsection.

Theorem 2.3

Let $U$ be bounded convex set in $\mathbb{L}_{p}$, $p>1$. Let $f_{0}$ be internal point of $U$. Then there is sequence $\rho_{n}\to 0$ as $n\to\infty$ such that sets of alternatives $V(\rho_{n})$ are uniformly consistent, if and only if, set $U$ is relatively compact.

Proof of Theorem 2.3 completely coincides with the proof of Theorem 2.1. It suffices only to implement Theorem 5.9 in [5] instead of Theorem 2.2.

2.3 Signal detection in Gaussian white noise

Suppose we observe a realization of stochastic process $Y_{\epsilon}(t),t\in(0,1)$, defined by the stochastic differential equation

[TABLE]

where $S\in\mathbb{L}_{2}(0,1)$ is unknown signal and $dw(t)$ is Gaussian white noise.

Let $U$ be bounded convex set in $\mathbb{L}_{2}(0,1)$. We say that $S_{0}\in U$ is internal point of convex set $U$ if, for any $S\in U$, there is $\lambda<0$ such that $S_{0}+\lambda(S-S_{0})\in U$.

Denote $B_{r}(S)$ ball in $\mathbb{L}_{p}$, $p>1$, having radius $r>0$ and center $S\in\mathbb{L}_{2}$.

We wish to test a hypothesis

[TABLE]

versus alternatives

[TABLE]

For any test $K_{\varepsilon}$, $\varepsilon>0$, denote $\alpha(K_{\varepsilon})=\mathbf{E}_{0}[K_{\varepsilon}]$ its type I error probability and denote $\beta(K_{\varepsilon},S)$ its type II error probability for alternative $S\in V(\rho_{\varepsilon})$.

We say that family of tests $K_{\varepsilon}$, $0<\alpha(K_{\varepsilon})\leq\alpha<1$, $\varepsilon>0$, is uniformly consistent if there is $\varepsilon_{0}>0$ and $\delta>0$ such that, for all $0<\varepsilon<\varepsilon_{0}$, we have

[TABLE]

Theorem 2.4

Let $U$ be bounded convex set in $\mathbb{L}_{p}$, $p>1$. Let $S_{0}$ be internal point of $U$. Then there is $\rho_{\varepsilon}\to 0$ as $\varepsilon\to 0$ such that sets of alternatives $V(\rho_{\varepsilon})$ are uniformly consistent, if and only if, the set $U$ is relatively compact.

Proof of Theorem 2.3 completely coincides with proof of Theorem 2.1. It suffices to implement Theorem 5.3 in [5] instead of Theorem 2.2.

2.4 Hypothesis testing on a solution of linear ill-posed problem

In Hilbert space $\mathbb{H}$ we observe a realization of Gaussian random vector

[TABLE]

Hereafter $A:\mathbb{H}\to\mathbb{H}$ is known linear operator and $\xi$ is Gaussian random vector having known covariance operator $R:\mathbb{H}\to\mathbb{H}$ and $\mathbf{E}\xi=0$.

Denote $B_{r}(S)$ ball in $\mathbb{H}$, having radius $r>0$ and center $S\in\mathbb{H}$.

Let $U$ be bounded convex set in $\mathbb{H}$.

We wish to test a hypothesis

[TABLE]

versus alternatives

[TABLE]

Below we use the same definition of uniform consistency and similar definition of interior point as in subsection 2.3.

For any operator $K:\mathbb{H}\to\mathbb{H}$ denote $\mathfrak{R}(K)$ the rangespace of $K$.

Suppose that the nullspaces of $A$ and $R$ equal zero and $\mathfrak{R}(A)\subseteq\mathfrak{R}(R^{1/2})$.

Theorem 2.5

Let the operator $R^{-1/2}A$ be bounded in $\mathbb{H}$. Let $S_{0}$ be internal point of $U$. Then there is $\rho_{\varepsilon}\to 0$ as $\varepsilon\to 0$ such that sets of alternatives $V(\rho_{\varepsilon})$ are uniformly consistent, if and only if, the set $U$ is relatively compact.

Proof 2.1

The statement of Theorem 2.4 hold for the sets of alternatives $R^{-1/2}A\,V(\rho_{n})$. Thus it suffices to implement the inverse map $A^{-1}R^{1/2}$ to (2.1) (see [5]).

2.5 Hypothesis testing on an operator known with a random error

Let $\Theta$ be the set of all bounded linear operators $\mathbb{H}\to\mathbb{H}$ where $\mathbb{H}$ is separable Hilbert space.

Let $A\in\Theta$ be an unknown linear operator.

Let for all $S\in H$ we can observe random Gaussian vector

[TABLE]

with $\mathbf{E}\xi_{S}=0$ and $\mathbf{E}<\xi_{S_{1}},\xi_{S_{2}}>=<S_{1},S_{2}>$ where $<S_{1},S_{2}>$ is inner product of vectors $S_{1},S_{2}\in\mathbb{H}$.

Let $U$ be convex set in $\Theta$.

We say that $A_{1}\in U$ is internal point of $U$ if, for any $A\in U$, there is $\lambda<0$ such that $A_{1}+\lambda(A-A_{1})\in U$.

Denote $A_{0}\in\Theta$ linear operator such that, for any $S\in\mathbb{H}$, we have $A_{0}S=0$.

For Hilbert-Schmidt norm in $\Theta$, denote $B(r)$ ball in $\Theta$ having radius $r>0$ and center $A_{0}$.

Let $A_{0}$ be interior point of $U$.

The problem is to test hypothesis $H_{0}\,:\,A=A_{0}$ versus alternatives $H_{1}\,:\,A\in V(\rho_{\varepsilon})\doteq U\setminus B(\rho_{\varepsilon})$ with $\rho_{\varepsilon}>0$.

The definition of uniform consistency is akin to one in subsection 2.3.

In the following Theorem we consider the boundedness and compactness for Hilbert-Schmidt norm.

Theorem 2.6

Let $U$ be bounded convex set and let $A_{0}$ be internal point of $U$. Then there is $\rho_{\varepsilon}\to 0$ as $\varepsilon\to 0$ such that sets of alternatives $V(\rho_{\varepsilon})$ are uniformly consistent, if and only if, the set $U$ is relatively compact.

The reasoning are akin to proof of Theorem 2.1. It suffices only to implement Theorem 5.7 in [5].

3 Proof of Theorem 2.1

Suppose that there is uniformly consistent sequence of alternatives $V(\rho_{n})$, $\rho_{n}\to 0$ as $n\to\infty$. Then we implement Theorem 2.2 and point out compact set $W$ such that $U\subset W$.

Fix $\varepsilon_{1}>0$. For each $i>1$, we put $\varepsilon_{i}=\frac{\varepsilon_{i-1}}{2}$.

By Theorem 2.2, for each $i$, there are functions $g_{i1},\ldots,g_{ik_{i}}\in\mathbb{L}_{q}$, $\|g_{ij}\|_{q}=1$ for $1\leq j\leq k_{i}$, $\frac{1}{p}+\frac{1}{q}=1$, and constants $c_{i1},\ldots,c_{ik_{i}}$ such that

[TABLE]

and

[TABLE]

where

[TABLE]

Denote $\Gamma(g_{lj},c_{lj})$ the boundary hyperplane of $\Pi(g_{lj},c_{lj})$.

Define sets $G_{i}=\{g_{i1},\ldots,g_{ik_{i}}\}$, $1\leq i<\infty$.

It is clear, that we can suppose additionally that, if $g\in G_{i}$, then $-g\in G_{i}$, $1\leq i<\infty$. Otherwise we can add such functions. Moreover, we can suppose that, if $g_{ij_{1}}=-g_{ij_{2}}$, $1\leq j_{1}\neq j_{2}\leq k_{i}$, then $c_{ij_{1}}=c_{ij_{2}}$.

We can also suppose that $g_{ij}$ are step functions and therefore $g_{ij}\in\mathbb{L}_{p}\cap\mathbb{L}_{q}$, $1\leq j\leq k_{i}$, $1\leq i<\infty$. Otherwise we can replace functions functions $g_{ij}$ by their step approximations and change constants $c_{ij}$ insignificantly. We can also choose functions $g_{ij}$ such that function $g_{i_{1}j_{1}}$ is orthogonal to function $g_{i_{2},j_{2}}$ in $\mathbb{L}_{2}$, for any $1\leq i_{1}\neq i_{2}<\infty$ and for any $1\leq j_{1}<k_{i_{1}}$, $1\leq j_{2}<k_{i_{2}}$. Suppose also that $\|g_{ij}\|_{q}=1$ for $1\leq j\leq k_{i}$, $1n\leq i<\infty$.

For $i$, $1\leq i<\infty$, denote $M_{i}$ linear subspace generated functions $g_{i1},\ldots,g_{ik_{i}}$.

Denote $B(M_{i},\rho)$ ball in subspace $M_{i}$ of $\mathbb{L}_{p}$ having center zero and radius $\rho>0$.

Denote $\Omega_{i}=M_{i}\cap\cap_{j=1}^{k_{i}}\bar{\Pi}(g_{ij},\varepsilon_{i-1})$

Fix $\delta>0$. We can take the number of functions $g_{ij}$, $\|g_{ij}\|_{q}=1$, $1\leq j\leq k_{i}$, so large that, for any $f\in\Omega_{i}$, we have

[TABLE]

Denote $d_{i}\doteq\frac{\varepsilon_{i-1}}{1-\delta}$.

For definition of set $W$ we implement induction by $i$.

Since set $U$ is bounded there is $d_{1}$ such that for any $j$, $1\leq j\leq k_{1}$, and, for any $f\in U$, we have $f\in\bar{\Pi}(g_{1j},d_{1})$ where $\bar{\Pi}(g_{1j},d_{1j})=\mathbb{L}_{p}\setminus\Pi(g_{1j},d_{1})$.

Let $v\in U\setminus B_{\varepsilon_{1}}(0)$, $\int_{S}g_{11}vd\,\mathbf{P}_{0}>c_{11}$, and let $v$ be bounded function. Then there is $\lambda<0$ such that $h\doteq\lambda v\in U$ and $1+\lambda v$ is a density.

Denote \Pi_{0}=\Bigl{\{}\,f\,:\,\int_{S}g_{11}fd\,\mathbf{P}_{0}>0,f\in\mathbb{L}_{p}\Bigr{\}}. Let $\Gamma_{0}$ be the boundary hyperplane of $\Pi_{0}$.

Denote $W_{1}=\cap_{j=1}^{k_{1}}\bar{\Pi}(g_{1j},d_{1})$.

Let $i>1$. For each $j$, $1\leq j\leq k_{i}$, we draw hyperplane $\Upsilon_{ij}$ passing through the point $h$ and $\Gamma(g_{ij},d_{i})\cap\Gamma_{0}$. Denote $\Psi_{i,j}$ the set having hyperplane $\Upsilon_{i,j}$ as a boundary and containing $(U\setminus B(\varepsilon_{i-1}))\cap\Pi_{0}$.

Denote $W_{i}=\cap_{j=1}^{k_{i}}\Psi_{ij}\cap W_{i-1}$.

Then $U\cap\Pi_{0}\subset W_{i}$.

Denote $L_{i}$ linear space generated functions from linear spaces $M_{1},\ldots,M_{i}$.

Let $R>0$ be such that $U\subset B_{R}(0)$.

Using triangle similarity considerations and (3.3), we have

[TABLE]

for any $i$, $1<i<\infty$, and any $j$, $1\leq j\leq k_{i}$.

Denote $W$ the closure of $\cap_{i=1}^{\infty}W_{i}$. By (3.4), $n$-width of W tends to zero, therefore $W$ is relatively compact. Since $U\cap\Pi_{0}\subset W$ then $U\cap\Pi_{0}$ is relatively compact as well. The case $U\setminus\Pi_{0}$ is explored similarly.

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