Distributions Escaping to Infinity and the Limiting Power of the Cliff-Ord Test for Autocorrelation

Abstract

We consider a family of proper random variables which converges to an improper random variable. The limit in distribution is found and applied to obtain a closed-form expression for the limiting power of the Cliff-Ord test for autocorrelation. The applications include the theory of characteristic functions of proper random variables, the theory of almost periodic functions, and the test for spatial correlation in a linear regression model.

1. Introduction

Improper random variables do not satisfy the condition 𝑃(𝑋∈𝑅)=1; that is, they may take values outside the real line 𝑅. They are not used much by themselves, but there are situations when they arise as limits of proper random variables. In such cases, we say that a distribution escapes to infinity. The main problem considered in this paper is illustrated in the following in a simplified situation.

Throughout the paper we denote by 𝜒_𝐴 the indicator of a set 𝐴. Let 𝑔=(1/2)𝜒_[−1,1] be a uniform distribution on the segment [−1,1]. Consider the family of densities 𝑔_𝜆(𝑡)=𝜆𝑔(𝜆𝑡), where 𝜆>0. The total mass is constant: ∫_𝑅𝑔_𝜆(𝑡)𝑑𝑡=1 for any 𝜆>0. In Bayesian estimation, improper priors are obtained by letting 𝜆→0. In this case, there are two effects at work: the support supp𝑔_𝜆=[−1/𝜆,1/𝜆] stretches out indefinitely and the height of the density max𝑔_𝜆=𝜆/2 goes to zero. One might be tempted to think of the limit 𝐺 of {𝑔_𝜆}, as 𝜆→0, as an infinitesimally thin layer smeared over the whole real line. This notion would be wrong because _𝑅𝑔_𝜆(𝑡)𝜑(𝑡)≤𝜆𝑑𝑡2_supp𝜑𝜑(𝑡)𝑑𝑡⟶0,𝜆→0, (1.1) for any 𝜑∈𝐶₀(𝑅) (the set of continuous functions on 𝑅 with bounded support supp𝜑). Thus, if the limit in distribution 𝐺 of {𝑔_𝜆} exists, it vanishes on all elements of 𝐶₀(𝑅). By the definition of the support of a distribution [1, Chapter 1, Section 13], the support of 𝐺 does not contain 𝑅. So instead of spreading the mass over 𝑅, it is more correct to say that the mass escapes to infinity.

In this paper we provide a rigorous framework for treating more complex situations. To illustrate the arising complexities, let us look at the standard normal variable 𝑋 with the density 𝑔(𝑥)=(2𝜋)^−1/2exp(−𝑥²/2). Let 𝑋_𝜆 have the density 𝑔_𝜆(𝑥)=𝜆𝑔(𝜆𝑥)=𝜆(2𝜋)^−1/2𝑒^{−(𝜆𝑥)2/2}. Its characteristic function is 𝐸𝑒^{𝑖𝑡𝑋𝜆}=_𝑅𝑒^{𝑖(𝑡/𝜆)𝑥}𝑔(𝑥)𝑑𝑥=𝐸𝑒^{𝑖(𝑡/𝜆)𝑋}=𝑒^{−(𝑡/𝜆)2/2}. (1.2) The moment generating function of 𝑋_𝜆 is 𝐸𝑒^𝑡𝑋𝜆=𝑒^{(𝑡/𝜆)2/2}. None of these expressions is good for characterizing the limit as 𝜆→0. Further, let 𝜑 denote an arbitrary continuous and bounded function on 𝑅. In the expression _𝑅𝑔_𝜆(𝑡)𝜑(𝑡)𝑑𝑡=𝜆_𝑅𝑔(𝜆𝑡)𝜑(𝑡)𝑑𝑡, (1.3) the height of the density goes to zero, so the integrand converges to zero everywhere. However, the graph of the density stretches out from the origin. Therefore, the best majorant for the integrand is |𝑔(0)𝜑(𝑡)| which is generally not integrable. Thus, the dominated convergence theorem cannot be used to obtain convergence in distribution. While the case of collapsing density (𝜆→∞) is easily handled by the existing theory, the case of stretching out (𝜆→0) requires new tools which we develop here.

Problem 1. Describe the limit in distribution when the stretching-out is applied to a density along all or some variables.
This problem is solved in Theorem 2.2 in Section 2 (case of all variables) and Theorem 2.6 (case of some variables). In their simplest form, those results reveal the main ideas. Let 𝑔 be any summable even function on 𝑅 (it may change sign and not integrate to unity). Let 𝑀𝜑 denote the generalized mean of 𝜑 over 𝑅: 𝑀𝜑=lim_𝑟→∞12𝑟^𝑟_−𝑟𝜑(𝑡)𝑑𝑡. (1.4) Then for a continuous and bounded function 𝜑 on 𝑅 such that 𝑀𝜑 exists, one has 𝜆𝑔(_𝑅𝜆𝑡)𝜑(𝑡)𝑑𝑡⟶𝑀𝜑_𝑅𝑔(𝑡)𝑑𝑡,𝜆⟶0. (1.5)

Now suppose that the function 𝑔 depends on two variables and the stretching-out is applied with respect to the first of them. Let 𝜑 be continuous and bounded on 𝑅². Denote 𝑀₁𝜑, the result of application of (1.4) to 𝜑 with respect to the first argument,𝑀₁𝜑𝑡₂=lim_𝑟→∞12𝑟^𝑟_−𝑟𝜑𝑡₁,𝑡₂𝑑𝑡₁, (1.6) and let 𝑔₁(𝑡₂)=∫_𝑅𝑔(𝑡₁,𝑡₂)𝑑𝑡₁. In cases when 𝑔 is not a density, we call this function a marginal “density.”   Then, (1.5) can be used to prove _𝑅2𝜆𝑔𝜆𝑡₁,𝑡₂𝜑(𝑡)𝑑𝑡=_𝑅𝑅𝜆𝑔𝜆𝑡₁,𝑡₂𝜑𝑡₁,𝑡₂𝑑𝑡₁𝑑𝑡₂⟶_𝑅𝑀₁𝜑𝑡₂𝑔₁𝑡₂𝑑𝑡₂,𝜆⟶0, (1.7) if 𝑔 is integrable and spherically symmetric. As we show, the right sides of (1.5) and (1.7) determine distributions supported at infinity.

Theorem 2.2 is followed by two applications. One generalizes results from [2] on the link between jumps of a distribution function of a (proper) random variable and its characteristic function. Another application is to the fundamental theorem of H. Bohr on the Parseval identity for almost-periodic functions; see [3]. Theorem 2.6 is applied to the limiting power of the Cliff-Ord test for autocorrelation. To formulate the related problems, we need some notation.

Consider a linear regression model 𝑦=𝑋𝛽+𝑢,𝐸𝑢=0,var(𝑢)=𝜎²Σ𝜌, (1.8) where 𝑋 is an 𝑛×𝑘 matrix of rank 𝑘<𝑛; a vector 𝛽∈𝑅^𝑘 and a number 𝜎²>0 are unknown parameters. The matrix Σ(𝜌) is assumed to be a nonnegative function of the parameter 𝜌∈[0,𝑎], with some 𝑎>0. 𝜌 in applications characterizes the degree of autocorrelation. Testing for autocorrelation takes the form 𝐻₀∶𝜌=0 versus 𝐻_𝑎∶𝜌>0. The case 𝜌→𝑎 is of special interest for determining the limiting power of tests.

Assuming that Σ(𝜌) is positive definite for 𝜌∈[0,𝑎), denote 𝜀=Σ^−1/2(𝜌)𝑢 and let 𝑔(𝜀) be the density of 𝜀. The density of 𝑦 is then given by [4, Equation (D.2)] 𝑓_𝜌𝑦=detΣ^−1/2𝜌𝑔Σ^−1/2𝜌𝑦−𝑋𝛽. (1.9)

Problem 2. Assuming that rankΣ⁻¹(𝑎)=𝑛−1, (1.10) describe the limit in distribution of 𝑓_𝜌.
This problem, as an intermediate step in the proof, was considered by Martellosio [4], and his answer is reproduced as follows. Let 𝑁(𝐴) denote the null space of a matrix 𝐴 and denote 𝐴(𝑎)=Σ⁻¹(𝑎). Let 𝑋𝛽+𝑁(𝐴(𝑎)) be the translation by 𝑋𝛽 of the null space of 𝐴(𝑎). Reference [4] states that in case (1.10) as𝜌⟶𝑎,𝑓_𝜌tendstoadegeneratedensitysupportedon𝑋𝛽+𝑁(𝐴(𝑎)), (1.11) see page 182. Equation (1.11) arises from the confusion between the stretching-out and collapsing. On page 159, rows 10 and 11, he remarks that (1.11) can be extended to the case rank(Σ⁻¹(𝑎))<𝑛−1. He does not specify the mode of convergence, but, as we argue in Section 2, the convergence in distribution is the right one for the problem of the behavior of the limiting power of tests for autocorrelation. Unfortunately, (1.11) does not hold under the convergence in distribution. We prove this and solve Problem 2 in Theorem 2.7 for the general case 0<rank(Σ⁻¹(𝑎))<𝑛.

Let Φ be a critical region for rejecting 𝐻₀ in favor of 𝐻_𝑎. Denote 𝛽_𝜌(Φ)=_Φ𝑓_𝜌𝑦𝑑𝑦, (1.12) the probability content of Φ under density (1.9) and define the limiting power 𝛽(Φ) as the limit of this probability content: 𝛽(Φ)=lim_𝜌→𝑎𝛽_𝜌(Φ). Specifically, we consider critical regions that arise from the Cliff-Ord test, to be described now. With the regressor matrix 𝑋 from (1.8) denote 𝐿_𝑋=𝑋(𝑋^′𝑋)⁻¹𝑋^′,𝑀_𝑋=𝐼−𝐿_𝑋. Under the spatial autocorrelation assumption, the regression disturbances follow 𝑢=𝜌𝑊𝑢+𝜀,𝐸𝜀=0,var(𝜀)=𝜎²_𝜀𝐼, (1.13) where 𝜀 is a new disturbance and 𝑊 is some known 𝑛×𝑛 matrix. The scalar 𝜌, which is unknown, determines the degree of correlation among the components of 𝑢. For testing the null 𝐻₀∶𝜌=0 against the alternative 𝐻_𝑎∶𝜌>0, Cliff and Ord [5] proposed a test that rejects the null if 𝑢^′𝑀_𝑋𝑊𝑀_𝑋𝑢/𝑢^′𝑀_𝑋𝑢>𝑐. Denote Φ=𝑢∈𝑅^𝑛∶𝑢^′𝑀_𝑋𝑊𝑀_𝑋𝑢𝑢^′𝑀_𝑋𝑢>𝑐,Φ₋=𝑢∈𝑅^𝑛∶𝑢^′𝑀_𝑋𝑊𝑀_𝑋𝑢𝑢^′𝑀_𝑋𝑢<𝑐. (1.14)

Problem 3. Obtain a closed-form expression for 𝛽(Φ).
Theorem 2.7 is applied in Theorem 2.9 to solve this problem in case (1.10).

Problem 4. Describe the cases when the limiting power disappears, that is, when 𝛽(Φ)=0.
This problem has been the main motivation for this paper. Spatial models in general are peculiar in many respects, and the possibility of the limiting power to disappear is one of those peculiarities that has been attracting researchers' attention lately. Krämer [6] was the first to suggest that the limiting power of tests for spatial autocorrelation may vanish, for some combinations of the regressor matrix and the spatial matrix. Unfortunately, the terms in which Krämer expressed his results do not adequately reflect all the possibilities, and his proof contains an incorrect argument; see [7, Footnote 5]. Martellosio [4] was the first to suggest that the answer is better described in terms of the geometrical relationship between the eigenvectors of Σ⁻¹(𝑎) and the critical region Φ. However, both of his main result [4, Theorem 1] and its proof contain errors (see Remark 2.12 in Section 2). Our answer to Problem 4 is given in Theorem 2.11 and corrects [4, Theorem 1] in that where he excludes the extreme values 0 and 1 for the limiting power, we give examples showing that the extreme values are possible. In the light of our result, several statements in [4, 7] have to be reconsidered. The analysis for the Cliff-Ord test is very involved, and it is not feasible to indicate corrections for all Martellosio's main results that depend on the wrong intermediate statement or his Theorem 2.9. The complexity of our analysis necessitates a more detailed notation than that used by Martellosio. In particular, some of his verbal definitions and statements are cast in a more formal way. As a result, our citations are not word for word.
Despite the fact that the convergence in distribution is the right one for Problem 4, it would be interesting to know what kind of convergence produces the limit indicated by Martellosio, as is stated next.

Problem 5. Design the mode of convergence that leads to (1.11).

To this end, we introduce a new convergence concept (which, given its purpose, could be called a retrofit convergence), which may not look intuitively appealing but allows us to prove (1.11) in Theorem 2.13. Under this alternative convergence, there is no analog of (1.12). Therefore, we did not consider Problems 3 and 4 for this convergence.

The previous plan will be implemented under conditions much more restrictive than suggested by Martellosio. All main results that are stated in Sections 2 and 3 contains all proofs.

2. Main Statements

In the multidimensional version of the generalized mean (1.4) instead of averaging over segments [−𝑟,𝑟], we have to average over balls. The shape of those balls depends on the norm of 𝑅^𝑛. Let ‖⋅‖_𝑎 be an arbitrary norm in 𝑅^𝑛. The balls are defined by 𝐵_𝑎,𝑛(𝑥,𝑟)={𝑦∈𝑅^𝑛∶‖𝑥−𝑦‖_𝑎≤𝑟}, where the indication of the space dimension will be important when dealing with more than one space. As an example, one can think of the 𝑙_𝑝-norm defined by ‖𝑥‖_𝑝=^𝑛_𝑖∑₌₁𝑥_𝑖^𝑝1/𝑝,if1≤𝑝<∞,max_{1≤𝑖≤𝑛}𝑥_𝑖,if𝑝=∞. (2.1) In case of the Euclidean norm ‖⋅‖₂, we obtain usual balls; in cases 𝑝=1 and 𝑝=∞, the balls 𝐵_𝑝,𝑛(𝑥,𝑟) are cubes. Another useful example is ‖𝑥‖_𝐴=(𝑥^′𝐴𝑥)^1/2, where 𝐴 is a symmetric, positive definite matrix. We say that a function 𝑔 on 𝑅^𝑛 is ‖⋅‖_𝑎-spherically symmetric if 𝑔(𝑥)=𝑝(‖𝑥‖_𝑎) with some function 𝑝 defined on the half-axis [0,∞). Conditions involving spherical symmetry in the following are similar to Conditions (2.1) and (2.2) from [8].

Let 𝜎 denote an element of the unit sphere {𝜎∈𝑅^𝑛∶‖𝜎‖_𝑎=1} and let (𝜎,𝜌(𝜎)) be the representation of a point 𝑥∈𝑅^𝑛 in the polar system of coordinates such that 𝜎=𝑥/‖𝑥‖_𝑎 and 𝜌(𝜎)=‖𝑥‖_𝑎 for 𝑥≠0. Then, the Lebesgue measure of 𝐵_𝑎,𝑛(𝑥,𝑟) is mes𝐵_𝑎,𝑛(𝑥,𝑟)=mes𝐵_𝑎,𝑛(0,𝑟)=_{‖𝜎‖𝑎=10}^𝑟(𝜎)𝜌^𝑛−1𝑑𝜌𝑑𝜎=𝑟^𝑛_{‖𝜎‖𝑎=10}^{𝑟(𝜎)/𝑟}𝜌^𝑛−1𝑑𝜌𝑑𝜎=𝑣_𝑎𝑟^𝑛, (2.2) where 𝑣_𝑎=mes𝐵_𝑎,𝑛(0,1) is the volume of the unit ball. The norm ‖⋅‖_𝑎 gives rise to averages 𝑚𝜑,𝐵_𝑎,𝑛()=1𝑥,𝑟mes𝐵_𝑎,𝑛(𝑥,𝑟)_{𝐵𝑎,𝑛(𝑥,𝑟)}𝜑(𝑡)𝑑𝑡, (2.3) and to the generalized mean 𝑀_𝑎,𝑛𝜑=lim_𝑟→∞𝑚𝜑,𝐵_𝑎,𝑛(0,𝑟). (2.4) In the one-dimensional case all balls are segments, and we write simply 𝑀𝜑 instead of 𝑀_𝑎,1𝜑.

One of the basic properties of generalized means is that they do not depend on the behavior of 𝜑 in any fixed ball 𝐵_𝑎,𝑛(0,𝑟): 𝑀_𝑎,𝑛𝜑=lim_𝑟→∞1mes𝐵_𝑎,𝑛(0,𝑟)_{𝐵𝑎,𝑛(0,𝑟0)}𝜑(𝑡)𝑑𝑡+_{𝐵𝑎,𝑛(0,𝑟)⧵𝐵𝑎,𝑛(0,𝑟0)}𝜑(𝑡)𝑑𝑡=lim_𝑟→∞1mes𝐵_𝑎,𝑛(0,𝑟)_{𝐵𝑎,𝑛(0,𝑟)⧵𝐵𝑎,𝑛(0,𝑟0)}𝜑(𝑡)𝑑𝑡. (2.5) Other useful properties are (3.16), (3.37), and (3.42).

𝐶𝐿(𝑅^𝑛) denotes the set of continuous bounded functions on 𝑅^𝑛 that satisfy the Lipschitz condition 𝜑(𝑥)−𝜑𝑦≤𝑐₁𝑥−𝑦₂,𝑥,𝑦∈𝑅^𝑛; (2.6) By [9, Theorem  3.6.1], convergence in distribution of random elements 𝑋→−_𝑛^𝑑𝑋 is equivalent to the convergence of expected values 𝐸𝜑𝑋_𝑛⟶𝐸𝜑(𝑋),∀𝜑∈𝐶𝐿𝑅^𝑛. (2.7) [10, Theorem 2.1] asserts that here 𝐶𝐿(𝑅^𝑛) can be replaced by the set of bounded uniformly continuous functions. 𝐿_𝑝(𝑅^𝑛) is the space of 𝑝-summable functions on 𝑅^𝑛 provided with the norm ‖𝑔‖_𝐿𝑝=(∫_𝑅𝑛|𝑔(𝑡)|^𝑝𝑑𝑡)^1/𝑝,  1≤𝑝<∞.

For our applications, in the multidimensional version of (1.5), we need to allow 𝜑 to depend on the parameter 𝜆, as in lim_𝜆→0𝜆^𝑑_𝑅𝑑𝜑_𝜆(𝑡)𝑔(𝜆𝑡)𝑑𝑡=𝑀_𝑎,𝑑𝜑_0𝑅𝑑𝑔(𝑡)𝑑𝑡. (2.8) Here 𝜑∈𝐶𝐿(𝑅^𝑛),  𝜑_𝜆(𝑡)=𝜑(𝐹(𝜆)𝑡+𝐺(𝜆)). 𝐹(𝜆),  𝜆≥0, are 𝑛×𝑑 matrices and 𝐺(𝜆),  𝜆≥0, are 𝑛×1 vectors such that 𝐹(𝜆) and 𝐺(𝜆) tend to 𝐹(0) and 𝐺(0), respectively, sufficiently quickly, as stipulated in the next assumption.

Condition 1. (a)𝑔∈𝐿₁(𝑅^𝑑) is‖⋅‖_𝑎-spherically symmetric, 𝑔(𝑥)=𝑝(‖𝑥‖_𝑎).(b)‖𝐹(𝜆)−𝐹(0)‖=𝑜(𝜆) , ‖𝐺(𝜆)−𝐺(0)‖₂=𝑜(1), 𝜆→0.

Remark 2.1. Using polar coordinates we see that item (a) imposes a certain integrability restriction on 𝑝: _𝑅𝑑𝑔(𝑡)𝑑𝑡=_{‖𝜎‖𝑎=1}^∞₀𝑝𝜌𝜌^𝑑−1𝑑𝜌𝑑𝜎=|𝑆|_𝑎^∞₀𝑝𝜌𝜌^𝑑−1𝑑𝜌<∞, (2.9) where |𝑆|_𝑎 stands for the surface of the unit sphere {𝜎∶‖𝜎‖_𝑎=1}. The class of densities satisfying (a) includes contaminated normal distribution, multivariate 𝑡-distribution, multivariate Cauchy distribution, and see; [8]. It does not matter much which norms are used in item (b).

Theorem 2.2. Condition 1 is sufficient for (2.8) to hold on 𝜑∈𝐶𝐿(𝑅^𝑛) such that 𝑀_𝑎,𝑑𝜑₀ exists.

Remark 2.3. The triangle inequality ‖𝑥+𝑦‖_𝑎≤‖𝑥‖_𝑎+‖𝑦‖_𝑎 is not used in the proof, and a slight generalization of Theorem 2.2 in terms of the geometry of balls is possible. The Lipschitz condition (2.6) can be omitted if 𝐹,𝐺 are constant (𝜑 can be assumed just bounded and continuous).

The first application of Theorem 2.2 is to the theory of characteristic functions. Let 𝐹 be the distribution function of a (proper, real-valued) random variable 𝑋. Denote 𝑗(𝑥) the jump of 𝐹 at point 𝑥 and let 𝜑(𝑡)=𝐸𝑒^𝑖𝑡𝑋 be the characteristic function.

Corollary 2.4. If 𝑔∈𝐿₁(𝑅) is even, then lim_𝜆→0𝑅𝑒^{−𝑖𝑡𝑥}𝜑(𝑡)𝜆𝑔(𝜆𝑡)𝑑𝑡=𝑗(𝑥)_𝑅𝑔(𝑡)𝑑𝑡,lim_𝜆→0𝑅𝜑(𝑡)²𝜆𝑔(𝜆𝑡)𝑑𝑡=_𝑘𝑗²𝑥_𝑘𝑅𝑔(𝑡)𝑑𝑡, (2.10) where the sum on the right is over all jump points 𝑥_𝑘 of 𝐹.

Theorems 3.2.3 and 3.3.4 in [2] are a special case of this corollary with 𝑔=(1/2)𝜒_[−1,1]. The proof of Corollary 2.4 is obtained by combining those theorems with our Theorem 2.2.

The second application is to the theory of almost-periodic functions. A complex-valued continuous function 𝑓 on 𝑅 is called almost-periodic if for each 𝜀>0 there exists 𝑙(𝜀)>0 such that each interval (𝑎,𝑎+𝑙(𝜀)) of length 𝑙(𝜀) contains at least one number 𝜏 for which sup_𝑡∈𝑅|𝑓(𝑡+𝜏)−𝑓(𝑡)|<𝜀. In the space of almost-periodic functions, the formula (𝜑,𝜓)=𝑀𝜑𝜓 defines a scalar product (𝜓 is a complex conjugate of 𝜓), and the numbers 𝛼(𝜆)=(𝜑(⋅),𝑒^𝑖𝜆) are called Fourier coefficients of 𝜑. It is proved that for each 𝜑 at most; a countable number of these coefficients are nonzero. Denoting them 𝛼(𝜆_𝑘),  𝑘=1,2,…,𝐻. Bohr's theorem states that 𝑀|𝜑|²=∑_𝑘|𝛼(𝜆_𝑘)|²; see [3, page 134]. This fact together with our Theorem 2.2 gives the next corollary.

Corollary 2.5. If 𝑔∈𝐿₁(𝑅) is even, then for any almost-periodic function 𝜑lim_𝜆→0𝑅𝜑(𝑡)²𝜆𝑔(𝜆𝑡)𝑑𝑡=_𝑘𝛼𝜆_𝑘²_𝑅𝑔(𝑡)𝑑𝑡. (2.11) Further, if ∫_𝑅𝑔(𝑡)𝑑𝑡>0, the formula (𝜑,𝜓)_𝑔=lim_𝜆→0∫_𝑅𝜑(𝑡)𝜓(𝑡)𝜆𝑔(𝜆𝑡)𝑑𝑡 defines a scalar product which is equivalent to (𝜑,𝜓).

Note that the first part of this corollary applies to characteristic functions of purely discrete distribution functions because from [2, Corollary  2 of Theorem  3.2.3] any such characteristic function is almost periodic. Now, we turn to the multidimensional version of (1.7). For the density, we assume a stronger condition than Condition 1(a).

Condition 2. 𝑔∈𝐿₁(𝑅^𝑛) is ‖⋅‖₂- spherically symmetric,  𝑔(𝑥)=𝑝(‖𝑥‖₂).

This assumption allows us to show that when some coordinates of 𝑥 are fixed, 𝑔(𝑥) as a function of the remaining coordinates satisfies Condition 1(a).

Now, we provide the intuition for the next condition. The stretching-out applied in (1.7) is described by the transformation 𝑠=𝐴(𝜆)𝑡 where 𝐴(𝜆)=𝜆𝑜01,𝜆>0. (2.12)𝑔_𝜆(𝑡)=det𝐴(𝜆)𝑔(𝐴(𝜆)𝑡) is the analog of 𝑔_𝜆(𝑡)=𝜆𝑔(𝜆𝑡) because ∫_𝑅2𝑔_𝜆(𝑠)𝑑𝑠=∫_𝑅2𝑔(𝑡)𝑑𝑡. Here, the matrix 𝐴(𝜆) has two eigenvalues. The limit lim_𝜆→0𝐴(𝜆)=𝐴(0) is a singular matrix because one of these eigenvalues tends to zero as 𝜆→0. Generalizing upon this situation and also thinking of applications to invariant tests, in the 𝑛-dimensional case we consider a symmetric nonnegative matrix 𝐴(𝜌) of size 𝑛×𝑛, where the parameter 𝜌 belongs to the segment [0,𝑎]. Denote its eigenvalues 0≤𝜆₁(𝜌)≤⋯≤𝜆_𝑛(𝜌) and let 𝐴(𝜌) be diagonalized as 𝐴(𝜌)=𝑃(𝜌)Λ(𝜌)𝑃^′(𝜌), where Λ(𝜌)=diag[𝜆₁(𝜌),…,𝜆_𝑛(𝜌)] and 𝑃(𝜌) is an orthogonal matrix. 𝐴(𝜌) degenerates at the right end of the segment [0,𝑎], owing to the following assumption.

Condition 3. The matrix 𝐴(𝜌) is positive definite for 0≤𝜌<𝑎.
The first 𝑑 eigenvalues tend to zero at the same rate, 𝜆_𝑗(𝜌)=𝜆₁(𝜌)(1+𝑜(1)),𝑗=1,…,𝑑;  𝜆₁(𝜌)→0 as 𝜌→𝑎; the remaining ones have positive limits: 𝜆_𝑗(𝜌)→𝜆_𝑗(𝑎)>0 as 𝜌→𝑎,𝑗=𝑑+1,…,𝑛.
The matrices 𝑃(𝜌) and Λ(𝜌) converge sufficiently quickly as 𝜌→𝑎. Namely, with the matrix Λ𝜌=diag𝜆₁𝜌,…,𝜆₁𝜌_𝑑times,𝜆_𝑑+1(𝑎),…,𝜆_𝑛(𝑎) (2.13) one has ‖Λ⁻¹(𝜌)Λ(𝜌)−𝐼‖=𝑜(𝜆₁(𝜌)),  ‖𝑃(𝜌)−𝑃(𝑎)‖=𝑜(𝜆₁(𝜌)),  𝜌→𝑎.

Note that Λ⁻¹(𝑎) does not exist because of part (b), but part (c) allows us to set Λ⁻¹(𝑎)Λ(𝑎)=𝐼 by continuity. In line with part (b), we use the partitions 𝑡𝑡=⁽¹⁾𝑡⁽²⁾,𝑡⁽¹⁾=𝑡₁,…,𝑡_𝑑^′,𝑡⁽²⁾=𝑡_𝑑+1,…,𝑡_𝑛^′Λ𝜌=Λ₁,𝜌00Λ₂𝜌,Λ₁𝜌is𝑑×𝑑,Λ₂𝜌is(𝑛−𝑑)×(𝑛−𝑑). (2.14)(2.15) Also, partition 𝑃(𝑎)=(𝑃₁(𝑎),𝑃₂(𝑎)) conformably with (2.14) (𝑃₁(𝑎) is 𝑛×𝑑 and 𝑃₂(𝑎) is 𝑛×(𝑛−𝑑)). Define a transformation 𝑇 by (𝑇𝜑)(𝑡)=𝜑(𝑃₁(𝑎)𝑡⁽¹⁾+𝑃₂(𝑎)Λ₂⁻¹(𝑎)𝑡⁽²⁾).  (𝑀_2,𝑑𝑇𝜑)(𝑡⁽²⁾) denotes the result of application of the generalized mean operator with respect to 𝑡⁽¹⁾, keeping 𝑡⁽²⁾ fixed: 𝑀_2,𝑑𝑇𝜑𝑡⁽²⁾=lim_𝑟→∞𝑚𝑇𝜑⋅,𝑡⁽²⁾,𝐵_2,𝑑(0,𝑟), (2.16) and 𝑔₁ denotes a marginal "density": 𝑔₁𝑡⁽²⁾=_𝑅𝑑𝑔𝑡⁽¹⁾,𝑡⁽²⁾𝑑𝑡⁽¹⁾. (2.17)

Theorem 2.6. Let Conditions 2 and 3 hold and let 𝜑∈𝐶𝐿(𝑅^𝑛) be such that the limit (2.16) exists for almost all 𝑡⁽²⁾∈𝑅^𝑛−𝑑. Then, lim_{𝜌→𝑎𝑅𝑛}𝜑(𝑡)det𝐴𝜌𝑔𝐴𝜌𝑡𝑑𝑡=_{𝑅𝑛−𝑑}𝑀_2,𝑑𝑇𝜑𝑡⁽²⁾𝑔₁𝑡⁽²⁾𝑑𝑡⁽²⁾. (2.18)

Next, we turn to the solution of Problem 2. Denoting 𝐴(𝜌)=Σ^−1/2(𝜌) and assuming that 𝐴(𝜌) is positive definite for 𝜌∈[0,𝑎), we see that condition (1.10) corresponds to the case 𝑑=1 of our Condition 3. The density (1.9) fits the framework of our Theorem 2.6 because the stretching-out is applied along one variable. Thus, in Theorem 2.7 we apply Theorem 2.6 to (a) characterize the limit in distribution of 𝑓_𝜌 in case 1≤rank(Σ⁻¹(𝑎))≤𝑛−1 and (b) show that (1.11) does not hold under the weak convergence. By implication, (1.11) is wrong if any convergence stronger than the weak one is considered (e.g., uniform, almost sure, in probability and in 𝐿_𝑝). Even though we use Theorem 2.6, the assumption on the density in part (b) of the next theorem is weaker than that in Theorem 2.6.

Condition 4. 𝑔 is a density on 𝑅^𝑛 bounded by an integrable, ‖⋅‖₂-spherically symmetric function, 𝑔(𝑥)≤𝑝(‖𝑥‖₂).

An example of such a density is 𝑔(𝑥)=𝑝(‖𝑥‖_𝑎), where 𝑝 is a nonnegative and nonincreasing function on [0,∞), which decays at infinity sufficiently quickly for 𝑔 to be integrable. By monotonicity of 𝑝 inequality (3.14) in the following implies that 𝑔(𝑥)=𝑝(‖𝑥‖_𝑎)≤𝑝(‖𝑥‖₂/𝑐₃).

Theorem 2.7. Let Condition 3 hold in which 𝐴(𝜌)=Σ^−1/2(𝜌).(a)Denote 𝑄=𝑃₂(𝑎)𝑃^′₂(𝑎) the projector onto the subspace spanned by the last 𝑛−𝑑 eigenvectors of 𝐴(𝑎) and let 𝑇₁ be defined by 𝑇₁𝜑(𝑡)=𝜑𝑃₁(𝑎)𝑡⁽¹⁾+𝑃₂(𝑎)Λ₂⁻¹(𝑎)𝑡⁽²⁾+𝑄𝑋𝛽. (2.19)𝑀_2,𝑑𝑇₁𝜑 is obtained by replacing 𝑇 with 𝑇₁ in (2.16). If Condition 2 holds and 𝜑∈𝐶𝐿(𝑅^𝑛) is such that (𝑀_2,𝑑𝑇₁𝜑)(𝑡⁽²⁾) exists for almost all 𝑡⁽²⁾∈𝑅^𝑛−𝑑, then lim_{𝜌→𝑎𝑅𝑛}𝜑(𝑡)𝑓_𝜌(𝑡)𝑑𝑡=_{𝑅𝑛−𝑑}𝑀_2,𝑑𝑇₁𝜑𝑡⁽²⁾𝑔₁𝑡⁽²⁾𝑑𝑡⁽²⁾, (2.20) with 𝑔₁ defined in (2.17).(b)If Condition 4 holds, then (1.11) cannot be true if the convergence in distribution is understood.

Remark 2.8. Because of the identity 𝑋𝛽+𝑁(𝐴(𝑎))=(𝐼−𝑄)𝑋𝛽+𝑄𝑋𝛽+𝑁(𝐴(𝑎))=𝑄𝑋𝛽+𝑁(𝐴(𝑎)), (1.11) correctly captures one feature of the limit distribution: it depends on 𝑋𝛽 only through 𝑄𝑋𝛽.

Before giving the solution to Problem 3, we need more notation and definitions. Obtaining a closed-form formula for 𝛽(Φ) involves a meticulous analysis of Φ based on the representation of Φ given in the following.

Let Im𝑋={𝑋𝛽∶𝛽∈𝑅^𝑘} be the image of 𝑋. If for a given set Φ⊂𝑅^𝑛 (a) the space 𝑅^𝑛 is represented as an orthogonal sum 𝑅^𝑛=𝐸₁⊕𝐸₂ of two subspaces 𝐸₁ and 𝐸₂ and (b) there is a set 𝐵⊂𝐸₁ such that Φ=𝐵+𝐸₂={𝑏+𝑒∶𝑏∈𝐵,𝑒∈𝐸₂}, then we say that Φ is a cylinder with the base 𝐵 and element 𝐸₂. For any set 𝑆⊂𝑅^𝑛, denote Γ𝑆={𝛾𝑠∶𝑠∈𝑆,𝛾∈𝑅,𝛾≠0}. Using 𝑠=1⋅𝑠, we see that 𝑆⊂Γ𝑆. We say that 𝑆 is cone-like if 𝑆=Γ𝑆. In particular, a cone-like set with each of its element 𝑠 contains its opposite −𝑠. The next representation is a stronger statement than saying that Φ defined in (1.14) is invariant with respect to transformations 𝑦→𝛾𝑦+𝑋𝛽,  𝛾∈𝑅,  𝛾≠0,  𝛽∈𝑅^𝑘.

Representation of Φ
The rejection region Φ for the Cliff-Ord test is a cylinder Φ=Γ𝑆+Im𝑋, (2.21) with a cone-like base 𝑀_𝑋Φ=Γ𝑆, where 𝑆={𝑠∈Im𝑀_𝑋∶‖𝑠‖₂=1,𝑠^′𝑊𝑠>𝑐}.

It is convenient to call an aperture the set 𝑆 in representation (2.21). The function 𝑠^′𝑊𝑠 is continuous on the unit sphere of Im𝑀_𝑋 and the set (𝑐,∞) is open. By the general property of continuous mappings [11, Chapter 2, § 5, Section 5, Theorem 6], the preimage of an open set under a continuous mapping is open. Thus, the aperture is an open set (in the relative topology) of the unit sphere of Im𝑀_𝑋. We need the notations of the interior int(Φ) (defined as the set of points of Φ that belong to Φ with some neighborhood), closure cl(Φ) (the set of all limit points of Φ) and boundary bd(Φ)=cl(Φ)⧵Φ.

Writing (1.12) in the form 𝛽_𝜌(Φ)=_𝑅𝑛𝜒_Φ𝑦𝑓_𝜌𝑦𝑑𝑦, (2.22) we see that Theorem 2.7(a) will be applicable if we manage to extend it from continuous Lipschitz functions to discontinuous functions of type 𝜒_Φ. This is done in Theorem 2.9. In Theorems 2.9 and 2.11, we assume that 𝑑=1. Up to the notation, this is the same assumption as (1.10). At least within our method, generalizations of the results in the following to the case 𝑑>1 are hard to obtain.

Let Condition 3 hold for 𝐴(𝜌)=Σ^−1/2(𝜌) and let 𝑑=1. Then, 𝜆₁(𝜌)→0 as 𝜌→𝑎 and all other eigenvalues have positive limits. Denote 𝑓₁,...,𝑓_𝑛 the orthonormal eigenvectors of 𝐴(𝑎)=Σ^−1/2(𝑎) corresponding to the eigenvalues 𝜆₁(𝑎)=0,𝜆₂(𝑎),…,𝜆_𝑛(𝑎). The partitions (2.14) become 𝑡⁽¹⁾=𝑡₁, 𝑡⁽²⁾=(𝑡₂,…,𝑡_𝑛)^′, 𝑃₁(𝑎)=𝑓₁, 𝑃₂(𝑎)=(𝑓₂,…,𝑓_𝑛). The vector 𝑧(𝑡⁽²⁾)=𝑃₂(𝑎)Λ₂⁻¹(𝑎)𝑡⁽²⁾+𝑄𝑋𝛽 will be called a shift because its role is to shift the line 𝑃₁(𝑎)𝑡⁽¹⁾+𝑃₂(𝑎)Λ₂⁻¹(𝑎)𝑡⁽²⁾+𝑄𝑋𝛽=𝑓₁𝑡₁+𝑧(𝑡⁽²⁾). In the next theorem, we extend (2.20) to 𝜑=𝜒_Φ with Φ described by (2.21). In the notation of marginal densities, the subscript will indicate the number of integrated-out variables. For example, 𝑔₁(𝑡⁽²⁾)=∫_𝑅𝑔(𝑡₁,𝑡⁽²⁾)𝑑𝑡₁,  𝑔_𝑘(𝑡)=∫_𝑅𝑘𝑝((𝑡²+‖𝑣‖²₂)^1/2)𝑑𝑣.

Theorem 2.9 (representation of 𝛽(Φ)). Let Conditions 2 and 3 hold with 𝑑=1.(a)If 𝑓₁∈Im𝑋 and ¹₀𝑔_𝑘(𝑡)𝑑𝑡<∞, (2.23) then 𝛽(Φ)=_𝑅𝑛−1𝑀𝑇₁𝜒_Φ𝑡⁽²⁾𝑔₁𝑡⁽²⁾𝑑𝑡⁽²⁾, (2.24) where 𝑇₁𝜒_Φ is defined by (𝑇₁𝜒_Φ)(𝑡)=𝜒_Φ(𝑓₁𝑡₁+𝑧(𝑡⁽²⁾)), and the generalized mean is applied over 𝑡₁.(b)If 𝑓₁∉Im𝑋 and ¹₀𝑔_𝑘+1(𝑡)𝑑𝑡<∞, (2.25) then (2.24) is true.

Remark 2.10. (a) As can be seen from the proof, the theorem holds for any test with the critical region satisfying (2.21). (b) Conditions (2.23) and (2.25) are technical assumptions that provide integrability in the neighborhood of the origin of marginal densities that arise in the course of the proof.

To avoid triviality, in the following theorem we assume that the inclusion ∅⊂Φ⊂𝑅^𝑛 is strict. This implies that in representation (2.21) the set Γ𝑆 is a nonempty proper subset of Im𝑀_𝑋.

Theorem 2.11. Let conditions of Theorem 2.9 hold, with (2.23) accompanying 𝑓₁∈Im𝑋 and (2.25) accompanying 𝑓₁∉Im𝑋. Besides, we require the function 𝑝 from Condition 2 to be positive on [0,∞).(1)If 𝑓₁∈Φ, then 𝛽(Φ)=1.(2)Suppose 𝑓₁∈bd(Φ).(2.1) If 𝑓₁∈Im𝑋, then 𝛽(Φ)∈(0,1).(2.2) If 𝑓₁∉Im𝑋, then examples can be presented such that 𝛽(Φ)=0, 𝛽(Φ)∈(0,1) or 𝛽(Φ)=1.(3)If 𝑓₁∈Φ₋, then 𝛽(Φ)=0.

Remark 2.12. Here, we compare this theorem with [4, Theorem 1].
(1.4) Our conditions on the density and critical region are much more restrictive. Our proof reveals the distinction between the cases 𝑓₁∈Im𝑋 and 𝑓₁∉Im𝑋. In particular, in case (2.2), Martellosio excludes the extreme values 𝛽(Φ)=0 and 𝛽(Φ)=1, while we provide counter examples showing that they are possible. Note also that we do not impose any conditions on the structure of 𝑊. What happens to the limiting power in case of 𝑊 that arises in practice needs additional investigation.
(1.5) Martellosio's proof is based on (1.11) which we disprove in Theorem 2.7(b). A series of other propositions from the same paper (see Lemmas D.2, D.3 and E.4, Corollary 1 and Propositions 1, 2 and 5), as well as from [7] (see Lemma 3.2, Theorems 3.3, 3.5 and 4.1, and Proposition 3.6), depend on (1.11) and need a revision. In particular, his claim that his results are true for any invariant critical region and any continuous density 𝑢 that is unimodal at the origin is unwarranted.
(1.7) Even if statement (1.11) were right, the proof of [4, Theorem 1] would be incomplete because it incorrectly uses the 𝑛-dimensional Lebesgue measure. Its use is inappropriate because for a degenerate density even one point may carry a positive mass. In our proof, we justify the use of (𝑛−1)-dimensional Lebesgue measure.

Now, we turn to the description of the alternative approach (solution to Problem 5). A mode is not a very good characteristic of a distribution when two unimodal densities with the same modes have very different spreads. A set of points where the density is close to its maximum might be a better characteristic in this case, at least for bell-shaped distributions. Let 0<𝜀<1 and let 𝑚=max𝑓(𝑥) be the maximum of a continuous density 𝑓(𝑥). We call 𝑀_𝜀={𝑥∶𝑓(𝑥)≥(1−𝜀)𝑚} an 𝜀-maximizing set of 𝑓. The idea of this density-maximizing approach is close to the maximum likelihood principle.

Suppose a decision 𝑑 is taken if the statistic 𝑥 belongs to a set 𝐷. In case of a favorable decision, 𝐷 is chosen in such a way that the probability 𝑃(𝑥∈𝐷) is high. The use of probability in this decision rule presumes that the statistic can be calculated repeatedly. However, in practice, especially in economics, the decision is based on just one value of the statistic in question. In such a case it may be preferable to choose 𝐷 so that min_𝑥∈𝐷𝑓(𝑥) is sufficiently close to 𝑚=max𝑓(𝑥). For the 𝜀-maximizing set, we have min_{𝑥∈𝑀𝜀}𝑓(𝑥)≥(1−𝜀)𝑚. Requiring 𝜀 to be close to 1 in the density-maximizing approach is similar to requiring 𝑃(𝑥∈𝐷) to be close to 1 in the probability-maximizing approach (although normally 𝑃(𝑥∈𝑀_𝜀)→0 as 𝜀→0).

As before, we assume that the matrix 𝐴(𝜌)=Σ^−1/2(𝜌) of size 𝑛×𝑛 is symmetric for 0≤𝜌≤𝑎 and positive definite for 0≤𝜌<𝑎. The idea is to impose conditions ensuring that 𝜀-maximizing sets are ellipsoids which in the limit give a set of the desired shape. This idea is realized through a delicate balance of the limit behavior of the eigenvalues and density contained in Conditions 5 through 7 and (2.32).

Condition 5. In the diagonal representation of 𝐴(𝜌), the orthogonal matrix 𝑃 satisfies 𝑃(𝑎)=lim_𝜌→𝑎𝑃𝜌, (2.26) the first 𝑑 eigenvalues vanish as power functions 𝜆_𝑗𝜌=𝑐_𝜆𝑎−𝜌^𝛼𝜆(1+𝑜(1)),𝑗=1,…,𝑑,as𝜆⟶𝑎, (2.27) with positive constants 𝑐_𝜆,𝛼_𝜆; the remaining eigenvalues tend to positive constants 𝜆_𝑗𝜌=𝜆_𝑗(𝑎)(1+𝑜(1)),𝑗=𝑑+1,…,𝑛,as𝜆⟶𝑎, (2.28) where 𝜆_𝑗(𝑎)>0.

Condition 6. The function 𝜀(𝜌) in the definition of the set 𝑀_𝜀(𝜌) vanishes as a power function 𝜀𝜌=𝑐_𝜀𝑎−𝜌^𝛼𝜀(1+𝑜(1)),𝜌⟶𝑎, (2.29) where 𝑐_𝜀,𝛼_𝜀 are positive constants.

Condition 7. The density 𝑔 is ‖⋅‖₂-spherically symmetric, 𝑔(𝑥)=𝑝(‖𝑥‖₂), where 𝑝 is continuous and monotonically decreasing on [0,∞) and such that 𝑝(𝑟)=𝑚1−𝑐_𝑝𝑟^𝛼𝑝1+𝑜𝑟^𝛼𝜀,𝑟⟶0, (2.30) where 𝑐_𝑝,𝛼_𝑝>0.

This assumption implies that 𝑚=max𝑔(𝑥)=𝑝(0), that the inverse function 𝑝⁻¹ is continuous and monotonically decreasing on (0,𝑚], and that 0=𝑝⁻¹(𝑚)=lim_𝑥→𝑚𝑝⁻¹(𝑥). (2.31)

Theorem 2.13. Denote 𝑁(𝐴(𝑎)) the null subspace of 𝐴(𝑎) (spanned by the eigenvectors corresponding to its zero eigenvalues). If Conditions 5–7 hold and 𝛼_𝜀𝛼_𝑝−𝛼_𝜆<0, (2.32) then the 𝜀(𝜌)-maximizing set 𝑀_𝜀(𝜌)={𝑥∶𝑓_𝜌(𝑥)≥(1−𝜀(𝜌))𝑚} converges to 𝑋𝛽+𝑁(𝐴(𝑎)) as 𝜌→𝑎.

3. Proofs

The idea of the proof is to approximate 𝑔 with a step function ℎ, prove the statement for ℎ and then pass to the limit to obtain the statement for 𝑔. A step function, by definition, is a finite linear combination of indicators of measurable sets. Due to the spherical symmetry of 𝑔, these sets turn out to be balls. For the method to work, the radii of the balls should be positive and finite. Approximation of 𝑝 by a continuous function in Step 1 is a trick to make sure that ℎ takes a finite number of values.

Step 1 (𝑝 can be assumed continuous on (0,∞)). Let Ω=(0,∞) and denote 𝑍(Ω) the space of measurable functions 𝑓 on Ω such that ‖𝑓‖_𝑍(Ω)=∫^∞₀|𝑓(𝜌)|𝜌^𝑑−1𝑑𝜌<∞.  𝑍(Ω) satisfies conditions of [12, Theorem 1]:(1)𝐶^∞₀(Ω)⊂𝑍(Ω)⊂𝐿₁^loc(Ω). (2)Minkowsky inequality: if 𝐴⊂𝑅^𝑚 is a measurable set and Φ(𝑥,𝑦) is a measurable on Ω×𝐴 function, then ‖∫_𝐴Φ(⋅,𝑦)𝑑𝑦‖_𝑍(Ω)≤∫_𝐴‖Φ(⋅,𝑦)‖_𝑍(Ω)𝑑𝑦.(3)A multiplication operator by a function 𝜑∈𝐶^∞₀(Ω) is bounded in 𝑍(Ω): ‖𝑓𝜑‖_𝑍(Ω)≤max|𝜑|‖𝑓‖_𝑍(Ω).(4)Any finite in Ω function 𝑓∈𝑍(Ω) is translation-continuous:  lim_ℎ→0‖𝑓(⋅+ℎ)−𝑓(⋅)‖_𝑍(Ω)=0.
The first three properties are standard facts of the theory of 𝐿_𝑝 spaces; the last one follows from the fact that if supp𝑓 is a compact subset of Ω, then the weight 𝜌^𝑑−1 in the definition of the norm of 𝑍(Ω) satisfies 𝑐₁≤𝜌^𝑑−1≤𝑐₂ on supp𝑓 and therefore for such an 𝑓 the norms ‖𝑓‖_𝑍(Ω) and ‖𝑓‖_𝐿1 are equivalent. Functions from 𝐿₁ are known to be translation-continuous.
By Burenkov’s theorem there exists a sequence {𝑝_𝑠}⊂𝐶^∞(Ω)∩𝑍(Ω) such that ‖𝑝_𝑠−𝑝‖_𝑍(Ω)→0. Defining 𝑔_𝑠(𝑥)=𝑝_𝑠(‖𝑥‖_𝑎), from (2.9) we have ‖𝑔_𝑠−𝑔‖_𝐿1=|𝑆|_𝑎‖𝑝_𝑠−𝑝‖_𝑍(Ω)→0. Hence, 𝜆^𝑑_𝑅𝑑𝜑_𝜆(𝑡)𝑔_𝑠(𝜆𝑡)𝑑𝑡−𝜆^𝑑_𝑅𝑑𝜑_𝜆(𝑡)𝑔(𝜆𝑡)𝑑𝑡≤sup𝜑𝜆^𝑑_𝑅𝑑𝑔_𝑠()(𝜑𝑔𝜆𝑡−𝑔𝜆𝑡)𝑑𝑡=sup_𝑠−𝑔_𝐿1⟶0 (3.1) uniformly in 𝜆. Thus, if we establish lim_𝜆→0𝜆^𝑑_𝑅𝑑𝜑_𝜆(𝑡)𝑔_𝑠(𝜆𝑡)𝑑𝑡=𝑀_𝑎,𝑑𝜑_0𝑅𝑑𝑔_𝑠(𝑡)𝑑𝑡∀𝑠, (3.2) then (2.8) will follow.

Step 2 (approximating 𝑔 with a Step Function). Take an arbitrary 𝜀>0. We can assume that 𝑝 is continuous on (0,∞).(a)Suppose ‖𝑔‖_𝐿1≥𝜀. We approximate 𝑔(𝑡) by a step function ℎ(𝑡), which will vanish where 𝑡 is large or small. By summability of 𝑔, there exist 0<𝑁₁=𝑁₁(𝜀)<𝑁₂=𝑁₂(𝜀)<∞ such that _{‖𝑡‖𝑎<𝑁1}𝑔(𝑡𝜀)𝑑𝑡≤3,_{‖𝑡‖𝑎>𝑁2}𝑔(𝑡𝜀)𝑑𝑡≤3 (3.3)𝑔 is uniformly continuous on the ring {𝑁₁≤‖𝑡‖_𝑎≤𝑁₂}. By Condition 1(a), for any natural 𝑚 we can find 𝛿>0 and split this ring into smaller rings 𝐴_𝑚,𝑙=𝑡∶𝑁₁+𝑙𝛿≤‖𝑡‖_𝑎<𝑁₁+()𝛿𝑁𝑙+1,𝑙=0,…,𝐿≡²−𝑁₁𝛿−1, (3.4) in such a way that 𝑔 in each ring is close to its value on the inner boundary: sup_{𝑡∈𝐴𝑚,𝑙}𝑔(𝑡)𝑙−𝑝𝑚≤1𝑚. (3.5) Put ℎ(𝑡)=0 if ‖𝑡‖_𝑎<𝑁₁ or ‖𝑡‖_𝑎>𝑁₂;  ℎ(𝑡)=𝑝(𝑙/𝑚) for 𝑡∈𝐴_𝑚,𝑙,  𝑙=0,…,𝐿. Combination of (3.3), (3.4), and (3.5) leads to ℎ−𝑔_𝐿1=_{‖𝑡‖𝑎<𝑁1}+_{‖𝑡‖𝑎>𝑁2}𝑔(𝑡)𝑑𝑡+^𝐿_{𝑙=0𝐴𝑚,𝑙}ℎ(𝑡)(𝑡≤−𝑔)𝑑𝑡2𝜀3+1𝑚mes𝐵_𝑎,𝑑0,𝑁₂=2𝜀3+𝑣_𝑎𝑁^𝑑₂(𝜀)𝑚. (3.6) Consequently, we can fix 𝑚=𝑚(𝜀) so that ℎ−𝑔_𝐿1≤𝜀. (3.7)(b)If ‖𝑔‖_𝐿1<𝜀, we just put ℎ≡0 to get (3.7).

Step 3 (replacing rings by balls in the representation of ℎ). Suppose ℎ is not identically zero. The sets 𝐴_𝑚,𝑙 in (3.4) are concentric rings with finite positive radii of the inner and outer boundaries, and by construction, ℎ is a finite linear combination of indicators of such rings. Hence, it can be written as ℎ=∑^𝐿_𝑙=0𝑎_𝑙(𝜒_{𝐵𝑎,𝑑(0,𝑟𝑙)}−𝜒_{𝐵𝑎,𝑑(0,𝑟𝑙−1)}), where 𝑎_𝑙 are some real numbers and the radii satisfy 𝑁₁=𝑟₀<⋯<𝑟_𝐿=𝑁₂. (3.8) Therefore, with some new constants 𝑏_𝑙, we can write ℎ as a linear combination of indicators of balls: ℎ=^𝐿_𝑙=0ℎ_𝑙,whereℎ_𝑙=𝑏_𝑙𝜒_{𝐵𝑎,𝑑(0,𝑟𝑙)}. (3.9)
If ℎ≡0, we put formally 𝑟₀=1,  𝑏₀=0,  𝐿=0. Note that 𝐿,  𝑏_𝑙 and 𝑟_𝑙 all depend on 𝜀 and that the constants 𝑏_𝑙 may deviate from the values of ℎ significantly.

Step 4 (introducing residuals for generalized means). Define the residual 𝑅₁(𝑟) by 𝑅₁(𝑟)=𝑚𝜑₀,𝐵_𝑎,𝑑(0,𝑟)/𝑀_𝑎,𝑑𝜑₀−1,if𝑀_𝑎,𝑑𝜑₀𝑚𝜑≠0,₀,𝐵_𝑎,𝑑(0,𝑟),if𝑀_𝑎,𝑑𝜑₀=0. (3.10) Then, 𝑚𝜑₀,𝐵_𝑎,𝑑()=𝑀0,𝑟^𝑎,𝑑𝜑₀𝑅₁(𝑟)+1,if𝑀_𝑎,𝑑𝜑₀𝑅≠0,₁(𝑟),if𝑀_𝑎,𝑑𝜑₀=0, (3.11) and in both cases lim_𝑟→∞𝑅₁(𝑟)=0. (3.12)
By the Lipschitz condition (2.6) and Condition 1(b), 𝜑(𝐹(𝜆)𝑡+𝐺(𝜆))−𝜑(𝐹(0)𝑡+𝐺(0))≤𝑐₁‖(𝐹(𝜆)−𝐹(0))𝑡+𝐺(𝜆)−𝐺(0)‖₂=𝑜(𝜆)‖𝑡‖_𝑎+𝑜(1)=𝑜(𝜆)𝑟+𝑜(1)for‖𝑡‖_𝑎≤𝑟. (3.13) Here, we have used the fact that on 𝑅^𝑑 any two norms are equivalent, so 𝑐₂‖𝑡‖_𝑎≤‖𝑡‖₂≤𝑐₃‖𝑡‖_𝑎, (3.14) with some 𝑐₂,𝑐₃>0. Equations (2.3) and (3.13) imply that 𝑚𝜑(𝐹(𝜆)⋅+𝐺(𝜆))−𝜑₀(⋅),𝐵_𝑎,𝑑(0,𝑟)=𝑜(𝜆)𝑟+𝑜(1)∀𝑟>0,as𝜆→0, (3.15) where 𝑜(𝜆) and 𝑜(1) do not depend on 𝑟. From this bound and (3.11), it follows that 𝑚𝜑(𝐹(𝜆)⋅+𝐺(𝜆)),𝐵_𝑎,𝑑()=𝑀0,𝑟^𝑎,𝑑𝜑₀𝑅₁(𝑟)+1+𝑅₂(𝜆,𝑟),if𝑀_𝑎,𝑑𝜑₀𝑅≠0,₁(𝑟)+𝑅₂(𝜆,𝑟),if𝑀_𝑎,𝑑𝜑₀=0, (3.16) where 𝑅₂(𝜆,𝑟) is a new residual satisfying 𝑅₂(𝜆,𝑟)=𝑜(𝜆)𝑟+𝑜(1)∀𝑟>0,as𝜆→0. (3.17)

Step 5 (proving (2.8) for the approximating function). Let 𝑀_𝑎,𝑑𝜑₀≠0. For one term in (3.9) by (2.2), and the first equation in (3.16) we have 𝜆^𝑑_𝑅𝑑𝜑(𝐹(𝜆)𝑡+𝐺(𝜆))ℎ_𝑙(𝜆𝑡)𝑑𝑡=𝜆^𝑑𝑏_{𝑙𝐵𝑎,𝑑0,𝑟𝑙/𝜆}𝜑(𝐹(𝜆)(𝜆=𝑣𝑡+𝐺))𝑑𝑡_𝑎𝑟^𝑑_𝑙𝑏_𝑙𝑣_𝑎𝑟_𝑙/𝜆^𝑑𝐵_𝑎,𝑑^0,𝑟_𝑙^/𝜆𝜑(𝐹(𝜆)𝑡+𝐺(𝜆))𝑑𝑡=_𝑅𝑑ℎ_𝑙(𝑠)𝑑𝑠𝑀_𝑎,𝑑𝜑₀𝑅₁𝑟_𝑙𝜆+1+𝑅₂𝑟𝜆,^𝑙𝜆. (3.18) Summation of these equations produces 𝜆^𝑑_𝑅𝑑𝜑(𝐹(𝜆)(𝜆ℎ()=𝑡+𝐺))𝜆𝑡𝑑𝑡_{𝐿𝑙=0𝑅𝑑}ℎ_𝑙(𝑠)𝑑𝑠𝑀_𝑎,𝑑𝜑₀𝑅₁𝑟_𝑙𝜆+1+𝑅₂𝑟𝜆,^𝑙𝜆=𝑀_𝑎,𝑑𝜑_0𝑅𝑑ℎ(𝑠)+𝑑𝑠_{𝐿𝑙=0𝑅𝑑}ℎ_𝑙(𝑠)𝑑𝑠𝑀_𝑎,𝑑𝜑₀𝑅₁𝑟_𝑙𝜆+𝑅₂𝑟𝜆,^𝑙𝜆. (3.19)
Here, by (3.8), (3.12), and (3.17), 𝑅₁(𝑟_𝑙/𝜆)→0,  𝑅₂(𝜆,(𝑟_𝑙/𝜆))=𝑜(𝑟_𝑙)+𝑜(1)=𝑜(1),  𝑙=0,…,𝐿, as 𝜆→0. From these relations and (3.19), we see that for the given 𝜀 there exists 𝜆(𝜀) such that 𝜆^𝑑_𝑅𝑑𝜑_𝜆(𝑡)ℎ(𝜆𝑡)𝑑𝑡−𝑀_𝑎,𝑑𝜑_0𝑅𝑑ℎ(𝑠)𝑑𝑠≤𝜀for0<𝜆≤𝜆(𝜀). (3.20)
In case 𝑀_𝑎,𝑑𝜑₀=0, the only difference consists in application of the second equation in (3.16). The conclusion is (3.20) with 𝑀_𝑎,𝑑𝜑₀=0.

Step 6 (proving (2.8)). Using (3.7) and (3.20), we have 𝜆^𝑑_𝑅𝑑𝜑_𝜆(𝑡)𝑔(𝜆𝑡)𝑑𝑡−𝑀_𝑎,𝑑𝜑_0𝑅𝑑𝑔(𝑡)≤𝜆𝑑𝑡_𝑑𝑅𝑑𝜑_𝜆(𝑡𝑔()(+𝜆)𝜆𝑡−ℎ𝜆𝑡)𝑑𝑡_𝑑𝑅𝑑𝜑_𝜆(𝑡)ℎ(𝜆𝑡)𝑑𝑡−𝑀_𝑎,𝑑𝜑_0𝑅𝑑ℎ(𝑡)+𝑀𝑑𝑡_𝑎,𝑑𝜑_0𝑅𝑑ℎ(𝑡)(𝑡≤𝜑−𝑔)𝑑𝑡_𝐶𝑔−ℎ_𝐿1+𝜀+𝑀_𝑎,𝑑𝜑₀𝑔−ℎ_𝐿1≤𝜑_𝐶+1+𝑀_𝑎,𝑑𝜑₀𝜀,0<𝜆≤𝜆(𝜀). (3.21) Equation (2.8) follows because 𝜀>0 is arbitrary.

3.1. Proof of Theorem 2.6

Denote 𝐽_𝜌(𝜑)=∫_𝑅𝑛𝜑(𝑡)det𝐴(𝜌)𝑔(𝐴(𝜌)𝑡)𝑑𝑡.

Step 1 (replacing lim𝐽_𝜌(𝜑) by an equivalent limit). Using constancy of 𝑔 on spheres (Condition 2), write 𝐽_𝜌(𝜑) as 𝐽_𝜌𝜑=_𝑅𝑛𝜑𝑦detΛ𝜌𝑔𝑃𝜌Λ𝜌𝑃^′𝜌𝑦𝜌𝑑𝑦=detΛ_𝑅𝑛𝜑𝑦𝑔Λ𝜌𝑃^′𝜌𝑦𝜌𝑑𝑦=detΛ_𝑅𝑛𝜑𝑦𝑔Λ𝜌Λ⁻¹𝜌Λ𝜌𝑃^′𝜌𝑦𝑑𝑦. (3.22) Denote 𝐻⁻¹(𝜌)=Λ⁻¹(𝜌)Λ(𝜌)𝑃^′(𝜌) and replace 𝐻⁻¹(𝜌)𝑦=𝑡 to get 𝐽_𝜌𝜑=detΛ𝜌det𝐻𝜌_𝑅𝑛𝜑𝐻𝜌𝑡𝑔Λ𝜌𝑡𝑑𝑡. (3.23) By Condition 3(b,c), detΛ𝜌=𝜆^𝑑₁𝜌detΛ₂(𝑎(1,𝜌Λ)(1+𝑜))det𝐻=det⁻¹𝜌Λ𝜌det𝑃^′𝜌=1+𝑜(1). (3.24) From (3.23) and (3.24) we deduce 𝐽_𝜌𝜑=𝜆^𝑑₁𝜌detΛ₂(𝑎)(1+𝑜(1))_𝑅𝑛𝜑𝐻𝜌𝑡𝑔Λ𝜌𝑡𝑑𝑡=(1+𝑜(1))×detΛ₂(𝑎)_{𝑅𝑛−𝑑}𝜆^𝑑₁𝜌_𝑅𝑑𝜑𝐻𝜌𝑡𝑔𝜆₁𝜌𝑡⁽¹⁾,Λ₂(𝑎)𝑡⁽²⁾𝑑𝑡⁽¹⁾𝑑𝑡⁽²⁾. (3.25)

Step 2 (changing the limit variable). Let us have a closer look at the expression in the square brackets in (3.25): 𝐽_𝜌𝜑,𝑡⁽²⁾≡𝜆^𝑑₁𝜌_𝑅𝑑𝜑𝐻𝜌𝑡𝑔𝜆₁𝜌𝑡⁽¹⁾,Λ₂(𝑎)𝑡⁽²⁾𝑑𝑡⁽¹⁾. (3.26) Since Λ⁻¹(𝑎)Λ(𝑎)=𝐼, here 𝐻(𝑎)=𝑃(𝑎), and by Condition 3(c), 𝐻𝜌−𝑃(𝑎)=𝑃𝜌Λ⁻¹𝜌Λ𝜌𝜌𝜌(𝑎≤𝑃𝜌Λ−𝑃+𝑃−𝑃)₋₁𝜌Λ𝜌+𝑃𝜌(𝑎𝜆−𝐼−𝑃)=𝑜₁𝜌,𝜌⟶𝑎. (3.27) Considering 𝜆=𝜆₁ the independent variable, take 𝜌=𝜌(𝜆) as its function. The limit relation 𝜌→𝑎 translates to 𝜆→0, and (3.26) and (3.27) become, 𝐽_𝜌(𝜆)𝜑,𝑡⁽²⁾=𝜆^𝑑_𝑅𝑑𝜑𝐻𝜌(𝜆)𝑡𝑔𝜆𝑡⁽¹⁾,Λ₂(𝑎)𝑡⁽²⁾𝑑𝑡⁽¹⁾,𝐻𝜌(𝜆)−𝑃(𝑎)=𝑜(𝜆),𝜆⟶0, (3.28)(3.29) respectively.

Step 3 (proving a preliminary version of (2.18)). Now we check that Theorem 2.2 applies to (3.28) for any fixed 𝑡⁽²⁾. Letting 𝑝(𝑧)=𝑝𝑧²+Λ₂(𝑎)𝑡⁽²⁾²₂^1/2,𝑧≥0,𝑔𝑡⁽¹⁾=𝑔𝑡⁽¹⁾,Λ₂(𝑎)𝑡⁽²⁾, (3.30) we see that 𝑔∈𝐿₁(𝑅^𝑑) is ‖⋅‖₂-spherically symmetric: 𝑔𝑡⁽¹⁾=𝑔𝑡⁽¹⁾,Λ₂(𝑎)𝑡⁽²⁾=𝑝𝑡⁽¹⁾²₂+Λ₂(𝑎)𝑡⁽²⁾²₂^1/2=𝑝𝑡⁽¹⁾₂. (3.31) Condition 1(a) is satisfied. As for Condition 1(b), partition 𝐻=(𝐻₁,𝐻₂) conformably with (2.14) and put 𝐹(𝜆)=𝐻₁(𝜌(𝜆)),  𝐺(𝜆)=𝐻₂(𝜌(𝜆))𝑡⁽²⁾,  𝐹(0)=𝐻₁(𝑎)=𝑃₁(𝑎),  𝐺(0)=𝑃₂(𝑎)𝑡⁽²⁾. Then, 𝐻(𝜌(𝜆))𝑡=𝐻₁(𝜌(𝜆))𝑡⁽¹⁾+𝐻₂(𝜌(𝜆))𝑡⁽²⁾=𝐹(𝜆)𝑡⁽¹⁾+𝐺(𝜆), and Condition 1(b) follows from (3.29).
By Theorem 2.2, 𝐽_𝜌(𝜆)𝜑,𝑡⁽²⁾⟶𝐽_𝜌(0)𝜑,𝑡⁽²⁾≡𝑀_2,𝑑𝜑𝑡⁽²⁾_𝑅𝑑𝑔𝑡⁽¹⁾,Λ₂(𝑎)𝑡⁽²⁾𝑑𝑡⁽¹⁾,𝑡⁽²⁾∈𝑅^𝑛−𝑑, (3.32) where (𝑀_2,𝑑𝜑)(𝑡⁽²⁾) is obtained by applying 𝑀_2,𝑑 to 𝜑(𝑡)≡𝜑(𝑃₁(𝑎)𝑡⁽¹⁾+𝑃₂(𝑎)𝑡⁽²⁾) with respect to 𝑡⁽¹⁾. Functions (3.28) are bounded by an integrable function of 𝑡⁽²⁾: 𝐽_𝜌(𝜆)𝜑,𝑡⁽²⁾≤𝜆^𝑑𝜑_𝐶𝑅𝑑𝑔𝜆𝑡⁽¹⁾,Λ₂(𝑎)𝑡⁽²⁾𝑑𝑡⁽¹⁾=𝜑_𝐶𝑅𝑑𝑔𝑡⁽¹⁾,Λ₂(𝑎)𝑡⁽²⁾𝑑𝑡⁽¹⁾. (3.33) By the dominated convergence theorem, (3.32) and (3.33) imply that _{𝑅𝑛−𝑑}𝐽_𝜌(𝜆)𝜑,𝑡⁽²⁾𝑑𝑡⁽²⁾⟶_{𝑅𝑛−𝑑}𝐽_𝜌(0)𝜑,𝑡⁽²⁾𝑑𝑡⁽²⁾,𝜆→0. (3.34) Now, we combine (3.25), (3.28), (3.32), and (3.34) to get lim_𝜌→𝑎𝐽_𝜌𝜑=detΛ₂(𝑎)_{𝑅𝑛−𝑑}𝑀_2,𝑑𝜑𝑡⁽²⁾_𝑅𝑑𝑔𝑡⁽¹⁾,Λ₂(𝑎)𝑡⁽²⁾𝑑𝑡⁽¹⁾𝑑𝑡⁽²⁾=_{𝑅𝑛−𝑑}𝑀_2,𝑑𝜑Λ₂⁻¹(𝑎)𝑠⁽²⁾_𝑅𝑑𝑔𝑡⁽¹⁾,𝑠⁽²⁾𝑑𝑡⁽¹⁾𝑑𝑠⁽²⁾=_{𝑅𝑛−𝑑}𝑀_2,𝑑𝑇𝜑𝑠⁽²⁾𝑔₁𝑠⁽²⁾𝑑𝑠⁽²⁾ (3.35) because 𝑀_2,𝑑𝜑Λ₂⁻¹(𝑎)𝑠⁽²⁾=lim_𝑟→∞𝑚𝜑𝑃₁(𝑎)⋅+𝑃₂(𝑎)Λ₂⁻¹(𝑎)𝑠⁽²⁾,𝐵_2,𝑑(0,𝑟)=𝑀_2,𝑑𝑇𝜑𝑠⁽²⁾. (3.36)

3.2. Proof of Theorem 2.7

3.2.1. Part (a)

Step 1 (𝑀_𝑎,𝑑𝜑 does not depend on the center of balls). Let us make definition (2.4) more precise: (𝑀_𝑎,𝑑𝜑)(𝑥)=lim_𝑟→∞𝑚(𝜑,𝐵_𝑎,𝑑(𝑥,𝑟)),  𝑥∈𝑅^𝑑. Here, we prove that 𝑀_𝑎,𝑑𝜑(𝑥)=𝑀_𝑎,𝑑𝜑(0)forany𝑥∈𝑅^𝑑. (3.37) For 𝑟≥‖𝑥‖_𝑎, one has implications ‖𝑦‖_𝑎≤𝑟−‖𝑥‖_𝑎⇒‖𝑦−𝑥‖_𝑎≤𝑟⇒‖𝑦‖_𝑎≤‖𝑦−𝑥‖_𝑎+‖𝑥‖_𝑎≤𝑟+‖𝑥‖_𝑎. Geometrically, they mean the inclusion 𝐵_𝑎,𝑑(0,𝑟−‖𝑥‖_𝑎)⊂𝐵_𝑎,𝑑(𝑥,𝑟)⊂𝐵_𝑎,𝑑(0,𝑟+‖𝑥‖_𝑎). This inclusion implies that 𝑟−‖𝑥‖_𝑎𝑟^𝑑𝑚𝜑,𝐵_𝑎,𝑑0,𝑟−‖𝑥‖_𝑎=𝑟−‖𝑥‖_𝑎𝑟^𝑑1𝑣_𝑎𝑟−‖𝑥‖_𝑎^𝑑𝐵_𝑎,𝑑^{(0,𝑟−‖𝑥‖}_𝑎⁾𝜑𝑦≤1𝑑𝑦𝑣_𝑎𝑟^𝑑_{𝐵𝑎,𝑑(𝑥,𝑟)}𝜑𝑦𝑑𝑦=𝑚𝜑,𝐵_𝑎,𝑑()≤‖𝑥‖𝑥,𝑟𝑟+_𝑎𝑟^𝑑1𝑣_𝑎𝑟+‖𝑥‖_𝑎^𝑑𝐵_𝑎,𝑑^{(0,𝑟+‖𝑥‖}_𝑎⁾𝜑𝑦=‖𝑥‖𝑑𝑦𝑟+_𝑎𝑟^𝑑𝑚𝜑,𝐵_𝑎,𝑑0,𝑟+‖𝑥‖_𝑎. (3.38) It remains to let 𝑟→∞ to prove (3.37).

Step 2 (proving (2.20)). If 𝜑∈𝐶𝐿(𝑅^𝑛), then 𝜑(⋅+𝑋𝛽)∈𝐶𝐿(𝑅^𝑛). By (2.16), we need to look at 𝑀_2,𝑑𝑇𝜑⋅+𝑋𝛽𝑡⁽²⁾=lim_𝑟→∞1𝑣_𝑎𝑟^𝑑_{𝐵2,𝑑(0,𝑟)}𝜑𝑃₁(𝑎)𝑡⁽¹⁾+𝑃₂(𝑎)Λ₂⁻¹(𝑎)𝑡⁽²⁾+𝑋𝛽𝑑𝑡⁽¹⁾. (3.39) Using 𝐼=𝑃(𝑎)𝑃^′(𝑎)=𝑃₁(𝑎)𝑃^′₁(𝑎)+𝑄, we see that 𝑃₁(𝑎)𝑡⁽¹⁾+𝑋𝛽=𝑃₁(𝑎)𝑡⁽¹⁾+𝑃₁(𝑎)𝑃^′₁(𝑎)𝑋𝛽+𝑄𝑋𝛽=𝑃₁(𝑎)𝑡⁽¹⁾+𝑃^′₁(𝑎)𝑋𝛽+𝑄𝑋𝛽. (3.40) Therefore, replacing 𝑡⁽¹⁾+𝑃^′₁(𝑎)𝑋𝛽=𝑠⁽¹⁾, we can continue (3.39) as 𝑀_2,𝑑𝑇𝜑⋅+𝑋𝛽𝑡⁽²⁾=lim_𝑟→∞1𝑣_𝑎𝑟^𝑑_{𝐵2,𝑑(𝑃}^′_{1(𝑎)𝑋𝛽,𝑟)}𝜑𝑃₁(𝑎)𝑠⁽¹⁾+𝑃₂(𝑎)Λ₂⁻¹(𝑎)𝑡⁽²⁾+𝑄𝑋𝛽𝑑𝑠⁽¹⁾. (3.41) Using the definition of 𝑇₁ and (3.37), we see that this is the same as (𝑀_2,𝑑𝑇₁𝜑)(𝑡⁽²⁾). Hence, the existence of (𝑀_2,𝑑𝑇𝜑(⋅+𝑋𝛽))(𝑡⁽²⁾) is equivalent to that of (𝑀_2,𝑑𝑇₁𝜑)(𝑡⁽²⁾). Application of Theorem 2.6 proves (2.20).

3.2.2. Part (b)

Step 1 (the generalized mean vanishes on 𝑝-integrable functions). If 𝜑∈𝐿_𝑝(𝑅^𝑑) with some 𝑝∈[1,∞), then by Hölder’s inequality 𝑚𝜑,𝐵_𝑎,𝑑()≤10,𝑟𝑣_𝑎𝑟^𝑑_{𝐵𝑎,𝑑(0,𝑟)}𝜑(𝑡≤1)𝑑𝑡𝑣_𝑎𝑟^𝑑mes𝐵_𝑎,𝑑(0,𝑟)^1−1/𝑝_{𝐵𝑎,𝑑(0,𝑟)}𝜑(𝑡)^𝑝𝑑𝑡^1/𝑝=𝑣_𝑎𝑟^{𝑑−1/𝑝}𝜑_𝐿𝑝⟶0,𝑟⟶∞. (3.42)

Step 2 (lim𝐽_𝜌(𝜑) vanishes on functions with compact support). By Condition 4, _𝑅𝑛𝜑(𝑡)𝑓_𝜌(𝑡)𝑑𝑡≤_𝑅𝑛𝜑(𝑡)𝑓_𝜌(𝑡)𝑑𝑡, (3.43) where 𝜑(𝑡)=|𝜑(𝑡)| and 𝑓_𝜌 is obtained from (1.9) by replacing 𝑔(𝑥) with 𝑔(𝑥)=𝑝(‖𝑥‖₂). The function 𝜑 satisfies the Lipschitz condition, |𝜑(𝑥)−𝜑(𝑦)|=||𝜑(𝑥)|−|𝜑(𝑦)||≤𝑐₁‖𝑥−𝑦‖₂, and 𝑔 satisfies Condition 2. Part (a) of this theorem allows us to use (2.20) with 𝜑 in place of 𝜑 and 𝑔 instead of 𝑔 whenever (𝑀_2,𝑑𝑇₁𝜑)(𝑡⁽²⁾) exists for almost all 𝑡⁽²⁾∈𝑅^𝑛−𝑑.
Let 𝐶𝐿₀(𝑅^𝑛) denote the set of functions 𝜑∈𝐶𝐿(𝑅^𝑛) with compact support supp𝜑. Suppose 𝜑∈𝐶𝐿₀(𝑅^𝑛). By Step 1, (𝑀_2,𝑑𝑇₁𝜑)(𝑡⁽²⁾)≡0. Equations (2.20) and (3.43) imply that lim_{𝜌→𝑎𝑅𝑛}𝜑(𝑡)𝑓_𝜌(𝑡)𝑑𝑡=0,𝜑∈𝐶𝐿₀𝑅^𝑛. (3.44) Suppose (1.11) is true in the weak sense or, more generally, the family {𝑓_𝜌∶0≤𝜌<𝑎} converges in distribution, as 𝜌→𝑎, to a probabilistic measure 𝜇 supported on some set 𝑆⊂𝑅^𝑛. Then by (2.7) ∫_𝑅𝑛𝜑(𝑡)𝑓_𝜌(𝑡)𝑑𝑡→∫_𝑆𝜑𝑑𝜇,𝜑∈𝐶𝐿(𝑅^𝑛). This equation and (3.44) imply ∫_𝑆𝜑𝑑𝜇=0,𝜑∈𝐶𝐿₀(𝑅^𝑛). Taking here 𝜑∈𝐶𝐿₀(𝑅^𝑛) such that 𝜑≥0 and 𝜑=1 on 𝐵_2,𝑛(0,𝑟) we have ∫_{𝑆∩𝐵2,𝑛(0,𝑟)}𝑑𝜇=0 for any 𝑟>0. Thus, 𝜇(𝑆)=0 which is impossible.

3.3. Proof of Theorem 2.9

The proof of Theorem 2.9 will be preceded by a series of lemmas. The main difficulty consists in estimating integrals of the marginal density 𝑔₁ over cylindrical sets of an infinite Lebesgue measure.

3.3.1. Preliminary Results

Lemma 3.1. For the rejection region Φ defined in (1.14), representation (2.21) holds.

Proof. Since 𝐿_𝑋,  𝑀_𝑋 are complementary orthoprojectors, the images Im𝑀_𝑋 and Im𝐿_𝑋=Im𝑋 are orthogonal to one another and 𝑅^𝑛=Im𝑀_𝑋⊕Im𝑋. Representing 𝑢∈Φ as 𝑢=𝐿_𝑋𝑢+𝑀_𝑋𝑢 and using 𝑀^′_𝑋=𝑀_𝑋,  𝑀²_𝑋=𝑀_𝑋 we get 𝑢^′𝑀_𝑋𝑊𝑀_𝑋𝑢𝑢^′𝑀_𝑋𝑢=𝑀_𝑋𝑢^′𝑊𝑀_𝑋𝑢𝑀_𝑋𝑢^′𝑀_𝑋𝑢=𝑣^′𝑊𝑣𝑣^′𝑣, (3.45) where we have denoted 𝑣=𝑀_𝑋𝑢. Hence, only the component 𝑣∈𝑀_𝑋Φ determines whether 𝑢 belongs to Φ or not, while the component 𝐿_𝑋𝑢∈Im𝑋 is arbitrary. This proves the equation Φ=𝑀_𝑋Φ+Im𝑋. Finally, the equation 𝑀_𝑋Φ=Γ𝑆 follows by homogeneity from (3.45).

Now we review a few facts about the Moore-Penrose inverse in a form adapted to our needs. Let 𝐴 be a matrix of size 𝑚×𝑛, not necessarily square, and denote its rank 𝑟=rank𝐴. Its Moore-Penrose inverse 𝐴⁺ is of size 𝑛×𝑚.

Lemma 3.2 (see, [13, pages 24-25]). 𝐴⁺ is unique and has the following properties.(a)𝐴𝐴⁺𝐴=𝐴 (𝐴⁺ is a generalized inverse of 𝐴).(b)𝐴⁺𝐴𝐴⁺=𝐴⁺ (𝐴 is a generalized inverse of 𝐴⁺).(c)The matrix 𝐻=𝐴⁺𝐴 is an orthoprojector, 𝐻²=𝐻, 𝐻^′=𝐻, and rank𝐻=rank𝐴⁺=𝑟.(d)For a given 𝑦∈Im𝐴, a general solution of the equation 𝐴𝑥=𝑦 is given by 𝑥=𝐴⁺𝑦+(𝐼−𝐻)𝑧, (3.46)where 𝑧∈𝑅^𝑛 is arbitrary. (Rao writes (𝐻−𝐼)𝑧 instead of (𝐼−𝐻)𝑧, but our form is more convenient).

Regarding (3.46), we can remark the following.(i)The two terms on the right are orthogonal because by (b) and (c) 𝐴⁺𝑦,(𝐼−𝐻)𝑧=𝐻𝐴⁺𝑦,(𝐼−𝐻)𝑧=𝐴⁺𝑦,𝐻−𝐻²𝑧=0. (3.47) Here and in the sequel, (⋅,⋅) stands for the scalar product.(ii)Instead of writing the second term in the form (𝐼−𝐻)𝑧,  𝑧∈𝑅^𝑛, we can equivalently write it as 𝑣∈Im(𝐼−𝐻).(iii)𝐴⁺𝑦=0 implies that 𝑦=0. This is because if 𝐴⁺𝑦=0 and 𝑦=𝐴𝑥, then by (a) 𝑦=𝐴𝑥=𝐴𝐴⁺(𝐴𝑥)=𝐴(𝐴⁺𝑦)=0.

Recalling that a general solution of the equation 𝐴𝑥=𝑦 is, by definition, the preimage 𝐴⁻¹𝑦, we state our conclusion in a lemma.

Lemma 3.3. Let 𝑌⊆Im𝐴. The preimage 𝐴⁻¹𝑌 is a cylinder 𝐴⁺𝑌+Im(𝐼−𝐻) with the base 𝐴⁺𝑌⊆Im𝐻 and element Im(𝐼−𝐻). Further, 𝐴⁺𝑦=0 for 𝑦∈Im𝐴 implies that 𝑦=0, that is, 𝐴⁺ on Im𝐻 is one-to-one.

Lemma 3.4. Suppose 𝑓₁∈Im𝑋. Fix a set 𝑌⊆Im𝑀_𝑋 and a vector 𝜏∈Im𝑀_𝑋. Denote 𝐴=𝑀_𝑋𝑃₂(𝑎)Λ₂⁻¹(𝑎) and let 𝐴⁺ and 𝐻 be the accompanying matrices from Lemma 3.2. Then, the set 𝑇⁽²⁾ defined by 𝑇⁽²⁾={𝑡⁽²⁾∈𝑅^𝑛−1∶𝐴𝑡⁽²⁾+𝜏∈𝑌} is a cylinder: 𝑇⁽²⁾=𝐴⁺(𝑌−𝜏)+Im(𝐼−𝐻). (3.48)
Suppose 𝑓₁∉Im𝑋 and denote 𝑝=𝑀_𝑋𝑓₁/‖𝑀_𝑋𝑓₁‖₂. Then 𝑀_𝑋−𝑝𝑝^′ is an orthoprojector with the image Im𝑀_𝑋−𝑝𝑝^′=𝑥∈Im𝑀_𝑋∶𝑥⟂𝑝. (3.49) In part (a) replace 𝑌,𝜏,𝐴 by 𝑌⊆Im(𝑀_𝑋−𝑝𝑝^′), 𝜏∈Im(𝑀_𝑋−𝑝𝑝^′), and 𝐴=(𝑀_𝑋−𝑝𝑝^′)𝑃₂(𝑎)Λ₂⁻¹(𝑎), respectively. Then the set 𝑇⁽²⁾ has representation (3.48).

Proof. Let us show that 𝑓₁∈Im𝑋 implies that Im𝐴=Im𝑀_𝑋. (3.50) The inclusion Im𝐴⊂Im𝑀_𝑋 is obvious. Conversely, let 𝑥∈Im𝑀_𝑋. Then, 𝑥=𝑀_𝑋𝑥. Since 𝑓₁,…,𝑓_𝑛 is an orthonormal basis, we have 𝑥=∑^𝑛_𝑖=1(𝑥,𝑓_𝑖)𝑓_𝑖. Denoting 𝑡⁽²⁾=((𝑥,𝑓₂),…,(𝑥,𝑓_𝑛))^′, it follows that 𝑥=𝑀_𝑋𝑥=^𝑛_𝑖=1𝑥,𝑓_𝑖𝑀_𝑋𝑓_𝑖=^𝑛_𝑖=2𝑥,𝑓_𝑖𝑀_𝑋𝑓_𝑖=𝑀_𝑋𝑓₂,…,𝑀_𝑋𝑓_𝑛𝑡⁽²⁾=𝑀_𝑋𝑃₂(𝑎)𝑡⁽²⁾=𝐴Λ₂(𝑎)𝑡⁽²⁾∈Im𝐴. (3.51) We have proved (3.50) which implies 𝑌−𝜏⊂Im𝐴. Application of Lemma 3.3 proves (3.48).
It is easy to check that 𝑀_𝑋−𝑝𝑝^′ is an orthoprojector and its image satisfies (3.49). From (3.49), we conclude that 𝑀_𝑋 and 𝑀_𝑋−𝑝𝑝^′ commute and 𝑀_𝑋−𝑝𝑝^′𝑀_𝑋=𝑀_𝑋−𝑝𝑝^′, (3.52) see [14, Chapter VI, Section 3, Lemma 2(a)]. Applying 𝑀_𝑋−𝑝𝑝^′ to both sides of (3.50) and using the last equation we get Im𝑀_𝑋−𝑝𝑝^′𝑃₂(𝑎)Λ₂⁻¹(𝑎)=Im𝑀_𝑋−𝑝𝑝^′𝑀_𝑋𝑃₂(𝑎)Λ₂⁻¹(𝑎=𝑀_𝑋)−𝑝𝑝^′Im𝑀_𝑋𝑃₂(𝑎)Λ₂⁻¹(𝑎=𝑀_𝑋)−𝑝𝑝^′Im𝑀_𝑋=Im𝑀_𝑋−𝑝𝑝^′. (3.53) We have proved an analog of (3.50) for the matrix under consideration: Im𝐴=Im𝑀_𝑋−𝑝𝑝^′. (3.54) As previously mentioned, application of Lemma 3.3 proves (3.48).

Lemma 3.5. Let 𝑓₁,𝜏,𝐴,𝐴⁺ and 𝐻 be the same as in Lemma 3.4(a).(a)Suppose 𝑌⊆Im𝑀_𝑋 is of the form 𝑌=ΓΔ where Δ⊆{𝑠∈Im𝑀_𝑋∶‖𝑠‖₂=1}. If (2.23) holds, then for the set 𝑇⁽²⁾ from (3.48) one has 𝛼(𝑌,𝜏)≡_𝑇(2)𝑔₁𝑡⁽²⁾𝑑𝑡⁽²⁾≤𝑐‖𝜏‖₂mes(Δ). (3.55)(b)If 𝑌⊆{𝑠∈Im𝑀_𝑋∶‖𝑠‖₂≤1}, then 𝛼(𝑌,𝜏)→0,as mes(𝑌)→0, uniformly in 𝜏.

Proof. Part (a)
Step  1 (proving an auxiliary bound). Denote 𝛽(𝑌,𝜏)=_{𝐴+(𝑌−𝜏)}𝑔_𝑘‖𝑢‖₂𝑑𝑢. (3.56) We intend to prove that 𝛽(ΓΔ,𝜏)≤𝑐‖𝜏‖₂mes(Δ). (3.57) By Lemma 3.2, (3.50) implies that 𝑟≡rank𝐴=𝑛−𝑘. (3.58) By Lemma 3.3, 𝐴⁺ is a one-to-one mapping from Im𝐴 to Im𝐴⁺, both of which are of dimension 𝑟. We use this fact twice.
Firstly, selecting two bases, one in Im𝐴 and another in Im𝐴⁺, we can identify 𝐴⁺ with its matrix representation in these bases. This matrix is square of order 𝑟 and rank 𝑟, so the Jacobian 𝑐₁ of the transformation 𝑢=𝐴⁺𝑣,  𝑣∈ΓΔ−𝜏, is positive. Therefore, 𝛽(ΓΔ,𝜏)=𝑐_1ΓΔ−𝜏𝑔_𝑘𝐴⁺𝑣₂𝑑𝑣=𝑐_1ΓΔ𝑔_𝑘𝐴⁺(𝑤−𝜏)₂𝑑𝑤. (3.59)
Secondly, ‖𝐴⁺𝑣‖₂=0 implies that 𝐴⁺𝑣=0 which for 𝑣∈Im𝐴 implies that 𝑣=0. Thus, ‖𝐴⁺𝑣‖₂ is a norm on Im𝐴 and by (3.14), 𝐴⁺(𝑤−𝜏)₂≥𝑐₂‖𝑤−𝜏‖₂≥max𝑐₂‖𝑤‖₂−‖𝜏‖₂,0,𝑤∈ΓΔ. (3.60)
As 𝑔 is assumed nonincreasing, 𝑔_𝑘 is also nonincreasing, and (3.59) leads to 𝛽(ΓΔ,𝜏)≤𝑐_1ΓΔ𝑔_𝑘max𝑐₂‖𝑤‖₂−‖𝜏‖₂,0𝑑𝑤. (3.61) We bound this using polar coordinates: 𝛽(ΓΔ,𝜏)≤𝑐_1Δ^∞_‖𝜏‖2𝑔_𝑘𝑐₂𝜌−‖𝜏‖₂𝜌^𝑟−1𝑑𝜌𝑑𝜎=𝑐₃mes(Δ). (3.62) Here, the quantity ^∞_‖𝜏‖2𝑔_𝑘𝑐₂𝜌−‖𝜏‖₂𝜌^𝑟−11𝑑𝜌=𝑐₂^∞₀𝑔_𝑘𝜌𝜌𝑐₂+‖𝜏‖₂^𝑟−1𝑑𝜌 (3.63) is finite because by condition (2.23), 𝑔_𝑘 is integrable in the neighborhood of zero and (𝜌/𝑐₂+‖𝜏‖₂)^𝑟−1 is of order 𝜌^𝑟−1 at infinity.
Step  2 (proving the main bound). The matrix 𝐴 from Lemma 3.4(a) is of size 𝑛×(𝑛−1) and 𝐴⁺ is of size (𝑛−1)×𝑛. Equation (3.50) and 𝑅^𝑛−1=Im𝐻⊕Im(𝐼−𝐻) imply that rank(𝐼−𝐻)=𝑛−1−𝑟=𝑘−1. (3.64) Equation (3.48) written point-wise is 𝑡⁽²⁾=𝑢+𝑣,  𝑡⁽²⁾∈𝑇⁽²⁾,  𝑢∈𝐴⁺(ΓΔ−𝜏),  𝑣∈Im(𝐼−𝐻). Since 𝑢,𝑣 here are orthogonal, we have 𝑔₁𝑡⁽²⁾=_𝑅𝑝𝑡²₁+‖𝑢‖²₂+‖𝑣‖²₂^1/2𝑑𝑡₁≡ℎ‖𝑢‖₂,‖𝑣‖₂, (3.65) where the right side defines the function ℎ.
The transition from 𝑡⁽²⁾ to the coordinates (𝑢,𝑣) is described by an orthogonal matrix. This can be shown as follows. Take in Im𝐻, an orthonormal basis and identify 𝑢 with the set of its coordinates in that basis. Do the same with Im(𝐼−𝐻) and 𝑣. Join the two bases. The equation 𝑡⁽²⁾=𝑢+𝑣 induces a transformation between the original coordinates of 𝑡⁽²⁾ and the ones in the joint basis. Since Im𝐻⟂Im(𝐼−𝐻), the Jacobian of the resulting transformation is unity. Hence, by (3.48), (3.64), (3.65), and (3.57), 𝛼(𝑌,𝜏)=_{𝐴+(ΓΔ−𝜏)+Im(𝐼−𝐻)}𝑔₁()=𝑢+𝑣𝑑𝑢𝑑𝑣_{𝐴+(ΓΔ−𝜏)𝑅𝑘−1}ℎ‖𝑢‖₂,‖𝑣‖₂=𝑑𝑣𝑑𝑢_{𝐴+(ΓΔ−𝜏)}𝑔_𝑘‖𝑢‖₂()‖𝜏‖𝑑𝑢=𝛽ΓΔ,𝜏≤𝑐₂mes(Δ). (3.66)
Part (b). In the bound (see (3.59) and (3.60)), 𝛽(𝑌,𝜏)=𝑐_1𝑌−𝜏𝑔_𝑘𝐴⁺𝑣₂𝑑𝑣≤𝑐_1𝑌−𝜏𝑔_𝑘𝑐₂‖𝑣‖₂𝑑𝑣, (3.67) we have mes(𝑌−𝜏)=mes(𝑌)→0, so 𝛽(𝑌,𝜏)→0 uniformly in 𝜏. As in (3.66), it follows that𝛼(𝑌,𝜏)=_{𝐴+(𝑌−𝜏)+Im(𝐼−𝐻)}𝑔₁(𝑢+𝑣)𝑑𝑢𝑑𝑣=𝛽(𝑌,𝜏)⟶0. (3.68)

Lemma 3.6. Let 𝑓₁,𝑝,𝜏,𝐴,𝐴⁺, and 𝐻 be the same as in Lemma 3.4(b).(a)Suppose 𝑌=ΓΔ where Δ⊆{𝑠∈Im(𝑀_𝑋−𝑝𝑝^′)∶‖𝑠‖₂=1}. If (2.25) holds, then (3.55) is true with 𝑇⁽²⁾ from Lemma 3.4(b).(b)If 𝑌⊆{𝑠∈Im(𝑀_𝑋−𝑝𝑝^′)∶‖𝑠‖₂≤1}, then 𝛼(𝑌,𝜏)→0, as mes(𝑌)→0, uniformly in 𝜏.

Proof. The proof essentially follows that of Lemma 3.5(a). From (3.49), and (3.54) we see that instead of (3.58) we have 𝑟=𝑛−𝑘−1. Instead of (3.64) we have rank(𝐼−𝐻)=𝑘. This leads to the definition of the marginal density used in (2.25). With these changes, (3.62), and (3.66) are true and prove part (a).
With the same changes the proof of Lemma 3.5(b) goes through.

In the course of the proof, we approximate 𝜒_Φ by functions 𝜑^±_𝜀∈𝐶𝐿(𝑅^𝑛) from below and above. Then, we show that the rate of approximation improves as 𝜀→0 and 𝜌→𝑎.

Step 1 (constructing the approximating functions). Denote 𝜌(𝑥,𝐴)=inf_𝑦∈𝐴‖𝑥−𝑦‖₂ the distance from a point 𝑥 to a set 𝐴. For a given 𝜀>0, set Ψ⁺_𝜀={𝑥∈𝑅^𝑛∶𝜌(𝑥,Φ)≤𝜀},  Ψ⁻_𝜀={𝑥∈Φ∶𝜌(𝑥,bd(Φ))>𝜀}.

Obviously, Ψ⁻_𝜀⊂Φ⊂Ψ⁺_𝜀. As in [10, page 12 proof of (ii)⇒(iii)], we can construct functions 𝜑^±_𝜀 that are uniformly continuous on 𝑅^𝑛 and satisfy 𝜑⁺_𝜀=1onΦ,𝜑⁺_𝜀=0outsideΨ⁺_𝜀,0≤𝜑⁺_𝜀≤1on𝑅^𝑛,𝜑⁻_𝜀=1onΨ⁻_𝜀,𝜑⁻_𝜀=0outsideΦ,0≤𝜑⁻_𝜀≤1on𝑅^𝑛. (3.69)

Step 2 (reducing the convergence statement to a special case). Denote 𝐼𝜑=_𝑅𝑛−1𝑀𝑇₁𝜑𝑡⁽²⁾𝑔₁𝑡⁽²⁾𝑑𝑡⁽²⁾. (3.70) Here, we show that lim_𝜀→0𝐼𝜑^±_𝜀=𝐼𝜒_Φ (3.71) implies (2.24). From the inequality 𝜑⁻_𝜀≤𝜒_Φ≤𝜑⁺_𝜀 we have 𝐽_𝜌𝜑⁻_𝜀≤𝐽_𝜌𝜒_Φ≤𝐽_𝜌𝜑⁺_𝜀. (3.72) By Theorem 2.7, lim_𝜌→𝑎𝐽_𝜌𝜑^±_𝜀=𝐼𝜑^±_𝜀,forany𝜀>0. (3.73) By (3.71), for any 𝛿>0, there exists 𝜀(𝛿)>0 such that 𝐼𝜑⁺_𝜀−𝛿≤𝐼𝜒_Φ≤𝐼𝜑⁻_𝜀+𝛿,0<𝜀≤𝜀(𝛿). (3.74) By (3.73), for the same 𝛿, there is 𝜌(𝛿)∈(0,𝑎) such that 𝐽_𝜌𝜑^±_𝜀−𝐼𝜑^±_𝜀≤𝛿,𝜌(𝛿)≤𝜌<𝑎. (3.75) Equations (3.72), (3.74), and (3.75) imply that 𝐼𝜒_Φ−2𝛿≤𝐼𝜑⁻_𝜀−𝛿≤𝐽_𝜌𝜑⁻_𝜀≤𝐽_𝜌𝜒_Φ≤𝐽_𝜌𝜑⁺_𝜀≤𝐼𝜑⁺_𝜀+𝛿≤𝐼𝜒_Φ+2𝛿,𝜌(𝛿)≤𝜌<𝑎. (3.76) Since 𝛿>0 is arbitrary, this proves (2.24).

Step 3 (further reduction to an indicator function). Let us call 𝐵(𝜀)=Ψ⁺_𝜀⧵Ψ⁻_𝜀 a band. Here, we show that (3.71) will follow if we prove that lim_𝜀→0𝐼𝜒_𝐵(𝜀)=0. (3.77) Note the identity 𝐼𝜒_𝐵(𝜀)=𝐼𝜒_Ψ+𝜀−𝐼𝜒_Ψ−𝜀=𝐼𝜒_Ψ+𝜀−𝐼𝜒_Φ+𝐼𝜒_Φ−𝐼𝜒_Ψ−𝜀. (3.78) Since each of the square brackets on the right is nonnegative, (3.77) gives lim_𝜀→0𝐼(𝜒_Ψ±𝜀)=𝐼(𝜒_Φ). Now, it suffices to take into account the inequality 𝐼(𝜒_Ψ−𝜀)≤𝐼(𝜑⁻_𝜀)≤𝐼(𝜒_Φ)≤𝐼(𝜑⁺_𝜀)≤𝐼(𝜒_Ψ+𝜀) to see that (3.71) follows.

Step 4 (geometric description of the task). Denote 𝑙(𝑠,𝑖)={𝑠𝑡+𝑖∶𝑡∈𝑅} a straight line with a slope 𝑠∈𝑅^𝑛 and intercept 𝑖∈𝑅^𝑛 and put 𝑏_𝜀𝑡⁽²⁾=𝑀𝑇₁𝜒_𝐵(𝜀)𝑡⁽²⁾=lim_𝑟→∞12𝑟^𝑟_−𝑟𝜒_𝐵(𝜀)𝑓₁𝑡₁+𝑧𝑡⁽²⁾𝑑𝑡₁. (3.79) This expression shows that we need to study the set {𝑡₁∈𝑅∶𝑓₁𝑡₁+𝑧(𝑡⁽²⁾)∈𝐵(𝜀)} of points common to the straight line 𝑙(𝑓₁,𝑧(𝑡⁽²⁾)) and the band 𝐵(𝜀). The cylindrical nature of the sets considered implies that the band admits an equivalent definition as a cylinder with the base 𝑀_𝑋𝐵(𝜀)=𝑥∈Im𝑀_𝑋∶𝜌𝑥,bd𝑀_𝑋Φ≤𝜀 (3.80) and element Im𝑋. From this definition, we see that the condition 𝑓₁𝑡₁+𝑧(𝑡⁽²⁾)∈𝐵(𝜀) is equivalent to 𝑀_𝑋𝑓₁𝑡₁+𝑀_𝑋𝑧𝑡⁽²⁾∈𝑀_𝑋𝐵(𝜀). (3.81) We have to sort out points 𝑡⁽²⁾∈𝑅^𝑛−1 into (a) those for which the set of points 𝑡₁ satisfying (3.81) is so small that (3.79) is zero and (b) that set is large enough for (3.79) to be positive.

Step 5 (case 𝑓₁∈Im𝑋). In this case, 𝑀_𝑋𝑓₁=0, and (3.81) reduces to 𝑀_𝑋𝑃₂(𝑎)Λ₂⁻¹(𝑎)𝑡⁽²⁾+𝑀_𝑋𝑄𝑋𝛽∈𝑀_𝑋𝐵(𝜀). (3.82) This condition does not contain 𝑡₁. Therefore, if it is not satisfied, the integrand in (3.79) is identically zero and 𝑏_𝜀(𝑡⁽²⁾)=0 for such 𝑡⁽²⁾. If it is satisfied, the integrand in (3.79) is identically 1 and 𝑏_𝜀(𝑡⁽²⁾)=1 for such 𝑡⁽²⁾. Hence, 𝐼𝜒_𝐵(𝜀)=_(3.82)istrue𝑔₁𝑡⁽²⁾𝑑𝑡⁽²⁾. (3.83) The set of 𝑡⁽²⁾ satisfying (3.82) has an infinite Lebesgue measure which precludes us from application of the continuity property of the Lebesgue measure (for an integrable ℎ,∫_𝐴ℎ(𝑡)𝑑𝑡→0 as mes(𝐴)→0).
As one can see from Figure 1, 𝑀_𝑋𝐵(𝜀) is not a cone-like set. We divide it into two parts: 𝛼ⁱⁿ_𝜀={𝑠∈𝑀_𝑋𝐵(𝜀)∶‖𝑠‖₂≤1} includes the points inside the unit ball and on the unit sphere and 𝛼^out_𝜀={𝑠∈𝑀_𝑋𝐵(𝜀)∶‖𝑠‖₂>1} includes the points of 𝑀_𝑋𝐵(𝜀) outside the unit ball. Put Δ_𝜀={𝑠∈Im𝑀_𝑋∶‖𝑠‖₂=1,𝜌(𝑠,bd(𝑀_𝑋Φ))≤𝜀} and let us show that 𝛼^out_𝜀⊂ΓΔ_𝜀. (3.84) Suppose 𝑥∈𝛼^out_𝜀. Then, 𝑥∈Im𝑀_𝑋,‖𝑥‖₂>1 and by (3.80), there exists a sequence {𝑦^(𝑘)}⊂bd(𝑀_𝑋Φ) such that limsup‖𝑥−𝑦^(𝑘)‖₂≤𝜀. The sequence {𝑦^(𝑘)} is bounded and therefore precompact, while bd(𝑀_𝑋Φ)=𝑀_𝑋(cl(Φ)⧵Φ) is closed by construction. Hence, we can assume that 𝑦^(𝑘)→𝑦∈bd(𝑀_𝑋Φ) (passing to a subsequence, if necessary). Then, ‖𝑥−𝑦‖₂≤𝜀. This implies  ‖𝑥/‖𝑥‖₂−𝑦/‖𝑥‖₂‖₂≤𝜀/‖𝑥‖₂<𝜀. Let 𝑠=𝑥/‖𝑥‖₂. Since bd(𝑀_𝑋Φ) is cone-like, it follows that 𝑦/‖𝑥‖₂∈bd(𝑀_𝑋Φ) and 𝜌(𝑠,bd(𝑀_𝑋Φ))<𝜀. This proves that 𝑠∈Δ_𝜀,  𝑥=‖𝑥‖₂𝑠∈ΓΔ_𝜀 and (3.84) is true.
From (3.84), we conclude that 𝑀_𝑋𝐵(𝜀)⊂𝛼ⁱⁿ_𝜀∪ΓΔ_𝜀. (3.85) This means that the set of solutions of (3.82) is contained in the union of two sets: 𝑇₁⁽²⁾={𝑡⁽²⁾∈𝑅^𝑛−1∶𝐴𝑡⁽²⁾+𝜏∈ΓΔ_𝜀} and 𝑇₂⁽²⁾={𝑡⁽²⁾∈𝑅^𝑛−1∶𝐴𝑡⁽²⁾+𝜏∈𝛼ⁱⁿ_𝜀} where 𝐴=𝑀_𝑋𝑃₂(𝑎)Λ₂⁻¹(𝑎) and 𝜏=𝑀_𝑋𝑄𝑋𝛽.
By Lemma 3.4(a), 𝑇₁⁽²⁾=𝐴⁺(ΓΔ_𝜀−𝜏)+Im(𝐼−𝐻). By Lemma 3.5(a) _𝑇1(2)𝑔₁𝑡⁽²⁾𝑑𝑡⁽²⁾≤𝑐‖𝜏‖₂mesΔ_𝜀⟶0,𝜀→0. (3.86) By Lemma 3.4(a), 𝑇₂⁽²⁾=𝐴⁺(𝛼ⁱⁿ_𝜀−𝜏)+Im(𝐼−𝐻). By Lemma 3.5(b) _𝑇2(2)𝑔₁𝑡⁽²⁾𝑑𝑡⁽²⁾⟶0asmes𝛼ⁱⁿ_𝜀⟶0. (3.87) Equations (3.86) and (3.87) prove that 𝐼(𝜒_𝐵(𝜀))→0,  𝜀→0.

Figure 1 

𝑀𝑋Φ consists of two cones bounded by solid lines. 𝑀𝑋𝐵(𝜀) is bounded by dashed lines.

Step 6 (case 𝑓₁∉Im𝑋). Equation (2.5) implies that if the straight line 𝑙(𝑀_𝑋𝑓₁,𝑀_𝑋𝑧(𝑡⁽²⁾)) has a bounded intersection with the band 𝑀_𝑋𝐵(𝜀), then (𝑀𝑇₁𝜒_𝐵(𝜀))(𝑡⁽²⁾)=0. Therefore, we are interested in the set of slopes: Sl=𝑀_𝑋𝑓₁∶𝑙𝑀_𝑋𝑓₁,𝑀_𝑋𝑧𝑡⁽²⁾∩𝑀_𝑋𝐵(𝜀)isunbounded. (3.88) Let us prove that 𝑀_𝑋𝑓₁∈Sl⇒𝑀_𝑋𝑓₁/𝑀_𝑋𝑓₁₂∈bd(𝑆). (3.89) Suppose 𝑀_𝑋𝑓₁∈Sl. Then, there exists a sequence of points 𝑝^(𝑘) which lie on the straight line, 𝑝^(𝑘)=(𝑀_𝑋𝑓₁)𝑡₁^(𝑘)+𝑀_𝑋𝑧(𝑡⁽²⁾), where |𝑡₁^(𝑘)|→∞, and which are 2𝜀-close to bd(𝑀_𝑋Φ): there exists a sequence 𝑦^(𝑘)∈bd(𝑀_𝑋Φ) such that 𝑝^(𝑘)−𝑦^(𝑘)₂≤2𝜀,𝑦^(𝑘)₂⟶∞. (3.90) Equation (2.21) implies that 𝑀_𝑋Φ=Γ𝑆 and bd(𝑀_𝑋Φ)=Γbd(𝑆)∪{0}. Therefore 𝑠^(𝑘)≡𝑦^(𝑘)/‖𝑦^(𝑘)‖₂∈bd(𝑆). Because bd(𝑆) is compact, we can assume that 𝑠^(𝑘)→𝑠∈bd(𝑆). From (3.90), we have ‖(𝑀_𝑋𝑓₁)𝑡₁^(𝑘)+𝑀_𝑋𝑧(𝑡⁽²⁾)−‖𝑦^(𝑘)‖₂𝑠^(𝑘)‖₂≤2𝜀 or, dividing both sides by ‖𝑦^(𝑘)‖₂, ‖(𝑀_𝑋𝑓₁)𝑡₁^(𝑘)/‖𝑦^(𝑘)‖₂−𝑠^(𝑘)‖₂→0. This shows that the sequence {𝑡₁^(𝑘)/‖𝑦^(𝑘)‖₂} is bounded and can be assumed convergent to a number 𝛿. Letting 𝑘→∞, we get (𝑀_𝑋𝑓₁)𝛿=𝑠∈bd(𝑆), and (3.89) is true.
Fix 𝑀_𝑋𝑓₁∈Sl and denote 𝑝=𝑀_𝑋𝑓₁/‖𝑀_𝑋𝑓₁‖₂∈bd(𝑆)⊂Im𝑀_𝑋. The matrix 𝑝𝑝^′ projects Im 𝑀_𝑋 onto the straight line 𝑙(𝑀_𝑋𝑓₁,0). The projector 𝑀_𝑋−𝑝𝑝^′ projects Im  𝑀_𝑋 onto Im(𝑀_𝑋−𝑝𝑝^′). Condition (3.81) implies two equations in projections: 𝑝𝑝^′𝑀_𝑋𝑓₁𝑡₁+𝑀_𝑋𝑧𝑡⁽²⁾∈𝑝𝑝^′𝑀_𝑋𝐵(𝜀),𝑀_𝑋−𝑝𝑝^′𝑀_𝑋𝑓₁𝑡₁+𝑀_𝑋𝑧𝑡⁽²⁾∈𝑀_𝑋−𝑝𝑝^′𝑀_𝑋𝐵(𝜀). (3.91) The first one can be safely omitted because the shift 𝑝𝑝^′𝑀_𝑋𝑧(𝑡⁽²⁾) along a line does not change the property of the line to have an unbounded intersection with 𝑀_𝑋𝐵(𝜀). Denoting 𝐴=(𝑀_𝑋−𝑝𝑝^′)𝑃₂(𝑎)Λ₂⁻¹(𝑎),  𝜏=(𝑀_𝑋−𝑝𝑝^′)𝑄𝑋𝛽, and using (3.52), we write the second equation as 𝐴𝑡⁽²⁾+𝜏∈𝑀_𝑋−𝑝𝑝^′𝐵(𝜀). (3.92) The set of points 𝑡⁽²⁾ such that (3.81) holds for an unbounded set of values 𝑡₁ is clearly contained in the set 𝑇⁽²⁾ of points 𝑡⁽²⁾ that satisfy (3.92). For 𝑡⁽²⁾∈𝑇⁽²⁾ we use an obvious bound 𝑏_𝜀(𝑡⁽²⁾)≤1 (see (3.79)), and, therefore, 𝐼(𝜒_𝐵(𝜀))≤∫_(3.92)istrue𝑔₁(𝑡⁽²⁾)𝑑𝑡⁽²⁾. The rest of the proof closely follows that for the case 𝑓₁∈Im𝑋. An inclusion of type (3.85) holds with 𝑀_𝑋 replaced by 𝑀_𝑋−𝑝𝑝^′ in all definitions. The reference to Lemma 3.4(a) is replaced by Lemma 3.4(b). Lemma 3.6 is used instead of Lemma 3.5. The conclusion is that 𝐼(𝜒_𝐵(𝜀))→0.

We shall use several times the fact that for the density 𝑔₁ in representation (2.24)will, the conditions ∫_𝐴𝑔₁(𝑡⁽²⁾)𝑑𝑡⁽²⁾>0 and mes(𝐴)>0 are equivalent. From (2.24), we see that the task is to calculate the function (3.79) with Φ in place of 𝐵(𝜀). Because of (2.21), the condition that 𝑓₁𝑡₁+𝑧(𝑡⁽²⁾)∈Φ is equivalent to the condition in projections: 𝑀_𝑋𝑓₁𝑡₁+𝑀_𝑋𝑧𝑡⁽²⁾∈Γ𝑆. (3.93)

The magnitude of 𝛽(Φ) depends on the set 𝐴𝑡⁽²⁾=𝑡₁∈𝑅∶𝑓₁𝑡₁+𝑧𝑡⁽²⁾∈Φ. (3.94)

The properties of 𝐴(𝑡⁽²⁾) depend on those of mes{𝑡⁽²⁾∈𝑅^𝑛−1∶𝑀_𝑋𝑧(𝑡⁽²⁾)∈𝐺} for certain sets 𝐺⊆Im𝑀_𝑋, as we will see.

Lemma 3.7. Suppose Im𝑄⊆Im𝑋 (𝑄 is the projector from Theorem 2.7). Then, 𝑀_𝑋𝑧(𝑡⁽²⁾)=0 for all 𝑡⁽²⁾∈𝑅^𝑛−1.
Let Im𝑄⊈Im𝑋. Then, mes{𝑡⁽²⁾∶𝑀_𝑋𝑧(𝑡⁽²⁾)=0}=0.

Proof. The assumption Im𝑄⊆Im𝑋 implies that 𝑀_𝑋𝑓_𝑖=0 for 𝑖=2,…,𝑛, which can be written as 𝑀_𝑋𝑃₂(𝑎)=0. Hence, 𝑀_𝑋𝑧(𝑡⁽²⁾)=𝑀_𝑋𝑃₂(𝑎)[Λ₂⁻¹(𝑎)𝑡⁽²⁾+𝑃^′₂(𝑎)𝑋𝛽]=0 for any 𝑡⁽²⁾∈𝑅^𝑛−1.
If Im𝑄⊈Im𝑋, then 𝑓_𝑖∉Im𝑋 and 𝑀_𝑋𝑓_𝑖≠0 for at least one 𝑖. The matrix 𝑀_𝑋𝑃₂(𝑎) is of size 𝑛×(𝑛−1), and of rank at least 1. From the equation dim𝑁(𝑀_𝑋𝑃₂(𝑎))+rank(𝑀_𝑋𝑃₂(𝑎))=𝑛−1 we see that the null space 𝑁(𝑀_𝑋𝑃₂(𝑎)) is of dimension at most 𝑛−2. The set of 𝑡⁽²⁾ satisfying 𝑀_𝑋𝑧(𝑡⁽²⁾)=𝑀_𝑋𝑃₂(𝑎)Λ₂⁻¹(𝑎)𝑡⁽²⁾+𝑀_𝑋𝑄𝑋𝛽=0 is a partial solution 𝑡₀⁽²⁾ plus an arbitrary element from the null space 𝑁(𝑀_𝑋𝑃₂(𝑎)). Thus the set of all solutions is a hyperplane in 𝑅^𝑛−1 of dimension at most 𝑛−2 and has (𝑛−1)-dimensional Lebesgue measure zero.

Lemma 3.8. Suppose we are given a set 𝐺⊆𝑅^𝑛−1 and for almost each 𝑡⁽²⁾∈𝐺, the set 𝐴(𝑡⁽²⁾)⊆𝑅 is defined by (3.94).(a)If 𝐴(𝑡⁽²⁾) is bounded for almost all 𝑡⁽²⁾∈𝐺 and mes(𝐺)>0, then 𝛽(Φ)<1. In particular, if 𝐺=𝑅^𝑛−1, then 𝛽(Φ)=0.(b)If 𝑅⧵𝐴𝑡⁽²⁾isboundedforalmostall𝑡⁽²⁾∈𝐺 (3.95) and mes(𝐺)>0, then 𝛽(Φ)>0. In particular, if 𝐺=𝑅^𝑛−1, then 𝛽(Φ)=1.(c)Suppose that 𝐺=𝑅^𝑛−1 and for almost all 𝑡⁽²⁾∈𝐺, theset𝐴𝑡⁽²⁾isahalf−axis (3.96) (i.e., either 𝐴(𝑡⁽²⁾)=(𝑎(𝑡⁽²⁾),∞) or 𝐴(𝑡⁽²⁾)=(−∞,𝑎(𝑡⁽²⁾)) with some 𝑎(𝑡⁽²⁾)∈𝑅). Then, 𝛽(Φ)=1/2.

Proof. If 𝐴(𝑡⁽²⁾) is bounded, then 𝜒_Φ(𝑓₁𝑡₁+𝑧(𝑡⁽²⁾))=0 outside a bounded set, and by (2.5), 𝑀𝜒_{𝐴(𝑡(2))}=𝑀𝑇₁𝜒_Φ𝑡⁽²⁾=0 (3.97) for almost all 𝑡⁽²⁾∈𝐺. Since 0≤𝑀𝜒_{𝐴(𝑡(2))}≤1 outside 𝐺, we have 𝛽(Φ)≤_{𝑅𝑛−1⧵𝐺}𝑔₁𝑡⁽²⁾𝑑𝑡⁽²⁾<1. (3.98) Obviously, the integral is null if 𝐺=𝑅^𝑛−1.
By (3.97), condition (3.95) implies that 𝑀𝜒_{𝐴(𝑡(2))}=𝑀1−𝜒_{𝑅⧵𝐴(𝑡(2))}=1a.e.in𝐺. (3.99) Since mes(𝐺)>0, we have 𝛽(Φ)≥_𝐺𝑔₁𝑡⁽²⁾𝑑𝑡⁽²⁾>0. (3.100) In case 𝐺=𝑅^𝑛−1, the last integral is 1.
Consider, for example, the case of the right half-axis: 𝑀𝜒_{𝐴(𝑡(2))}=lim_𝑟→∞12𝑟⁰_𝑎(𝑡⁽²⁾₎𝑑𝑡+^𝑟₀=1𝑑𝑡2. (3.101) Hence, 𝛽(Φ)=(1/2)∫_𝑅𝑛−1𝑔₁(𝑡⁽²⁾)𝑑𝑡⁽²⁾=1/2.

Case 1. Let 𝑓₁∈Φ. Then, 𝑓₁∉Im𝑋 and 𝑀_𝑋𝑓₁≠0. Define 𝑝=𝑀_𝑋𝑓₁/‖𝑀_𝑋𝑓₁‖₂. Obviously, 𝑀_𝑋𝑓₁∈Γ𝑆 and 𝑝∈𝑆. In the equation 𝑀_𝑋𝑓₁𝑡₁+𝑀_𝑋𝑧𝑡⁽²⁾=𝑀_𝑋𝑓₁₂𝑡₁𝑀𝑝+^𝑋𝑧𝑡⁽²⁾𝑀_𝑋𝑓₁₂𝑡₁, (3.102) for all large |𝑡₁|, the term in the square brackets is close to 𝑝 and, because 𝑀_𝑋Φ is open in Im𝑀_𝑋, this term belongs to 𝑀_𝑋Φ. Since 𝑀_𝑋Φ is cone-like, the left side of (3.102) belongs to 𝑀_𝑋Φ for all large |𝑡₁|. Hence, the set 𝐴(𝑡⁽²⁾) contains the neighborhood of infinity (more precisely, all 𝑡₁ with |𝑡₁|>𝑁 where 𝑁 depends on 𝑡⁽²⁾). By Lemma 3.8(b), where 𝐺=𝑅^𝑛−1, then 𝛽(Φ)=1.

Case 2. Suppose 𝑓₁∈bd(Φ).
Subcase 2.1. Let 𝑓₁∈Im𝑋 or, equivalently, 𝑀_𝑋𝑓₁=0. Condition (3.93) reduces to 𝑀_𝑋𝑧𝑡⁽²⁾=𝑀_𝑋𝑃₂(𝑎)Λ₂⁻¹(𝑎)𝑡⁽²⁾+𝑀_𝑋𝑄𝑋𝛽∈Γ𝑆. (3.103) By Lemma 3.4(a), the set of solutions of (3.103) is 𝑇⁽²⁾=𝐴⁺(Γ𝑆−𝜏)+Im(𝐼−𝐻) where 𝐴⁺ is the Moore-Penrose inverse of 𝐴=𝑀_𝑋𝑃₂(𝑎)Λ₂⁻¹(𝑎),𝜏=𝑀_𝑋𝑄𝑋𝛽, and 𝐻 is the projector from Lemma 3.2. Using Lemma 3.8(b), (3.66), and (3.59) we get 𝛽(Φ)=_𝑇(2)𝑔₁𝑡⁽²⁾𝑑𝑡⁽²⁾=_{𝐴+(Γ𝑆−𝜏)}𝑔_𝑘‖𝑢‖₂𝑑𝑢. (3.104) The density 𝑔_𝑘(‖𝑢‖₂) is spherically symmetric. It is positive everywhere because 𝑔_𝑘(𝑡)=_𝑅𝑘𝑝𝑡²+‖𝑣‖²₂^1/2𝑑𝑣=0 (3.105) would mean 𝑝((𝑡²+𝑡₁)^1/2)=0 for any 𝑡₁≥0, that is, 𝑝(𝑡₁)=0 for any 𝑡₁≥𝑡, which is impossible by assumption. 𝐴⁺ is one-to-one on Im𝐴=Im𝑀_𝑋, so 𝐴⁺(Γ𝑆−𝜏) is a nonempty proper subset of Im𝐴⁺. Therefore, 0<𝛽(Φ)<1.
Subcase 2.2. Suppose 𝑓₁∉Im𝑋. As in Case  1, 𝑝 is defined. Let us show that it satisfies 𝑝^′𝑊𝑝=𝑐. (3.106) Since 𝑓₁ is a limit point of Φ, there is a sequence 𝑥_𝑛∈Φ such that 𝑥_𝑛→𝑓₁. Hence, 𝑀_𝑋𝑥_𝑛→𝑀_𝑋𝑓₁ where ‖𝑀_𝑋𝑓₁‖₂≠0 because 𝑓₁∉Im𝑋. From (2.21), it follows that 𝑀_𝑋𝑥_𝑛∈Γ𝑆,𝑧_𝑛≡𝑀_𝑋𝑥_𝑛/‖𝑀_𝑋𝑥_𝑛‖₂∈𝑆 and 𝑧^′_𝑛𝑊𝑧_𝑛>𝑐. Letting 𝑛→∞ and using 𝑧_𝑛→𝑝, we get 𝑝^′𝑊𝑝≥𝑐. Since 𝑓₁∉Φ, we have 𝑀_𝑋𝑓₁∉𝑀_𝑋Φ and 𝑝∉𝑆. Thus, 𝑝^′𝑊𝑝>𝑐 is impossible and (3.106) is true.
In the three examples in the following we assume that 𝑛>𝑘+1. Then, Im𝑄⊈Im𝑋, and by Lemma 3.7(b), 𝑀_𝑋𝑧𝑡⁽²⁾≠0,foralmostall𝑡⁽²⁾∈𝑅^𝑛−1. (3.107)Sub-sub-case (a) (Example of 𝛽(Φ)=1). Let 𝑊 be symmetric. Assumptions (1.13) imply that var(𝑢)=𝜎²_𝜀(𝐼−𝜌𝑊)⁻² whenever det(𝐼−𝜌𝑊)≠0, so Σ⁻¹(𝜌)=(𝐼−𝜌𝑊)². If 𝜇_𝑖 denote eigenvalues of 𝑊, then the eigenvalues of 𝐼−𝜌𝑊 are 𝜈_𝑖(𝜌)=1−𝜌𝜇_𝑖. Let the constant 𝑐 in the definition of the critical region Φ be positive. Put 𝑎=1/𝑐 and with some integer 𝑖₀ and real numbers 𝜇₋,  𝜇₊ such that 1<𝑖₀<𝑛,  𝜇₋<1/𝑎<𝜇₊ define 𝜇_𝑖 by 𝜇₁=⋯=𝜇_𝑖0−1=𝜇₋<𝜇_𝑖0=1𝑎<𝜇_𝑖0+1=⋯=𝜇_𝑛=𝜇₊. (3.108) Then, for 0<𝜌≤𝑎, 𝜈₁𝜌=⋯=𝜈_𝑖0−1𝜌=1−𝜌𝜇₋>𝜈_𝑖0𝜌𝜌=1−𝑎>𝜈_𝑖0+1𝜌=…=𝜈_𝑛𝜌=1−𝜌𝜇₊, (3.109) and Σ^−1/2(𝑎) has eigenvalues 𝜈₁(𝑎)=⋯=𝜈_𝑖0−1(𝑎)=1−𝑎𝜇₋>𝜈_𝑖0(𝑎)<𝜈=0_𝑖0+1(𝑎)=⋯=𝜈_𝑛(𝑎)=1−𝑎𝜇₊. (3.110) Hence, in the notation of Condition 3,  𝜆₁(𝜌)=𝜈_𝑖0(𝜌) and 𝜆₂(𝜌),…,𝜆_𝑛(𝜌) are obtained from |𝜈_𝑖(𝜌)|,  𝑖≠𝑖₀, by reordering. For example, if 1−𝑎𝜇₋<|1−𝑎𝜇₊|, then 𝜆_𝑖(𝜌)=𝜈_𝑖−1(𝜌),𝑖=2,…,𝑖₀;  𝜆_𝑖(𝜌)=|𝜈_𝑖(𝜌)|,  𝑖=𝑖₀+1,…,𝑛. Assuming this ordering, denote the corresponding eigenvectors of Σ^−1/2(𝑎) by 𝑓₁,𝑓⁻₂,…,𝑓⁻_𝑖0,𝑓⁺_𝑖0+1,…,𝑓⁺_𝑛 where the minuses and pluses remind us that the corresponding eigenvalues depend on 𝜇₋ and 𝜇₊, respectively.
Let 𝑘=𝑖₀−1 and suppose that 𝑓⁻₂,…,𝑓⁻_𝑖0spanIm𝑋,𝑓₁,𝑓⁺_𝑖0+1,…,𝑓⁺_𝑛spanIm𝑀_𝑋. (3.111) Denote 𝑥=𝑀_𝑋𝑓₁𝑡₁+𝑀_𝑋𝑧𝑡⁽²⁾∈Im𝑀_𝑋. (3.112) Because of (3.111), here 𝑀_𝑋𝑓₁=𝑓₁. Besides, since 𝑧(𝑡⁽²⁾) is orthogonal to 𝑓₁, the decomposition 𝑀_𝑋𝑧𝑡⁽²⁾=^𝑛_𝑖=𝑖0+1𝑐_𝑖𝑓⁺_𝑖 (3.113) does not contain 𝑓₁. Hence, (3.112) becomes 𝑥=𝑡₁𝑓₁+^𝑛_𝑖=𝑖0+1𝑐_𝑖𝑓⁺_𝑖. (3.114) Remembering that the eigenvectors of Σ^−1/2(𝑎) are just reordered eigenvectors of 𝑊, we get 1𝑊𝑥=𝑎𝑡₁𝑓₁+𝜇₊^𝑛_𝑖=𝑖0+1𝑐_𝑖𝑓⁺_𝑖,𝑥^′1𝑊𝑥=𝑎𝑡²₁+𝜇₊^𝑛_𝑖=𝑖0+1𝑐²_𝑖=𝑐𝑡²₁+𝜇₊^𝑛_𝑖=𝑖0+1𝑐²_𝑖, (3.115) for any 𝑥 of form (3.112). From (3.107) and (3.113) we see that here ^𝑛_𝑖=𝑖0+1𝑐²_𝑖≠0a.e.in𝑅^𝑛−1. (3.116) Hence, normalizing 𝑥,  𝑠=𝑥/‖𝑥‖₂, and assuming that 𝜇₊>𝑐, we get from (3.115) 𝑠^′𝑊𝑠=𝑐𝑡²₁+𝜇₊∑^𝑛_𝑖=𝑖0+1𝑐²_𝑖𝑡²₁+∑^𝑛_𝑖=𝑖0+1𝑐²_𝑖>𝑐, (3.117) for all 𝑡₁∈𝑅 and almost all 𝑡⁽²⁾∈𝑅^𝑛−1. This shows that the set (3.94) coincides with 𝑅 for almost all 𝑡⁽²⁾∈𝑅^𝑛−1 and by Lemma 3.8(b) 𝛽(Φ)=1.
Sub-sub-case (b) Example of 𝛽(Φ)=0. We change (3.111) so as to obtain in (3.115) 𝜇₋ in place of 𝜇₊. Let 𝑘=𝑛−𝑖₀ and suppose that 𝑓⁺_𝑖0+1,…,𝑓⁺_𝑛spanIm𝑋,𝑓₁,𝑓⁻₂,…,𝑓⁻_𝑖0spanIm𝑀_𝑋. (3.118) Instead of (3.113), we get 𝑀_𝑋𝑧(𝑡⁽²⁾)=∑^𝑖0_𝑖=2𝑐_𝑖𝑓⁻_𝑖 which leads to an analog of (3.114): 𝑥=𝑡₁𝑓₁+∑^𝑖0_𝑖=2𝑐_𝑖𝑓⁻_𝑖. Equation(3.115) is replaced by 𝑥^′𝑊𝑥=𝑐𝑡²₁+𝜇₋^𝑖0_𝑖=2𝑐²_𝑖, (3.119) where ∑^𝑖0_𝑖=2𝑐²_𝑖≠0 a.e. in 𝑅^𝑛−1. The result is that 𝑠^′𝑊𝑠=𝑐𝑡²₁+𝜇₋∑^𝑖0_𝑖=2𝑐²_𝑖𝑡²₁+∑^𝑖0_𝑖=2𝑐²_𝑖<𝑐 (3.120) for all 𝑡₁∈𝑅 and almost all 𝑡⁽²⁾∈𝑅^𝑛−1. Hence, the set (3.94) is empty for almost all 𝑡⁽²⁾∈𝑅^𝑛−1 and by Lemma 3.8(a), 𝛽(Φ)=0.
Sub-sub-case (𝑐)Example of 𝛽(Φ)=1/2]. 𝑀_𝑋𝑧(𝑡⁽²⁾) runs over Im(𝑀_𝑋𝑃₂(𝑎)). If 𝑘>1, then 𝑛−1>𝑛−𝑘, and we can choose the eigenvectors 𝑓₂,…,𝑓_𝑛 of Σ^−1/2(𝑎) so that Im𝑄⊃Im𝑀_𝑋, and then 𝑀_𝑋𝑧(𝑡⁽²⁾) runs over Im𝑀_𝑋=Im(𝑀_𝑋𝑃₂(𝑎)). Let Γ𝑆 be a union of two cones, as in Figure 2. The line 𝑙(𝑀_𝑋𝑓₁,0) lies on the boundary of the cones. In Figure 2, 𝐿_𝑎=𝑙(𝑀_𝑋𝑓₁,0) and 𝐿_𝑏=𝑙(𝑏(𝑡⁽²⁾),0). For any nonzero shift 𝑀_𝑋𝑧(𝑡⁽²⁾), half of the shifted line is contained in 𝑀_𝑋Φ because 𝐿_𝑎 and 𝐿_𝑏 are not parallel. Here, “half” means “either for all large positive 𝑡₁ or for all large negative 𝑡₁.” By Lemma 3.8(c), 𝛽(Φ)=1/2.

Figure 2 

The base of Φ contains the area between two solid lines 𝐿𝑎 and  𝐿𝑏. The dashed line is 𝑙(𝑀𝑋𝑓1,𝑀𝑋𝑧(𝑡(2)))=𝐿𝑎+𝑀𝑋𝑧(𝑡(2)).

Case 3. Suppose 𝑓₁∈Φ₋. Then, as in part 1,𝑀_𝑋𝑓₁≠0 and 𝑝 is defined. It satisfies 𝑝^′𝑊𝑝<𝑐. The term in the square brackets in (3.102) for all large |𝑡₁| belongs to an open set 𝑀_𝑋Φ₋. Since this set is cone-like, the left side of (3.102) belongs to 𝑀_𝑋Φ₋ for all large |𝑡₁|. The set 𝐴(𝑡⁽²⁾) is bounded for all 𝑡⁽²⁾∈𝑅^𝑛−1, and by Lemma 3.8(a), 𝛽(Φ)=0.

Step 1 (the 𝜀(𝜌)-maximizing set is an ellipsoid). For 𝑦∈𝑀_𝜀(𝜌) by orthogonality of 𝑃 and Condition 7, we have 𝑓_𝜌𝑦=^𝑛_𝑗=1𝜆_𝑗𝜌𝑝Λ𝜌𝑃^′𝜌𝑦−𝑋𝛽₂≥1−𝜀𝜌^𝑛_𝑗=1𝜆_𝑗𝜌𝑚. (3.121) Since 𝑝⁻¹ is decreasing, this is equivalent to ‖Λ(𝜌)𝑃^′(𝜌)(𝑦−𝑋𝛽)‖₂≤𝑝⁻¹((1−𝜀(𝜌))𝑚). Denoting ℎ(𝜌)=𝑝⁻¹((1−𝜀(𝜌))𝑚),  𝑧=𝑃^′(𝜌)(𝑦−𝑋𝛽), we rewrite the resulting inequality as ^𝑑_𝑗=1𝑧²_𝑗ℎ𝜌/𝜆_𝑗𝜌²+^𝑛_𝑗=𝑑+1𝑧²_𝑗ℎ𝜌/𝜆_𝑗𝜌²≤1. (3.122) In 𝑧 coordinates, this is an ellipsoid with half-axes ℎ(𝜌)/𝜆_𝑗(𝜌).

Step 2 (characterizing the rate of decay of ℎ(𝜌)). First we determine the rate with which 𝑝⁻¹(𝑥) approaches 0 when 𝑥→𝑚. To approximately invert 𝑝, in the equation 𝑥=𝑝(𝑟), we use representation (2.30): 𝑥=𝑚(1−𝑐_𝑝𝑟^𝛼𝑝)(1+𝑜(𝑟^𝛼𝜀)). This is equivalent to (𝑥/𝑚)(1+𝑜(𝑟^𝛼𝜀))=1−𝑐_𝑝𝑟^𝛼𝑝. Solving this equation for the 𝑟 involved in the right-hand side produces 𝑝⁻¹(𝑥)1=𝑟=𝑐_𝑝𝑥1−𝑚1+𝑜𝑟^{𝛼𝜀1/𝛼𝑝},𝑟→0. (3.123) Since 𝜌→𝑎, we can put 𝑟=𝑎−𝜌 in the last equation to get 𝑝⁻¹(𝑥)=1𝑐_𝑝𝑥1−𝑚1+𝑜𝑎−𝜌^{𝛼𝜀1/𝛼𝑝},𝜌→𝑎. (3.124)
To obtain ℎ(𝜌), let 𝑥=(1−𝜀(𝜌))𝑚. Then, by (2.29), 𝑥1−𝑚1+𝑜𝑎−𝜌^𝛼𝜀=1−1−𝜀𝜌1+𝑜𝑎−𝜌^𝛼𝜀=1−1+𝑜𝑎−𝜌^𝛼𝜀+𝜀𝜌1+𝑜𝑎−𝜌^𝛼𝜀=𝜀𝜌(1+𝑜(1)). (3.125) Combining this with (3.124) and (2.29), we have ℎ𝜌=𝑝⁻¹(𝑥)=𝑐_𝜀𝑐_𝑝𝑎−𝜌^𝛼𝜀(1+𝑜(1))^1/𝛼𝑝=𝑐_𝜀𝑐_𝑝^1/𝛼𝑝𝑎−𝜌^{𝛼𝜀/𝛼𝑝}(1+𝑜(1)). (3.126)

Step 3 (behavior of half-axes). We intend to show that for 𝑗=𝑑+1,…,𝑛 the half-axes tend to zero, so that in the limit 𝑧_𝑑+1=⋯=𝑧_𝑛=0, (3.127) whereas for 𝑗=1,…,𝑑, they tend to ∞, and in the limit, 𝑧₁,…,𝑧_𝑑 can be any real numbers.
For 𝑗=𝑑+1,…,𝑛 from (2.28), we see that the condition ℎ(𝜌)/𝜆_𝑗(𝜌)→0 is equivalent to ℎ(𝜌)→0. Because of (2.31), this condition is the same as (1−𝜀(𝜌))𝑚→𝑚 which holds by (2.29).
For the remaining half-axes, we get by (2.27), (3.126), and (2.32) ℎ𝜌𝜆_𝑗𝜌=1𝑐_𝜆𝑐_𝜀𝑐_𝑝^1/𝛼𝑝𝑎−𝜌^{𝛼𝜀/𝛼𝑝−𝛼𝜆}(1+𝑜(1))⟶∞. (3.128) We have proved our statement on convergence of the half-axes. According to (2.26), (3.127) rewrites as (𝑃^′(𝑎)(𝑦−𝑋𝛽))_𝑗=0,  𝑗=𝑑+1,…,𝑛. Since the rows of 𝑃^′(𝑎) are eigenvectors of 𝐴(𝑎), the last equation means that 𝑦−𝑋𝛽 is orthogonal to eigenvectors numbered 𝑗=𝑑+1,…,𝑛. Hence, 𝑦−𝑋𝛽 belongs to the null space of 𝐴(𝑎), and the set of all such 𝑦’s is 𝑋𝛽+𝑁(𝐴(𝑎)).
In this proof, the set 𝑋𝛽+𝑁(𝐴(𝑎)) is obtained as the limit of the ridge of 𝑓_𝜌. In general, this ridge can have any shape and the limit will not be 𝑋𝛽+𝑁(𝐴(𝑎)).

4. Summary

Theorem 2.2 is the key result in this paper. We tried to obtain it under minimal conditions on 𝑔. It would be interesting to relax our conditions. In particular, densities that are not spherically symmetric are of interest in applications. Note that when the dependence on 𝜆 is more complex than in 𝑔_𝜆(𝑥)=𝜆𝑔(𝜆𝑥), one needs very different techniques and more detailed assumptions about 𝑔 to obtain an analog of Theorem 2.2. Such results can be found in [15]. Another promising direction of research is to consider tests other than the Cliff-Ord one.

Acknowledgments

The author would like to thank F. Martellosio for his remarks that helped him refine Theorem 2.11 and improve the exposition in general.