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Journal of Probability and Statistics
Volume 2012 (2012), Article ID 467187, 17 pages
http://dx.doi.org/10.1155/2012/467187
Research Article

A Class of Spherical and Elliptical Distributions with Gaussian-Like Limit Properties

Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK

Received 4 May 2011; Accepted 29 November 2011

Academic Editor: Mohammad Fraiwan Al-Saleh

Copyright © 2012 Chris Sherlock and Daniel Elton. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a class of spherically symmetric random variables defined by the property that as dimension increases to infinity the mass becomes concentrated in a hyperspherical shell, the width of which is negligible compared to its radius. We provide a sufficient condition for this property in terms of the functional form of the density and then show that the property carries through to equivalent elliptically symmetric distributions, provided that the contours are not too eccentric, in a sense which we make precise. Individual components of such distributions possess a number of appealing Gaussian-like limit properties, in particular that the limiting one-dimensional marginal distribution along any component is Gaussian.

1. Introduction

Any spherically symmetric random variable can be represented as a mixture of spherical “shells” with distribution function proportional to𝟏{𝐱𝑟}. We consider a class of the spherically symmetric random variables for which as dimension 𝑑 the effective range of the mixture of “shells” becomes infinitesimal relative to a typical scale from the mixture. We then generalise this class to include a subset of the corresponding elliptically symmetric random variables. This offers a relatively rich class of random variables, the components of which are shown to possess appealing Gaussian-like limit properties.

Specifically we consider sequences of spherically symmetric random variables {𝐗𝑑} which satisfy𝐗either𝑑𝑟𝑑𝑝1,(1.1)𝐗or𝑑𝑟𝑑m.s.1,(1.2) for some positive sequence {𝑟𝑑}. Here and throughout this paper refers to the Euclidean norm. The set of such sequences includes, for instance, the sequence of standard 𝑑-dimensional Gaussians, for which 𝐗𝑑/𝑑1/2m.s.1; indeed the Gaussian-like limit properties of the whole class arise from this fact. More generally, we provide a sufficient condition for (1.2) for sequences of random variables with densities of the form𝑓𝑑𝐱𝑑𝐱exp𝑔𝑑.(1.3)

We then consider elliptically symmetric random variables, which are obtained by a sequence of (potentially) random linear transformations of spherically symmetric random variables satisfying either (1.1) or (1.2) and show that the properties (1.1) and (1.2) are unaffected by the transformation provided that the eccentricity of the elliptically symmetric random variable is not too extreme, in a sense which we make precise. Finally we show Gaussian-like limiting behaviour for individual components of a random variable from this class, both in terms of their marginal distribution, and in terms of their maximum.

Section 2 presents the main results, which are briefly summarised and placed in context in Section 3; proofs are provided in Section 4.

2. Results

Our first result provides a class of densities and associated scaling constants that satisfy (1.2).

Theorem 2.1. Let {𝐗𝑑} be a sequence of spherically symmetric random variables with density given by (1.3). Let 𝑔𝐶2 satisfy 𝑟𝑑𝑑𝑟2𝑔(𝑟)𝑎𝑠𝑟,(2.1) and let 𝑟𝑑 be a solution to 𝑟𝑑𝑑𝑟𝑔(𝑟)=𝑑.(2.2) Then there is a sequence of solutions which satisfies 𝑟𝑑, where 𝑟𝑑 is unique for sufficiently large 𝑑. Elements of this sequence and 𝐗𝑑 together satisfy (1.2).

The class of interest therefore includes the exponential power family, which has densities proportional to exp(𝐱𝑎)(𝑎>0), and 𝑟𝑑=(𝑑/𝑎)1/𝑑; indeed the class includes any density with polynomial exponents.

Heuristically, the mass of 𝑅𝑑=𝐗𝑑 must concentrate around a particular radius, 𝑟𝑑, so that the effective width of the support becomes negligible compared to 𝑟𝑑 as 𝑑. Essentially (2.2) ensures that 𝑟𝑑 is at least a local mode of the density of log𝑅𝑑, and (2.1) together with the existence of a sequence of solutions 𝑟𝑑 forces the curvature (compared to the scale of log𝑟𝑑) of the log-density of log𝑅𝑑 at this sequence of modes to increase without bound.

Condition (2.1) fails for densities where the radial mass does not become concentrated, such as the log-normal, 𝑓(𝐱)𝐱1exp((log𝐱𝜇)2/(2𝜎2)). To see this explicitly for the log-normal form for the density of 𝐱, note that the marginal radial density, that is, the density of 𝑅=𝐗, is proportional to𝑟𝑑1[]exp𝑔(𝑟)𝑟11exp2𝜎2log𝑟𝜇+(𝑑1)𝜎22,(2.3) which is itself a log-normal density with parameters (𝜇+(𝑑1)𝜎2,𝜎2). Taking 𝑟𝑑=exp[𝜇+(𝑑1)𝜎2] we therefore find that for all 𝑑1||||𝐗𝑑||||𝑟𝑑LN0,𝜎2.(2.4)

Theorem 2.1 requires 𝑔𝐶2; however, other functional forms can also lead to the desired convergence, although not necessarily with 𝑟𝑑. For example, if exp𝑔(𝑟)=𝟏{𝑟1} then the marginal radial density is proportional to 𝑟𝑑1𝟏{𝑟1}; trivially, in this case, the mass therefore concentrates around 𝑟𝑑=1 as 𝑑.

We next show that (1.1) and (1.2) continue to hold after a linear transformation is applied to each 𝐗𝑑, providing that the resulting sequence of elliptically symmetric random variables is not too eccentric.

Theorem 2.2. Let {𝐗𝑑} be a sequence of spherically symmetric random variables and {𝑟𝑑} a sequence of positive constants. Further let {𝐓𝑑} be a sequence of random linear maps on 𝑑 which are independent of {𝐗𝑑}. Denote the eigenvalues of 𝐓𝑡𝑑𝐓𝑑 by 𝜆𝑑,1𝜆𝑑,2𝜆𝑑,𝑑0, and set 𝐵𝑑=𝑑𝑖=1𝜆𝑑,𝑖. If 𝜆𝑑,1𝐵𝑑𝑝0,(2.5) then 𝐗𝑑𝑟𝑑𝑝𝑑11/2𝐓𝑑𝐗𝑑𝐵𝑑1/2𝑟𝑑𝑝1,(2.6)𝐗𝑑𝑟𝑑m.s.𝑑11/2𝐓𝑑𝐗𝑑𝐵𝑑1/2𝑟𝑑m.s.1.(2.7)

The class of elliptically symmetric random variables therefore includes, for example, densities of the form exp((𝐱𝑡Λ𝐱)𝑎)(𝑎>0), for symmetric Λfor which the sum of the eigenvalues is much larger than their maximum.

Our final theorem demonstrates that even if the weaker condition (1.1) is satisfied by a spherically symmetric sequence, then any limiting one-dimensional marginal distribution is Gaussian; it also provides a slightly weaker result for elliptically symmetric sequences as well as a limiting bound on the maximum of all of the components.

Theorem 2.3. Let the sequence of spherically symmetric random variables {𝐗𝑑} and the sequence of positive constants {𝑟𝑑} satisfy (1.1), and let the sequence of 𝑑-dimensional linear maps, {𝐓𝑑}, satisfy (2.5).
(1)For any sequence of unit vectors {𝐞𝑑}, which may be random, but is independent of {𝐗𝑑},𝑑1/2𝑟𝑑𝐗𝑑𝐞𝑑𝐷𝑁(0,1).(2.8)(2)For any sequence of random unit vectors {𝐞𝑑}, with 𝐞𝑑𝑑 uniformly distributed on the surface of a unit 𝑑-sphere and independent of 𝐗𝑑 and 𝐓𝑑,𝑑𝐵𝑑1/2𝑟𝑑𝐓𝑑𝐗𝑑𝐞𝑑𝐷𝑁(0,1).(2.9)(3)Denote the 𝑖th component of 𝐗𝑑 as 𝑋𝑑,𝑖. Then𝑑1/2(2log𝑑)1/2𝑟𝑑max𝑖=1,,𝑑𝑋𝑝𝑑,𝑖1.(2.10)

It should be noted that the first part of Theorem 2.3 is not simply a standard consequence of the central limit theorem. Rather it results from the fact that the standard 𝑑-dimensional Gaussian satisfies condition (1.1), and hence any other sequence which satisfies (1.1) becomes in some sense “close” to a 𝑑-dimensional Gaussian as 𝑑, close enough that the marginal one-dimensional distributions start to resemble each other.

The resemblance to a standard multivariate Gaussian is sufficient for a similar deterministic limit on the maximum of all of the components (Part 3); however, the well-known limiting Gumbel distribution for the maximum of a set of independent Gaussians (see Section 4.3) is not shared by all members of this class.

3. Discussion

It is well known (e.g., [1]) that any given spherically (or elliptically) symmetric random variable can be represented as a mixture of Gaussians; the marginal distribution of any given component is therefore also a mixture of Gaussians. The authors in [2] consider spherically symmetric distributions with support confined to the surface of a sphere and show that the limiting distribution of any 𝑘 fixed components as total the number of components 𝑑 is muitivariate normal. Further, in [3] they show that for a sequence of independent and identically distributed components, the marginal one-dimensional distribution along all but a vanishingly small fraction of random unit vectors becomes closer and closer to Gaussian as dimension 𝑑.

In a sense we have presented an intersection of these ideas: a class of spherical and elliptical distributions, which are not confined to a spherical or elliptical surface, but which become concentrated about the surface as 𝑑, and for which the limiting marginal distribution is Gaussian, not a mixture. Moreover, the maximum component size is bounded in proportion to (log𝑑)1/2, in a similar manner to the maximum component size of a high-dimensional Gaussian. A sufficient condition for the functional form has been provided, and this is satisfied, for example, by the exponential power distribution.

The Gaussian-like limit properties are fundamental to results in [4, 5] where, it is shown that if the proposal distribution for a random walk Metropolis algorithm is chosen from this class then some aspects of the behaviour of the algorithm can become deterministic and, in particular, that the optimal acceptance rate approaches a known fixed value as 𝑑.

4. Proofs of Results

4.1. Proof of Theorem 2.1

It will be helpful to define 𝑅𝑑=𝐗𝑑 and 𝑈𝑑=log𝑅𝑑 and to transform the problem to that of approximating a single integral:(𝑅(𝑎,𝑏))𝑏𝑎𝑑𝑟𝑟𝑑1[]=exp𝑔(𝑟)log𝑏log𝑎𝑑𝑢exp𝑢𝑑𝑔(𝑒𝑢).(4.1) Here and elsewhere for clarity of exposition we sometimes omit the subscript, 𝑑.

Theorem 2.1 is proved in three parts.(i)We first show that, for 𝑑>𝑑 (for some 𝑑>0), the density exp[𝑢𝑑𝑔(𝑒𝑢)] attains a unique maximum in [𝑢,) for some fixed 𝑢. We will denote the value at which this maximum occurs as 𝑢𝑑. The required sequence of scalings will turn out to be 𝑟𝑑=exp(𝑢𝑑).(ii)Convexity arguments are then applied to show that 𝐗𝑑𝑟𝑑𝑝1.(4.2)(iii)It is then shown that for any fixed𝑘>01𝑟𝑘𝑑𝔼𝐗𝑑𝑘1.(4.3)

Applying this with 𝑘=1 and 𝑘=2 provides the required result.

4.1.1. Existence of a Unique Maximum in [𝑢,)

Define 𝜂(𝑢)=𝑔(𝑒𝑢). Clearly 𝜂and 𝜂𝐶2; also condition (2.1) is equivalent tolim𝑢𝜂(𝑢)=.(4.4) Hence, we may define𝑢=inf𝑢𝜂𝑢>1𝑢>𝑢.(4.5)

Lemma 4.1. Subject to condition (4.4), 𝑑 such that for all 𝑑>𝑑 there is a solution 𝑢𝑑>𝑢 to the equation 𝜂(𝑢)=𝑑 which is unique in [𝑢,). Moreover, 𝑢𝑑.

Proof. For 𝑢>𝑢, 𝜂(𝑢)>𝜂(𝑢)+𝑢𝑢. Let 𝑑 be the first positive integer greater than 𝜂(𝑢) then clearly there is a solution to 𝜂(𝑢)=𝑑forall𝑑𝑑.
If there are two such solutions, 𝑢 and 𝑢 with 𝑢>𝑢>𝑢, then we obtain a contradiction since, by the intermediate value theorem. 𝜂0=𝑢𝜂𝑢𝑢𝑢=𝜂𝑢forsome𝑢𝑢,𝑢.(4.6) Next consider successive solutions, 𝑢𝑑 and 𝑢𝑑+1 for 𝑑>𝑑 and again apply the intermediate value theorem. 1𝑢𝑑+1𝑢𝑑=𝜂𝑢𝑑+1𝜂𝑢𝑑𝑢𝑑+1𝑢𝑑=𝜂𝑢>0,(4.7) for some 𝑢, since 𝑢>𝑢. Therefore, 𝑢𝑑+1>𝑢𝑑 and the sequence {𝑢𝑑𝑑𝑑} is monotonic and therefore must approach a limit. Suppose that this limit is finite, 𝑢𝑑𝑐. Then, since 𝜂 is continuous, 𝜂(𝑢𝑑)𝜂(𝑐)<. This contradicts the fact that 𝜂(𝑢𝑑)=𝑑, hence 𝑢𝑑.

4.1.2. Convergence in Probability

Lemma 4.2. Let {𝐗𝑑} be a sequence of spherically symmetric random variables with density given by (1.3). If 𝑔𝐶2 and satisfies (2.1), then there is a sequence 𝑟𝑑 such that 𝐗𝑑𝑟𝑑𝑝1.(4.8)

In proving Lemma 4.2 we consider the log-density (up to a constant) of 𝑈𝑑:𝜓𝑑(𝑢)=𝑢𝑑𝜂(𝑢).(4.9)

Note that condition (4.4) implies that 𝜓𝑑(𝑢) as 𝑢, and 𝜓𝑑(𝑢)<1forall𝑢>𝑢.

We now assume 𝑑>𝑑 and consider the integral 𝑑𝑢exp[𝜓𝑑(𝑢)]. This integral must be finite for all 𝑑 greater than some 𝑑, since otherwise {𝑅𝑑} cannot be an infinite sequence of random variables. For a given 𝛿(0,1), the area of integration is partitioned into five separate regions:(i)𝑅1(𝑑)=(,𝑢];(ii)𝑅2(𝑑)=(𝑢,𝑢𝑑+log(1𝛿)];(iii)𝑅3(𝑑)=(𝑢𝑑+log(1𝛿),𝑢𝑑];(iv)𝑅4(𝑑)=(𝑢𝑑,𝑢𝑑+log(1+𝛿)];(v)𝑅5(𝑑)=(𝑢𝑑+log(1+𝛿),).

It will be convenient to define the respective integrals𝐼𝑖(𝑑)=𝑅𝑖(𝑑)𝜓𝑑𝑢exp𝑑((𝑢)𝑖=1,,5).(4.10)

Note that𝐼3(𝑑)+𝐼4(𝑑)(1+𝛿)exp(𝑢𝑑)𝑢(1𝛿)exp𝑑𝑑𝑟𝑓𝑟(𝑟),(4.11) where 𝑓𝑟(𝑟) is the density of 𝑅. The required convergence in probability will therefore be proven if we can show that, by taking 𝑑 large enough, each of 𝐼1(𝑑),𝐼2(𝑑), and 𝐼5(𝑑) can be made arbitrarily small compared with either 𝐼3(𝑑) or 𝐼4(𝑑).

The next three propositions arise from convexity arguments and will be applied repeatedly to bound certain ratios of integrals.

Proposition 4.3. Let 𝜓[𝑢,) have 𝜓(𝑢)<0. For any 𝑢0,𝑢1[𝑢,), 𝑢1𝑢0𝑑𝑢𝑒𝜓(𝑢)𝑒𝜓(𝑢1)𝑢1𝑢0𝜓𝑢0𝑢𝜓1𝑒𝜓(𝑢0)𝜓(𝑢1)1.(4.12)

Proof. Define the interval 𝐾=[𝑢0,𝑢1] if 𝑢1>𝑢0, and [𝑢1,𝑢0] otherwise. By the concavity of 𝜓, 𝑢𝜓(𝑢)𝜓1+𝜓𝑢1𝑢𝜓0𝑢1𝑢0𝑢𝑢1,𝑢𝐾.(4.13) Hence, 𝑢1𝑢0𝑑𝑢𝑒𝜓(𝑢)𝑒𝜓(𝑢1)𝑢1𝑢0𝜓𝑢𝑑𝑢exp1𝑢𝜓0𝑢1𝑢0𝑢𝑢1.(4.14) The result follows on evaluating the right-hand integral.

Proposition 4.4. Let 𝜓[𝑢,) have 𝜓(𝑢)0. For any 𝑢0,𝑢1[𝑢,) with 𝑢1>𝑢0 and 𝜓(𝑢0)>𝜓(𝑢1), 𝑢1𝑑𝑢𝑒𝜓(𝑢)𝑒𝜓(𝑢1)𝑢1𝑢0𝜓𝑢0𝑢𝜓1.(4.15)

Proof. By the concavity of 𝜓, 𝑢𝜓(𝑢)𝜓1+𝜓𝑢1𝑢𝑢1𝑢𝜓1+𝜓𝑢1𝑢𝜓0𝑢1𝑢0𝑢𝑢1𝑢,𝑢1.,(4.16) Hence, 𝑢1𝑑𝑢𝑒𝜓(𝑢)𝑒𝜓(𝑢1)𝑢1𝜓𝑢𝑑𝑢exp1𝑢𝜓0𝑢1𝑢0𝑢𝑢1.(4.17) Since (𝜓(𝑢1)𝜓(𝑢0))/(𝑢1𝑢0)is negative, the result follows on evaluating the right-hand integral.

The proof for the following is almost identical to that of Proposition 4.4 and is therefore omitted.

Proposition 4.5. Let 𝜓 have 𝜓(𝑢)0. For any 𝑢0,𝑢1 with 𝑢1<𝑢0 and 𝜓(𝑢0)>𝜓(𝑢1), 𝑢1𝑑𝑢𝑒𝜓(𝑢)𝑒𝜓(𝑢1)𝑢0𝑢1𝜓𝑢0𝑢𝜓1.(4.18)

Corollary 4.6. One has 𝐼5𝐼4+𝐼5𝜓𝑢exp𝑑𝑢+log(1+𝛿)𝜓𝑑.(4.19)

Proof. Set 𝑢0=𝑢𝑑 and 𝑢1=𝑢𝑑+log(1+𝛿) in Propositions 4.3 and 4.4 to obtain 𝐼4𝐼5𝜓𝑢exp𝑑𝑢𝜓𝑑+log(1+𝛿)1.(4.20) But 𝐼5𝐼4+𝐼5=11+𝐼4/𝐼5,(4.21) and so the result follows.

Corollary 4.7. One has 𝐼2𝐼2+𝐼3𝜓𝑢exp𝑑𝑢+log(1𝛿)𝜓𝑑.(4.22)

Proof. Define 𝜓𝑐(𝑢)=𝜓(𝑢),𝑢>𝑢,𝜓𝑢𝑢𝜓𝑢𝑢,𝑢<𝑢.(4.23) By definition, 𝜓𝑐(𝑢), and 𝜓(𝑢)0forall𝑢. Let 𝐼1𝑐=𝑅1𝜓𝑑𝑢exp𝑐(𝑢),(4.24) and note that 𝜓(𝑢)>0 since 𝜓(𝑢)0forall𝑢𝑢, and 𝜓(𝑢𝑑)=0 with 𝑢𝑑>𝑢. Hence, 𝑅1𝑑𝑢exp[𝜓𝑐(𝑢)] exists.
Set 𝑢0=𝑢𝑑 and 𝑢1=𝑢𝑑+log(1𝛿) in Propositions 4.3 and 4.5 to obtain 𝐼3𝐼1𝑐+𝐼2𝜓𝑢exp𝑑𝑢𝜓𝑑+log(1𝛿)1.(4.25) But 𝐼2𝐼2+𝐼3𝐼1𝑐+𝐼2𝐼1𝑐+𝐼2+𝐼3=11+𝐼3/𝐼1𝑐+𝐼2(4.26) and so the result follows.
We now consider 𝐼1(𝑑) and use the fact that 𝑅1𝑑𝑢exp[𝜓𝑑(𝑢)] must exist for all 𝑑>𝑑 (for some 𝑑>0) for {𝑅𝑑} to be an infinite sequence of random variables. Also note that 𝜓𝑑(𝑢)𝜓𝑘(𝑢)=(𝑑𝑘)𝑢, which is an increasing function for 𝑑>𝑘.

Corollary 4.8. If 𝐼1(𝑘)< for some 𝑘>0 and if for all 𝑑>𝑘, 𝜓𝑑(𝑢)𝜓𝑘(𝑢) is an increasing function of 𝑢, then 𝐼1(𝑑)𝐼2(𝑑)+𝐼3(𝑑)𝑒𝜓𝑘(𝑢)𝐼1(𝑘)𝑢𝑑𝑢.(4.27)

Proof. By the monotonicity of 𝜓𝑑𝜓𝑘, 𝐼1(𝑑)=𝑢𝑑𝑢𝑒𝜓𝑑(𝑢)𝜓𝑘(𝑢)𝑒𝜓𝑘(𝑢)𝑒𝜓𝑑(𝑢)𝜓𝑘(𝑢)𝐼1(𝑘).(4.28) By Proposition 4.3 with 𝑢0=𝑢𝑑 and 𝑢1=𝑢𝐼2(𝑑)+𝐼3(𝑑)𝑒𝜓𝑑(𝑢)𝑢𝑑𝑢𝜓𝑑𝑢𝑑𝜓𝑑𝑢𝑒𝜓𝑑(𝑢𝑑)𝜓𝑑(𝑢)1𝑒𝜓𝑑(𝑢)𝑢𝑑𝑢,(4.29) where the last statement follows since for 𝑥>0, 𝑒𝑥>1+𝑥.
The result follows from combining the two inequalities.

We next combine Corollaries (1.1), (1.2), and (1.3) to prove the sufficient condition for the required convergence in probability. We show that if Condition (4.4) is satisfied, then𝐼1(𝑑)+𝐼2(𝑑)+𝐼5(𝑑)𝐼1(𝑑)+𝐼2(𝑑)+𝐼3(𝑑)+𝐼4(𝑑)+𝐼5(𝑑)0as𝑑.(4.30)

By Lemma 4.1  𝑢𝑑 as 𝑑, and so from Corollary 4.8𝐼1(𝑑)𝐼2(𝑑)+𝐼3(𝑑)0.(4.31)

Since 𝑢𝑑, given some 𝛿(0,𝛿0) and any 𝑀>0, we may choose a 𝑑0 such that, for all 𝑑>𝑑0 and all 𝛿(0,𝛿),(log(1+𝛿))2𝜂𝑢𝑑+log1+𝛿𝑀.(4.32)

Taylor expand 𝜓𝑑 about 𝑢𝑑, recalling that 𝜓𝑑(𝑢𝑑)=0 and 𝜓𝑑(𝑢)=𝜂(𝑢):𝜓𝑑𝑢𝑑𝜓𝑑𝑢𝑑=1+log(1+𝛿)2(log(1+𝛿))2𝜂𝑢𝑑+log1+𝛿12𝑀,(4.33)

for some 𝛿(0,𝛿). From Corollary 4.6 we therefore see that𝐼5(𝑑)𝐼4(𝑑)+𝐼5(𝑑)𝑒(1/2)𝑀.(4.34)

Similarly, from Corollary 4.7𝐼2(𝑑)𝐼2(𝑑)+𝐼3(𝑑)𝑒(1/2)𝑀.(4.35)

But𝐼1(𝑑)+𝐼2(𝑑)+𝐼5(𝑑)𝐼1(𝑑)+𝐼2(𝑑)+𝐼3(𝑑)+𝐼4(𝑑)+𝐼5(𝑑)𝐼1(𝑑)𝐼2(𝑑)+𝐼3(𝑑)+𝐼2(𝑑)𝐼2(𝑑)+𝐼3(𝑑)+𝐼5(𝑑)𝐼4(𝑑)+𝐼5(𝑑),(4.36)

and each of the terms on the right-hand side can be made as small as desired by taking 𝑑 large enough.

4.1.3. Convergence of 𝑘th Moment

Proposition 4.9. Let 𝑟𝑑 be the (eventually) unique solution to the equation 𝑟𝑔(𝑟)=𝑑1.(4.37) If 𝑔(𝑟) satisfies (2.1) then for any fixed k>0lim𝑑𝑟𝑑𝑟𝑑+𝑘=1.(4.38)

Proof. Without loss of generality assume that 𝑟𝑑+𝑘>𝑟𝑑. Hence, by the Intermediate Value Theorem, there exists a value 𝑟[𝑟𝑑,𝑟𝑑+𝑘] such that 𝑘𝑟𝑑+𝑘𝑟𝑑+𝑘𝑟𝑑>𝑟𝑘𝑟𝑑+𝑘𝑟𝑑=𝑟𝑟𝑑+𝑘𝑔𝑟𝑑+𝑘𝑟𝑑𝑟𝑔𝑑𝑟𝑑+𝑘𝑟𝑑=𝑟𝑑𝑑𝑟𝑟𝑔|||(𝑟)𝑟.(4.39) Thus, 𝑟𝑑+𝑘𝑟𝑑𝑟𝑑+𝑘0,(4.40) and the result follows.

Lemma 4.10. For fixed 𝑘>0, 1𝑟𝑘𝑑𝔼𝑑𝑅𝑘1.(4.41)

Proof. Set 𝐼1=0𝑑𝑟𝑟𝑑1[],𝐼exp𝑔(𝑟)2=0𝑑𝑟𝑟𝑑1+𝑘[].exp𝑔(𝑟)(4.42) If 𝑔(𝑟) satisfies (2.1) then so does 𝑔(𝑟)𝑘log(𝑟). Therefore, from Lemma 4.2, given 𝜖>0 and 𝛿>0 there is a 𝑑1 such that, for all 𝑑>𝑑1, (1𝜖)𝐼2<𝑟𝑑+𝑘𝑟(1+2𝛿)𝑑+𝑘(12𝛿)𝑑𝑟𝑟𝑑1+𝑘[]exp𝑔(𝑟)<𝐼2.(4.43) Furthermore, by Proposition 4.9, there is a 𝑑2 such that, for all 𝑑>𝑑2, 𝑟𝑑(1𝛿)<𝑟𝑑+𝑘<(1+𝛿)𝑟𝑑.(4.44) Therefore, since the integrand is positive, for all 𝑑>𝑚𝑎𝑥(𝑑1,𝑑2), (1𝜖)𝐼2<𝑟𝑑𝑟(1+2𝛿)(1+𝛿)𝑑(12𝛿)(1𝛿)𝑑𝑟𝑟𝑑1+𝑘[]exp𝑔(𝑟)𝑟𝑘𝑑(1+2𝛿)𝑘(1+𝛿)𝑘𝑟𝑑𝑟(1+2𝛿)(1+𝛿)𝑑(12𝛿)(1𝛿)𝑑𝑟𝑟𝑑1[].exp𝑔(𝑟)(4.45) Similarly 𝐼2>𝑟𝑑𝑟(1+2𝛿)(1𝛿)𝑑(12𝛿)(1+𝛿)𝑑𝑟𝑟𝑑1+𝑘[]exp𝑔(𝑟)𝑟𝑘𝑑(12𝛿)𝑘(1+𝛿)𝑘𝑟𝑑𝑟(1+2𝛿)(1𝛿)𝑑(12𝛿)(1+𝛿)𝑑𝑟𝑟𝑑1[].exp𝑔(𝑟)(4.46) Applying Lemma 4.2 again, there is a 𝑑3 such that, for all 𝑑>𝑑3, (1𝜖)𝐼1<𝑟𝑑𝑟(1+2𝛿)(1𝛿)𝑑(12𝛿)(1+𝛿)𝑑𝑟𝑟𝑑1[]exp𝑔(𝑟)<𝐼1.(4.47) Therefore, for all 𝑑>𝑚𝑎𝑥(𝑑1,𝑑2,𝑑3), (1𝜖)𝐼2<𝑟𝑘𝑑(1+2𝛿)𝑘(1+𝛿)𝑘𝐼1,𝐼2>𝑟𝑘𝑑(12𝛿)𝑘(1+𝛿)𝑘(1𝜖)𝐼1.(4.48) Hence, (12𝛿)𝑘(1+𝛿)𝑘1(1𝜖)<𝑟𝑘𝑑𝐼2𝐼1<(1+2𝛿)𝑘(1+𝛿)𝑘(1𝜖)1.(4.49) The result follows since 𝛿 and 𝜖 can be made arbitrarily small.

4.2. Proof of Theorem 2.2

Any spherically symmetric random variable can be decomposed into a uniform angular component and a radial distribution. We may therefore create an invertible map from any 𝑑-dimensional spherically symmetric random variable 𝐕 with a continuous radial distribution function to a standard 𝑑-dimensional Gaussian, 𝐙. We will apply the following map: set𝐙=𝐹1𝐙𝐹𝐕(𝐕),(4.50) where 𝐹𝐕() and 𝐹𝐙() are the distribution functions of 𝐕 and 𝐙, respectively, and then fix𝐙=𝐙𝐕𝐕.(4.51)

This mapping is key both to the proofs of both Theorems 2.2 and 2.3. To simplify the exposition in both this section and Section 4.3 we define𝐕𝑑1=𝑟𝑑𝐗𝑑.(4.52)

The following is therefore equivalent to (2.6).

Lemma 4.11. Define {𝐕𝑑}, {𝐓𝑑}, {𝜆𝑑,𝑖}, and {𝐵𝑑} as in (4.52) and the statement of Theorem 2.2. If (2.5) holds and 𝐕𝑑𝑝1, then 𝑑𝐵𝑑𝐕𝑡𝑑𝐓𝑡𝑑𝐓𝑑𝐕𝑑𝑝1.(4.53)

Proof. For some 𝛿>0, let 𝐴𝑑𝐓=𝑑𝜆𝑑,1𝑑𝑖=1𝜆𝑑,𝑖<𝛿3.(4.54) For now fix 𝑑 and 𝐓𝑑𝐴𝑑, and suppress the subscript 𝑑. Denote the spectral decomposition of 𝐓𝑡𝐓 as 𝐋𝑡Λ𝐋, where Λ=diag(𝜆1,,𝜆𝑑). We will initially consider the Gaussian 𝐙 and define 𝐙=𝐋𝐙; since 𝐋 is orthonormal, it follows that 𝐙𝑁(𝟎,𝐈𝑑).
Define 𝐙𝑊=𝑡𝐓𝑡𝐓𝐙𝐵=𝐙𝑡𝚲𝐙𝐵.(4.55) Then, for fixed 𝑑, 𝔼𝐙[𝑊]=1𝐵𝔼𝐙𝑑𝑖=1𝜆𝑖𝑍𝑖2=1,Var𝐙[𝑊]=1𝐵2Var𝐙𝑑𝑖=1𝜆𝑖𝑍𝑖2=2𝑑𝑖=1𝜆2𝑖𝑑𝑖=1𝜆𝑖2𝜆21𝑑𝑖=1𝜆𝑖<2𝛿3.(4.56) Chebyshev’s inequality gives 𝐙||𝑊𝔼𝐙[𝑊]||<>𝛿Var𝐙[𝑊]𝛿2,thatis,𝐙||||𝑊1>𝛿<2𝛿.(4.57) By (2.5) there is a 𝑑0 such that, for all 𝑑>𝑑0, (𝐓𝑑𝐴𝑑)<𝛿. Thus, for all 𝑑>𝑑0, 𝐓,𝐙||||||||𝑊1>𝛿𝑊1>𝛿𝐓𝐴+(𝐓𝐴)<3𝛿.(4.58) Hence, 𝑊𝑑𝑝1. Now 𝑑𝐵𝑑𝐕𝑡𝑑𝐓𝑡𝑑𝐓𝑑𝐕𝑑=𝑑𝐙𝑑2𝐕𝑑2𝑊𝑑,(4.59) and since each of the three terms converge in probability to 1, so does the product.

We now turn to the proof of convergence in mean square and first show an equivalence of the expected second moments of the norms.

Proposition 4.12. For {𝐕𝑑}, {𝐓𝑑}, {𝜆𝑑,𝑖}, and {𝐵𝑑} to be as defined in (4.52) and the statement of Theorem 2.2, 𝔼𝐕𝑑2=𝑑𝐵𝑑𝔼𝐓𝑑𝐕𝑑2.(4.60)

Proof. For clarity of exposition we suppress the subscript 𝑑. Since 𝐕 is spherically symmetric we may without loss of generality consider it with axes along the principle components of 𝐓. Then 𝔼𝐓𝐕2=𝔼𝑑𝑖=1𝜆2𝑖𝑉2𝑖=𝑑𝑖=1𝜆2𝑖𝔼𝑉2𝑖.(4.61) But, again, 𝐕 is spherically symmetric so this is 𝑑𝑖=1𝜆2𝑖𝔼𝑉21=1𝑑𝑑𝑖=1𝜆2𝑖𝑑𝑗=1𝔼𝑉2𝑗=𝐵𝑑𝑑𝔼𝐕2.(4.62) Turning now to convergence in mean square itself, note that, by Proposition 4.12, 𝔼𝑑1/2𝐵𝑑𝐓𝑑𝐕𝑑12𝐕𝔼𝑑12𝔼𝑑=21/2𝐵𝑑𝐓𝑑𝐕𝑑𝐕𝔼𝑑.(4.63) But (1.2) implies that 𝐕𝑑m.s.1, and hence it is sufficient to show that 𝔼𝑑1/2𝐵𝑑𝐓𝑑𝐕𝑑𝐕𝔼𝑑0.(4.64) Now, by Lemma 4.11 and Proposition 4.12, 𝑑1/2𝐵𝑑𝐓𝑑𝐕𝑑𝑝𝑑1,𝔼1/2𝐵𝑑𝐓𝑑𝐕𝑑2𝐕=𝔼𝑑21.(4.65) We now require Scheffe’s Lemma, which states that, for any sequence of random variables {𝑌𝑑}, if 𝔼[𝑌2𝑑]1 and 𝑌𝑑𝑝1, then 𝔼[𝑌𝑑]1. Hence 𝔼[(𝑑1/2/𝐵𝑑)𝐓𝑑𝐕𝑑]1. Now (1.2) also implies that 𝔼[𝐕𝑑]1, and hence, (4.64) is satisfied.

4.3. Proof of Theorem 2.3

Throughout this section we define 𝐙𝑑 and 𝐕𝑑 as in Section 4.2. We first prove Part 1.

Given 𝛿>0, it will be convenient to define the following event:𝐴𝑑||||𝐙=𝑑𝑑1/21𝐕𝑑||||1<𝛿.(4.66)

Now, for 𝐞𝑑 independent of 𝐕𝑑 (and 𝐙𝑑),𝑑1/2𝐕𝑑𝐞𝑑𝐙𝑎=𝑑𝐞𝑑𝐙𝑎𝑑𝑑1/21𝐕𝑑𝑎𝐙=Φ𝑑𝑑1/21𝐕𝑑,(4.67) and soΦ𝑑(𝑎(1𝛿))<1/2𝐕𝑑𝐞𝑑𝑎𝐴(𝑑)<Φ(𝑎(1+𝛿)).(4.68)

For any event 𝐸,||||(𝐸)(𝐸𝐴)=(𝐴𝑐)||(𝐸𝐴𝑐||)(𝐸𝐴)(𝐴𝑐)(4.69) and, in particular,||𝑑1/2𝐕𝑑𝐞𝑑𝑑𝑎1/2𝐕𝑑𝐞𝑑||𝐴𝑎𝐴𝑐𝑑.(4.70)

Given 𝜖>0, by (1.1) we may define 𝑑0 such that, for all 𝑑>𝑑0, (𝐴𝑐𝑑)<𝜖. Thus, for all 𝑑>𝑑0,Φ(𝑎(1𝛿))𝜖<𝐕𝑑𝑑1/2𝐕𝑑𝐞(𝑑)𝑎<Φ(𝑎(1+𝛿))+𝜖.(4.71)

By taking 𝑑 large enough we can make 𝛿 and 𝜖 as small as desired. Moreover, since Φ() is bounded and monotonic, 𝛿>0 such that |Φ(𝑎(1+𝛿))Φ(𝑎)|<𝜖forall𝛿 with |𝛿|<𝛿, and hencelim𝑑𝑑1/2𝐕𝑑𝐞𝑑𝑎=Φ(𝑎).(4.72)

To prove Part 2, first note that, whereas 𝐙𝑑𝐞𝑑𝑁(0,1), (𝐓𝑑𝐙𝑑)𝐞𝑑𝑁(0,𝐓𝑑𝐞𝑑2), and so𝐕𝑑𝑑𝐵𝑑1/2𝐓𝑑𝐕𝑑𝐞𝑑𝑎=𝐙𝑑1𝐓𝑑𝐞𝑑𝐓𝑑𝐙𝑑𝐞𝑑𝐵𝑎𝑑1/2𝑑1/2𝐓𝑑𝐞𝑑𝐙𝑑𝑑1/21𝐕𝑑𝑎𝐵=Φ𝑑1/2𝑑1/2𝐓𝑑𝐞𝑑𝐙𝑑𝑑1/21𝐕𝑑.(4.73)

But a unit vector 𝐞𝑑 chosen uniformly at random can be written as 𝐙𝑑/||𝐙𝑑|| for some standard 𝑑-dimensional Gaussian 𝐙𝑑. Hence, by Theorem 2.2,𝑑1/2𝐵𝑑1/2𝐓𝑑𝐞𝑑=𝑑1/2𝐙𝑑𝐓𝑑𝐙𝑑𝐵𝑑𝑝1/21.(4.74)

We now define the event𝐴𝑑||||𝐵=𝑑1/2𝑑1/2𝐓𝑑𝐞𝑑𝐙𝑑𝑑1/21𝐕𝑑||||1<𝛿,(4.75) and the proof follows as for Part 1.

In proving Part 3 we require the following standard result (e.g., Theorem  1.5.3, [4]). Set𝑎𝑑=(2log𝑑)1/2,𝑏𝑑=(2log𝑑)1/212𝑎𝑑loglog𝑑+log(4𝜋).(4.76)

Also let 𝐺() be the distribution function of a Gumbel random variable, and let 𝑍1,,𝑍𝑑 be independent and identically distributed 𝑁(0,1) random variables. Then𝐺𝑑1(𝑐)=𝑎𝑑max𝑖=1,,𝑑𝑍𝑖𝑏𝑑𝑐𝐺(𝑐).(4.77)

Replacing 𝑐 with 𝑐𝑑=[loglog𝑑+log(4𝜋)]/2=((2log𝑑)1/2𝑏𝑑)/𝑎𝑑 or with 𝑐𝑑=𝛼(2log𝑑)1/2+[loglog𝑑+log(4𝜋)]/2=((1𝛼)(2log𝑑)1/2𝑏𝑑)/𝑎𝑑 (𝛼>0) givesmax𝑖=1,,𝑑𝑍𝑖(2log𝑑)1/2𝑐𝐺𝑑1,max𝑖=1,,𝑑𝑍𝑖(1𝛼)(2log𝑑)1/2𝑐𝐺𝑑0.(4.78)

Choose 𝛿 in (4.66) small enough that (1𝛿)(2+𝜖)1/2>21/2. Then𝑑1/2max𝑖=1,,𝑑𝑉𝑑,𝑖((2+𝜖)log𝑑)1/2𝐴𝑑=max𝑖=1,,𝑑𝑍𝑑,𝑖𝐙𝑑𝑑1/21𝐕𝑑((2+𝜖)log𝑑)1/2𝐴𝑑>max𝑖=1,,𝑑𝑍𝑑,𝑖(1𝛿)((2+𝜖)log𝑑)1/2𝐴𝑑>max𝑖=1,,𝑑𝑍𝑑,𝑖(2log𝑑)1/2𝐴𝑑>max𝑖=1,,𝑑𝑍𝑑,𝑖(2log𝑑)1/2𝐴𝑐𝑑.(4.79)

Similarly by choosing 𝛿 in (4.66) small enough that (1+𝛿)(2𝜖)1/2>21/2(1𝛼) for some small 𝛼>0,𝑑1/2max𝑖=1,,𝑑𝑉𝑑,𝑖((2𝜖)log𝑑)1/2𝐴𝑑>𝑃max𝑖=1,,𝑑𝑍𝑑,𝑖(1𝛼)(2log𝑑)1/2𝐴𝑐𝑑.(4.80)

In each case the first term tends to 1 and (𝐴𝑐𝑑)0, proving the desired result.

References

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