Journal of Probability and Statistics

Journal of Probability and Statistics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 467187 | https://doi.org/10.1155/2012/467187

Chris Sherlock, Daniel Elton, "A Class of Spherical and Elliptical Distributions with Gaussian-Like Limit Properties", Journal of Probability and Statistics, vol. 2012, Article ID 467187, 17 pages, 2012. https://doi.org/10.1155/2012/467187

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

Academic Editor: Mohammad Fraiwan Al-Saleh
Received04 May 2011
Accepted29 Nov 2011
Published28 Feb 2012

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 β€œshells” with distribution function proportional to𝟏{‖𝐱‖β‰₯π‘Ÿ}. We consider a class of the spherically symmetric random variables for which as dimension π‘‘β†’βˆž the effective range of the mixture of β€œshells” 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 satisfy‖‖𝐗eitherπ‘‘β€–β€–π‘Ÿπ‘‘π‘βŸΆ1,(1.1)‖‖𝐗orπ‘‘β€–β€–π‘Ÿπ‘‘m.s.⟢1,(1.2) for some positive sequence {π‘Ÿπ‘‘}. Here and throughout this paper β€–β‹…β€– 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.β†’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 formπ‘“π‘‘ξ€·π±π‘‘ξ€Έξ€Ίξ€·β€–β€–π±βˆexpβˆ’π‘”π‘‘β€–β€–ξ€Έξ€».(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 π‘”βˆˆπΆ2 satisfy ξ‚€π‘Ÿπ‘‘ξ‚π‘‘π‘Ÿ2𝑔(π‘Ÿ)βŸΆβˆžπ‘Žπ‘ π‘ŸβŸΆβˆž,(2.1) and let π‘Ÿπ‘‘ be a solution to π‘Ÿπ‘‘π‘‘π‘Ÿπ‘”(π‘Ÿ)=𝑑.(2.2) Then there is a sequence of solutions which satisfies π‘Ÿπ‘‘β†’βˆž, where π‘Ÿπ‘‘ 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(βˆ’β€–π±β€–π‘Ž)(π‘Ž>0), and π‘Ÿπ‘‘=(𝑑/π‘Ž)1/𝑑; indeed the class includes any density with polynomial exponents.

Heuristically, the mass of π‘…π‘‘βˆΆ=‖𝐗𝑑‖ must concentrate around a particular radius, π‘Ÿπ‘‘, so that the effective width of the support becomes negligible compared to π‘Ÿπ‘‘ as π‘‘β†’βˆž. Essentially (2.2) ensures that π‘Ÿπ‘‘ is at least a local mode of the density of log𝑅𝑑, and (2.1) together with the existence of a sequence of solutions π‘Ÿπ‘‘β†’βˆž forces the curvature (compared to the scale of logπ‘Ÿπ‘‘) 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)βˆβ€–π±β€–βˆ’1exp(βˆ’(logβ€–π±β€–βˆ’πœ‡)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 toπ‘Ÿπ‘‘βˆ’1[]expβˆ’π‘”(π‘Ÿ)βˆπ‘Ÿβˆ’1ξ‚ƒβˆ’1exp2𝜎2ξ€·ξ€·logπ‘Ÿβˆ’πœ‡+(π‘‘βˆ’1)𝜎2ξ€Έξ€Έ2ξ‚„,(2.3) which is itself a log-normal density with parameters (πœ‡+(π‘‘βˆ’1)𝜎2,𝜎2). Taking π‘Ÿπ‘‘=exp[πœ‡+(π‘‘βˆ’1)𝜎2] we therefore find that for all 𝑑β‰₯1||||𝐗𝑑||||π‘Ÿπ‘‘ξ€·βˆΌLN0,𝜎2ξ€Έ.(2.4)

Theorem 2.1 requires π‘”βˆˆπΆ2; however, other functional forms can also lead to the desired convergence, although not necessarily with π‘Ÿπ‘‘β†’βˆž. For example, if exp𝑔(π‘Ÿ)=𝟏{π‘Ÿβ‰€1} then the marginal radial density is proportional to π‘Ÿπ‘‘βˆ’1𝟏{π‘Ÿβ‰€1}; trivially, in this case, the mass therefore concentrates around π‘Ÿπ‘‘=1 as π‘‘β†’βˆž.

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 {π‘Ÿπ‘‘} a sequence of positive constants. Further let {𝐓𝑑} be a sequence of random linear maps on ℝ𝑑 which are independent of {𝐗𝑑}. Denote the eigenvalues of 𝐓𝑑𝑑𝐓𝑑 by πœ†π‘‘,1β‰₯πœ†π‘‘,2β‹―β‰₯πœ†π‘‘,𝑑β‰₯0, and set π΅π‘‘βˆ‘βˆΆ=𝑑𝑖=1πœ†π‘‘,𝑖. If πœ†π‘‘,1π΅π‘‘π‘βŸΆ0,(2.5) then β€–β€–π—π‘‘β€–β€–π‘Ÿπ‘‘π‘π‘‘βŸΆ1⟹1/2‖‖𝐓𝑑𝐗𝑑‖‖𝐡𝑑1/2π‘Ÿπ‘‘π‘βŸΆ1,(2.6)β€–β€–π—π‘‘β€–β€–π‘Ÿπ‘‘m.s.π‘‘βŸΆ1⟹1/2‖‖𝐓𝑑𝐗𝑑‖‖𝐡𝑑1/2π‘Ÿπ‘‘m.s.⟢1.(2.7)

The class of elliptically symmetric random variables therefore includes, for example, densities of the form exp(βˆ’(𝐱𝑑Λ𝐱)π‘Ž)(π‘Ž>0), for symmetric Ξ›for 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 {π‘Ÿπ‘‘} satisfy (1.1), and let the sequence of 𝑑-dimensional linear maps, {𝐓𝑑}, satisfy (2.5).
(1)For any sequence of unit vectors {πžπ‘‘}, which may be random, but is independent of {𝐗𝑑},𝑑1/2π‘Ÿπ‘‘π—π‘‘β‹…πžπ‘‘π·βŸΆπ‘(0,1).(2.8)(2)For any sequence of random unit vectors {πžπ‘‘}, with πžπ‘‘βˆˆβ„π‘‘ uniformly distributed on the surface of a unit 𝑑-sphere and independent of 𝐗𝑑 and 𝐓𝑑,𝑑𝐡𝑑1/2π‘Ÿπ‘‘ξ€·π“π‘‘π—π‘‘ξ€Έβ‹…πžπ‘‘π·βŸΆπ‘(0,1).(2.9)(3)Denote the 𝑖th component of 𝐗𝑑 as 𝑋𝑑,𝑖. Then𝑑1/2(2log𝑑)1/2π‘Ÿπ‘‘max𝑖=1,…,𝑑𝑋𝑝𝑑,π‘–βŸΆ1.(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 β€œclose” to a 𝑑-dimensional Gaussian as π‘‘β†’βˆž, 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 π‘˜ fixed components as total the number of components π‘‘β†’βˆž 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 π‘‘β†’βˆž.

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 π‘‘β†’βˆž, 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 π‘‘β†’βˆž.

4. Proofs of Results

4.1. Proof of Theorem 2.1

It will be helpful to define π‘…π‘‘βˆΆ=‖𝐗𝑑‖ and π‘ˆπ‘‘βˆΆ=log𝑅𝑑 and to transform the problem to that of approximating a single integral:β„™ξ€œ(π‘…βˆˆ(π‘Ž,𝑏))βˆπ‘π‘Žπ‘‘π‘Ÿπ‘Ÿπ‘‘βˆ’1[]=ξ€œexpβˆ’π‘”(π‘Ÿ)log𝑏logπ‘Žξ€Ίπ‘‘π‘’expπ‘’π‘‘βˆ’π‘”(𝑒𝑒)ξ€».(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>π‘‘βˆ— (for some π‘‘βˆ—>0), the density ∝exp[π‘’π‘‘βˆ’π‘”(𝑒𝑒)] attains a unique maximum in [π‘’βˆ—,∞) for some fixed π‘’βˆ—βˆˆβ„. We will denote the value at which this maximum occurs as 𝑒𝑑. The required sequence of scalings will turn out to be π‘Ÿπ‘‘=exp(𝑒𝑑).(ii)Convexity arguments are then applied to show that β€–β€–π—π‘‘β€–β€–π‘Ÿπ‘‘π‘βŸΆ1.(4.2)(iii)It is then shown that for any fixedπ‘˜>01π‘Ÿπ‘˜π‘‘π”Όξ‚ƒβ€–β€–π—π‘‘β€–β€–π‘˜ξ‚„βŸΆ1.(4.3)

Applying this with π‘˜=1 and π‘˜=2 provides the required result.

4.1.1. Existence of a Unique Maximum in [π‘’βˆ—,∞)

Define πœ‚(𝑒)∢=𝑔(𝑒𝑒). Clearly πœ‚βˆΆβ„β†’β„and πœ‚βˆˆπΆ2; also condition (2.1) is equivalent tolimπ‘’β†’βˆžπœ‚β€²ξ…ž(𝑒)=∞.(4.4) Hence, we may defineπ‘’βˆ—ξ€½βˆΆ=infπ‘’βˆΆπœ‚ξ…žξ…žξ€·π‘’ξ…žξ€Έ>1βˆ€π‘’ξ…žξ€Ύ>𝑒.(4.5)

Lemma 4.1. Subject to condition (4.4), βˆƒπ‘‘βˆ— such that for all 𝑑>π‘‘βˆ— there is a solution 𝑒𝑑>π‘’βˆ— to the equation πœ‚ξ…ž(𝑒)=𝑑 which is unique in [π‘’βˆ—,∞). Moreover, π‘’π‘‘β†’βˆž.

Proof. For 𝑒>π‘’βˆ—, πœ‚ξ…ž(𝑒)>πœ‚ξ…ž(π‘’βˆ—)+π‘’βˆ’π‘’βˆ—. Let π‘‘βˆ— be the first positive integer greater than πœ‚ξ…ž(π‘’βˆ—) then clearly there is a solution to πœ‚ξ…ž(𝑒)=𝑑forall𝑑β‰₯π‘‘βˆ—.
If there are two such solutions, 𝑒′ and π‘’ξ…žξ…ž with π‘’ξ…ž>π‘’ξ…žξ…ž>π‘’βˆ—, then we obtain a contradiction since, by the intermediate value theorem. πœ‚0=ξ…žξ€·π‘’ξ…žξ€Έβˆ’πœ‚ξ…žξ€·π‘’ξ…žξ…žξ€Έπ‘’ξ…žβˆ’π‘’ξ…žξ…ž=πœ‚ξ…žξ…žξ€·π‘’ξ…žξ…žξ…žξ€Έforsomeπ‘’β€²ξ…žξ…žβˆˆξ€Ίπ‘’ξ…ž,π‘’ξ…žξ…žξ€».(4.6) Next consider successive solutions, 𝑒𝑑 and 𝑒𝑑+1 for 𝑑>π‘‘βˆ— and again apply the intermediate value theorem. 1𝑒𝑑+1βˆ’π‘’π‘‘=πœ‚ξ…žξ€·π‘’π‘‘+1ξ€Έβˆ’πœ‚ξ…žξ€·π‘’π‘‘ξ€Έπ‘’π‘‘+1βˆ’π‘’π‘‘=πœ‚ξ…žξ…žξ€·π‘’ξ…žξ€Έ>0,(4.7) for some π‘’ξ…ž, since π‘’ξ…ž>π‘’βˆ—. Therefore, 𝑒𝑑+1>𝑒𝑑 and the sequence {π‘’π‘‘βˆΆπ‘‘β‰₯π‘‘βˆ—} is monotonic and therefore must approach a limit. Suppose that this limit is finite, 𝑒𝑑→𝑐. Then, since πœ‚ξ…ž is continuous, πœ‚ξ…ž(𝑒𝑑)β†’πœ‚ξ…ž(𝑐)<∞. This contradicts the fact that πœ‚ξ…ž(𝑒𝑑)=𝑑, hence π‘’π‘‘β†’βˆž.

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 π‘”βˆˆπΆ2 and satisfies (2.1), then there is a sequence π‘Ÿπ‘‘β†’βˆž such that β€–β€–π—π‘‘β€–β€–π‘Ÿπ‘‘π‘βŸΆ1.(4.8)

In proving Lemma 4.2 we consider the log-density (up to a constant) of π‘ˆπ‘‘:πœ“π‘‘(𝑒)=π‘’π‘‘βˆ’πœ‚(𝑒).(4.9)

Note that condition (4.4) implies that πœ“π‘‘ξ…žξ…ž(𝑒)β†’βˆ’βˆž as π‘’β†’βˆž, and πœ“π‘‘ξ…žξ…ž(𝑒)<βˆ’1forall𝑒>π‘’βˆ—.

We now assume 𝑑>π‘‘βˆ— and consider the integral βˆ«βˆžβˆ’βˆžπ‘‘π‘’exp[πœ“π‘‘(𝑒)]. This integral must be finite for all 𝑑 greater than some π‘‘βˆ—βˆ—, since otherwise {𝑅𝑑} cannot be an infinite sequence of random variables. For a given π›Ώβˆˆ(0,1), the area of integration is partitioned into five separate regions:(i)𝑅1(𝑑)∢=(βˆ’βˆž,π‘’βˆ—];(ii)𝑅2(𝑑)∢=(π‘’βˆ—,𝑒𝑑+log(1βˆ’π›Ώ)];(iii)𝑅3(𝑑)∢=(𝑒𝑑+log(1βˆ’π›Ώ),𝑒𝑑];(iv)𝑅4(𝑑)∢=(𝑒𝑑,𝑒𝑑+log(1+𝛿)];(v)𝑅5(𝑑)∢=(𝑒𝑑+log(1+𝛿),∞).

It will be convenient to define the respective integrals𝐼𝑖(𝑑)ξ€œβˆΆ=𝑅𝑖(𝑑)ξ€Ίπœ“π‘‘π‘’exp𝑑(ξ€»(𝑒)𝑖=1,…,5).(4.10)

Note that𝐼3(𝑑)+𝐼4(𝑑)βˆξ€œ(1+𝛿)exp(𝑒𝑑)𝑒(1βˆ’π›Ώ)expπ‘‘ξ€Έπ‘‘π‘Ÿπ‘“π‘Ÿ(π‘Ÿ),(4.11) where π‘“π‘Ÿ(π‘Ÿ) 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 πœ“βˆΆ[π‘’βˆ—,∞)→ℝ have πœ“β€²ξ…ž(𝑒)<0. For any 𝑒0,𝑒1∈[π‘’βˆ—,∞), ξ€œπ‘’1𝑒0π‘‘π‘’π‘’πœ“(𝑒)β‰₯π‘’πœ“(𝑒1)𝑒1βˆ’π‘’0πœ“ξ€·π‘’0ξ€Έξ€·π‘’βˆ’πœ“1ξ€Έξ€·π‘’πœ“(𝑒0)βˆ’πœ“(𝑒1)ξ€Έβˆ’1.(4.12)

Proof. Define the interval 𝐾∢=[𝑒0,𝑒1] if 𝑒1>𝑒0, and [𝑒1,𝑒0] otherwise. By the concavity of πœ“, ξ€·π‘’πœ“(𝑒)β‰₯πœ“1ξ€Έ+πœ“ξ€·π‘’1ξ€Έξ€·π‘’βˆ’πœ“0𝑒1βˆ’π‘’0ξ€·π‘’βˆ’π‘’1ξ€Έ,βˆ€π‘’βˆˆπΎ.(4.13) Hence, ξ€œπ‘’1𝑒0π‘‘π‘’π‘’πœ“(𝑒)β‰₯π‘’πœ“(𝑒1)ξ€œπ‘’1𝑒0ξƒ¬πœ“ξ€·π‘’π‘‘π‘’exp1ξ€Έξ€·π‘’βˆ’πœ“0𝑒1βˆ’π‘’0ξ€·π‘’βˆ’π‘’1ξ€Έξƒ­.(4.14) The result follows on evaluating the right-hand integral.

Proposition 4.4. Let πœ“βˆΆ[π‘’βˆ—,∞)→ℝ have πœ“ξ…žξ…ž(𝑒)≀0. For any 𝑒0,𝑒1∈[π‘’βˆ—,∞) with 𝑒1>𝑒0 and πœ“(𝑒0)>πœ“(𝑒1), ξ€œβˆžπ‘’1π‘‘π‘’π‘’πœ“(𝑒)β‰€π‘’πœ“(𝑒1)𝑒1βˆ’π‘’0πœ“ξ€·π‘’0ξ€Έξ€·π‘’βˆ’πœ“1ξ€Έ.(4.15)

Proof. By the concavity of πœ“, ξ€·π‘’πœ“(𝑒)β‰€πœ“1ξ€Έ+πœ“ξ…žξ€·π‘’1ξ€Έξ€·π‘’βˆ’π‘’1ξ€Έξ€·π‘’β‰€πœ“1ξ€Έ+πœ“ξ€·π‘’1ξ€Έξ€·π‘’βˆ’πœ“0𝑒1βˆ’π‘’0ξ€·π‘’βˆ’π‘’1𝑒,βˆ€π‘’βˆˆ1ξ€Έ.,∞(4.16) Hence, ξ€œβˆžπ‘’1π‘‘π‘’π‘’πœ“(𝑒)β‰€π‘’πœ“(𝑒1)ξ€œβˆžπ‘’1ξƒ¬πœ“ξ€·π‘’π‘‘π‘’exp1ξ€Έξ€·π‘’βˆ’πœ“0𝑒1βˆ’π‘’0ξ€·π‘’βˆ’π‘’1ξ€Έξƒ­.(4.17) Since (πœ“(𝑒1)βˆ’πœ“(𝑒0))/(𝑒1βˆ’π‘’0)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 πœ“βˆΆβ„β†’β„ have πœ“ξ…žξ…ž(𝑒)≀0. For any 𝑒0,𝑒1βˆˆβ„ with 𝑒1<𝑒0 and πœ“(𝑒0)>πœ“(𝑒1), ξ€œπ‘’1βˆ’βˆžπ‘‘π‘’π‘’πœ“(𝑒)β‰€π‘’πœ“(𝑒1)𝑒0βˆ’π‘’1πœ“ξ€·π‘’0ξ€Έξ€·π‘’βˆ’πœ“1ξ€Έ.(4.18)

Corollary 4.6. One has 𝐼5𝐼4+𝐼5ξ€Ίπœ“ξ€·π‘’β‰€exp𝑑𝑒+log(1+𝛿)βˆ’πœ“π‘‘ξ€Έξ€».(4.19)

Proof. Set 𝑒0=𝑒𝑑 and 𝑒1=𝑒𝑑+log(1+𝛿) in Propositions 4.3 and 4.4 to obtain 𝐼4𝐼5ξ€Ίπœ“ξ€·π‘’β‰₯expπ‘‘ξ€Έξ€·π‘’βˆ’πœ“π‘‘+log(1+𝛿)ξ€Έξ€»βˆ’1.(4.20) But 𝐼5𝐼4+𝐼5=11+𝐼4/𝐼5,(4.21) and so the result follows.

Corollary 4.7. One has 𝐼2𝐼2+𝐼3ξ€Ίπœ“ξ€·π‘’β‰€exp𝑑𝑒+log(1βˆ’π›Ώ)βˆ’πœ“π‘‘ξ€Έξ€».(4.22)

Proof. Define πœ“π‘ξ‚»(𝑒)∢=πœ“(𝑒),𝑒>π‘’βˆ—,πœ“ξ€·π‘’βˆ—ξ€Έβˆ’ξ€·π‘’βˆ—ξ€Έπœ“βˆ’π‘’ξ…žξ€·π‘’βˆ—ξ€Έ,𝑒<π‘’βˆ—.(4.23) By definition, πœ“π‘(𝑒)βˆΆβ„β†’β„, and πœ“ξ…žξ…ž(𝑒)≀0forallπ‘’βˆˆβ„. Let 𝐼1π‘ξ€œβˆΆ=𝑅1ξ€Ίπœ“π‘‘π‘’exp𝑐(𝑒),(4.24) and note that πœ“β€²(π‘’βˆ—)>0 since πœ“ξ…žξ…ž(𝑒)≀0forall𝑒β‰₯π‘’βˆ—, and πœ“β€²(𝑒𝑑)=0 with 𝑒𝑑>π‘’βˆ—. Hence, βˆ«π‘…1𝑑𝑒exp[πœ“π‘(𝑒)] exists.
Set 𝑒0=𝑒𝑑 and 𝑒1=𝑒𝑑+log(1βˆ’π›Ώ) in Propositions 4.3 and 4.5 to obtain 𝐼3𝐼1𝑐+𝐼2ξ€Ίπœ“ξ€·π‘’β‰₯expπ‘‘ξ€Έξ€·π‘’βˆ’πœ“π‘‘+log(1βˆ’π›Ώ)ξ€Έξ€»βˆ’1.(4.25) But 𝐼2𝐼2+𝐼3≀𝐼1𝑐+𝐼2𝐼1𝑐+𝐼2+𝐼3=11+𝐼3/𝐼1𝑐+𝐼2ξ€Έ(4.26) and so the result follows.
We now consider 𝐼1(𝑑) and use the fact that βˆ«π‘…1𝑑𝑒exp[πœ“π‘‘(𝑒)] must exist for all 𝑑>π‘‘βˆ—βˆ— (for some π‘‘βˆ—βˆ—>0) for {𝑅𝑑} to be an infinite sequence of random variables. Also note that πœ“π‘‘(𝑒)βˆ’πœ“π‘˜(𝑒)=(π‘‘βˆ’π‘˜)𝑒, which is an increasing function for 𝑑>π‘˜.

Corollary 4.8. If 𝐼1(π‘˜)<∞ for some π‘˜>0 and if for all 𝑑>π‘˜, πœ“π‘‘(𝑒)βˆ’πœ“π‘˜(𝑒) is an increasing function of 𝑒, then 𝐼1(𝑑)𝐼2(𝑑)+𝐼3(𝑑)β‰€π‘’βˆ’πœ“π‘˜(π‘’βˆ—)𝐼1(π‘˜)π‘’π‘‘βˆ’π‘’βˆ—.(4.27)

Proof. By the monotonicity of πœ“π‘‘βˆ’πœ“π‘˜, 𝐼1(𝑑)=ξ€œπ‘’βˆ—βˆ’βˆžπ‘‘π‘’π‘’πœ“π‘‘(𝑒)βˆ’πœ“π‘˜(𝑒)π‘’πœ“π‘˜(𝑒)β‰€π‘’πœ“π‘‘(π‘’βˆ—)βˆ’πœ“π‘˜(π‘’βˆ—)𝐼1(π‘˜).(4.28) By Proposition 4.3 with 𝑒0=𝑒𝑑 and 𝑒1=π‘’βˆ—πΌ2(𝑑)+𝐼3(𝑑)β‰₯π‘’πœ“π‘‘(π‘’βˆ—)π‘’π‘‘βˆ’π‘’βˆ—πœ“π‘‘ξ€·π‘’π‘‘ξ€Έβˆ’πœ“π‘‘ξ€·π‘’βˆ—ξ€Έξ€·π‘’πœ“π‘‘(𝑒𝑑)βˆ’πœ“π‘‘(π‘’βˆ—)ξ€Έβˆ’1β‰₯π‘’πœ“π‘‘(π‘’βˆ—)ξ€·π‘’π‘‘βˆ’π‘’βˆ—ξ€Έ,(4.29) where the last statement follows since for π‘₯>0, 𝑒π‘₯>1+π‘₯.
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)⟢0asπ‘‘βŸΆβˆž.(4.30)

By Lemma 4.1β€‰β€‰π‘’π‘‘β†’βˆž as π‘‘β†’βˆž, and so from Corollary 4.8𝐼1(𝑑)𝐼2(𝑑)+𝐼3(𝑑)⟢0.(4.31)

Since π‘’π‘‘β†’βˆž, given some π›Ώβˆˆ(0,𝛿0) and any 𝑀>0, we may choose a 𝑑0 such that, for all 𝑑>𝑑0 and all π›Ώβˆ—βˆˆ(0,𝛿),(log(1+𝛿))2πœ‚β€²ξ…žξ€·π‘’π‘‘ξ€·+log1+π›Ώβˆ—ξ€Έξ€Έβ‰₯𝑀.(4.32)

Taylor expand πœ“π‘‘ about 𝑒𝑑, recalling that πœ“ξ…žπ‘‘(𝑒𝑑)=0 and πœ“π‘‘ξ…žξ…ž(𝑒)=βˆ’πœ‚ξ…žξ…ž(𝑒):πœ“π‘‘ξ€·π‘’π‘‘ξ€Έβˆ’πœ“π‘‘ξ€·π‘’π‘‘ξ€Έ=1+log(1+𝛿)2(log(1+𝛿))2πœ‚ξ…žξ…žξ€·π‘’π‘‘ξ€·+log1+π›Ώβˆ—β‰₯1ξ€Έξ€Έ2𝑀,(4.33)

for some π›Ώβˆ—βˆˆ(0,𝛿). From Corollary 4.6 we therefore see that𝐼5(𝑑)𝐼4(𝑑)+𝐼5(𝑑)β‰€π‘’βˆ’(1/2)𝑀.(4.34)

Similarly, from Corollary 4.7𝐼2(𝑑)𝐼2(𝑑)+𝐼3(𝑑)β‰€π‘’βˆ’(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 π‘˜th Moment

Proposition 4.9. Let π‘Ÿπ‘‘ be the (eventually) unique solution to the equation π‘Ÿπ‘”ξ…ž(π‘Ÿ)=π‘‘βˆ’1.(4.37) If 𝑔(π‘Ÿ) satisfies (2.1) then for any fixed k>0limπ‘‘β†’βˆžπ‘Ÿπ‘‘π‘Ÿπ‘‘+π‘˜=1.(4.38)

Proof. Without loss of generality assume that π‘Ÿπ‘‘+π‘˜>π‘Ÿπ‘‘. Hence, by the Intermediate Value Theorem, there exists a value π‘Ÿβˆ—βˆˆ[π‘Ÿπ‘‘,π‘Ÿπ‘‘+π‘˜] such that π‘˜π‘Ÿπ‘‘+π‘˜π‘Ÿπ‘‘+π‘˜βˆ’π‘Ÿπ‘‘>π‘Ÿβˆ—π‘˜π‘Ÿπ‘‘+π‘˜βˆ’π‘Ÿπ‘‘=π‘Ÿβˆ—π‘Ÿπ‘‘+π‘˜π‘”β€²ξ€·π‘Ÿπ‘‘+π‘˜ξ€Έβˆ’π‘Ÿπ‘‘ξ€·π‘Ÿπ‘”β€²π‘‘ξ€Έπ‘Ÿπ‘‘+π‘˜βˆ’π‘Ÿπ‘‘=π‘Ÿβˆ—π‘‘ξ€·π‘‘π‘Ÿπ‘Ÿπ‘”ξ…žξ€Έ|||(π‘Ÿ)π‘Ÿβˆ—βŸΆβˆž.(4.39) Thus, π‘Ÿπ‘‘+π‘˜βˆ’π‘Ÿπ‘‘π‘Ÿπ‘‘+π‘˜βŸΆ0,(4.40) and the result follows.

Lemma 4.10. For fixed π‘˜>0, 1π‘Ÿπ‘˜π‘‘π”Όπ‘‘ξ€Ίπ‘…π‘˜ξ€»βŸΆ1.(4.41)

Proof. Set 𝐼1ξ€œβˆΆ=∞0π‘‘π‘Ÿπ‘Ÿπ‘‘βˆ’1[],𝐼expβˆ’π‘”(π‘Ÿ)2ξ€œβˆΆ=∞0π‘‘π‘Ÿπ‘Ÿπ‘‘βˆ’1+π‘˜[].expβˆ’π‘”(π‘Ÿ)(4.42) If 𝑔(π‘Ÿ) satisfies (2.1) then so does 𝑔(π‘Ÿ)βˆ’π‘˜log(π‘Ÿ). Therefore, from Lemma 4.2, given πœ–>0 and 𝛿>0 there is a 𝑑1 such that, for all 𝑑>𝑑1, (1βˆ’πœ–)𝐼2<ξ€œπ‘Ÿπ‘‘+π‘˜π‘Ÿ(1+2𝛿)𝑑+π‘˜(1βˆ’2𝛿)π‘‘π‘Ÿπ‘Ÿπ‘‘βˆ’1+π‘˜[]expβˆ’π‘”(π‘Ÿ)<𝐼2.(4.43) Furthermore, by Proposition 4.9, there is a 𝑑2 such that, for all 𝑑>𝑑2, π‘Ÿπ‘‘(1βˆ’π›Ώ)<π‘Ÿπ‘‘+π‘˜<(1+𝛿)π‘Ÿπ‘‘.(4.44) Therefore, since the integrand is positive, for all 𝑑>π‘šπ‘Žπ‘₯(𝑑1,𝑑2), (1βˆ’πœ–)𝐼2<ξ€œπ‘Ÿπ‘‘π‘Ÿ(1+2𝛿)(1+𝛿)𝑑(1βˆ’2𝛿)(1βˆ’π›Ώ)π‘‘π‘Ÿπ‘Ÿπ‘‘βˆ’1+π‘˜[]expβˆ’π‘”(π‘Ÿ)β‰€π‘Ÿπ‘˜π‘‘(1+2𝛿)π‘˜(1+𝛿)π‘˜ξ€œπ‘Ÿπ‘‘π‘Ÿ(1+2𝛿)(1+𝛿)𝑑(1βˆ’2𝛿)(1βˆ’π›Ώ)π‘‘π‘Ÿπ‘Ÿπ‘‘βˆ’1[].expβˆ’π‘”(π‘Ÿ)(4.45) Similarly 𝐼2>ξ€œπ‘Ÿπ‘‘π‘Ÿ(1+2𝛿)(1βˆ’π›Ώ)𝑑(1βˆ’2𝛿)(1+𝛿)π‘‘π‘Ÿπ‘Ÿπ‘‘βˆ’1+π‘˜[]expβˆ’π‘”(π‘Ÿ)β‰₯π‘Ÿπ‘˜π‘‘(1βˆ’2𝛿)π‘˜(1+𝛿)π‘˜ξ€œπ‘Ÿπ‘‘π‘Ÿ(1+2𝛿)(1βˆ’π›Ώ)𝑑(1βˆ’2𝛿)(1+𝛿)π‘‘π‘Ÿπ‘Ÿπ‘‘βˆ’1[].expβˆ’π‘”(π‘Ÿ)(4.46) Applying Lemma 4.2 again, there is a 𝑑3 such that, for all 𝑑>𝑑3, (1βˆ’πœ–)𝐼1<ξ€œπ‘Ÿπ‘‘π‘Ÿ(1+2𝛿)(1βˆ’π›Ώ)𝑑(1βˆ’2𝛿)(1+𝛿)π‘‘π‘Ÿπ‘Ÿπ‘‘βˆ’1[]expβˆ’π‘”(π‘Ÿ)<𝐼1.(4.47) Therefore, for all 𝑑>π‘šπ‘Žπ‘₯(𝑑1,𝑑2,𝑑3), (1βˆ’πœ–)𝐼2<π‘Ÿπ‘˜π‘‘(1+2𝛿)π‘˜(1+𝛿)π‘˜πΌ1,𝐼2>π‘Ÿπ‘˜π‘‘(1βˆ’2𝛿)π‘˜(1+𝛿)π‘˜(1βˆ’πœ–)𝐼1.(4.48) Hence, (1βˆ’2𝛿)π‘˜(1+𝛿)π‘˜1(1βˆ’πœ–)<π‘Ÿπ‘˜π‘‘πΌ2𝐼1<(1+2𝛿)π‘˜(1+𝛿)π‘˜(1βˆ’πœ–)βˆ’1.(4.49) The result follows since 𝛿 and πœ– 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: set‖𝐙‖=πΉβˆ’1‖𝐙‖𝐹‖𝐕‖(‖𝐕‖),(4.50) where 𝐹‖𝐕‖(β‹…) and 𝐹‖𝐙‖(β‹…) are the distribution functions of ‖𝐕‖ and ‖𝐙‖, respectively, and then fix𝐙=‖𝐙‖‖𝐕‖𝐕.(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∢=π‘Ÿπ‘‘π—π‘‘.(4.52)

The following is therefore equivalent to (2.6).

Lemma 4.11. Define {𝐕𝑑}, {𝐓𝑑}, {πœ†π‘‘,𝑖}, and {𝐡𝑑} as in (4.52) and the statement of Theorem 2.2. If (2.5) holds and β€–π•π‘‘β€–π‘βˆ’β†’1, then π‘‘π΅π‘‘π•π‘‘π‘‘π“π‘‘π‘‘π“π‘‘π•π‘‘π‘βŸΆ1.(4.53)

Proof. For some 𝛿>0, let π΄π‘‘ξƒ―π“βˆΆ=π‘‘βˆΆπœ†π‘‘,1βˆ‘π‘‘π‘–=1πœ†π‘‘,𝑖<𝛿3ξƒ°.(4.54) For now fix 𝑑 and π“π‘‘βˆˆπ΄π‘‘, and suppress the subscript 𝑑. Denote the spectral decomposition of 𝐓𝑑𝐓 as 𝐋𝑑Λ𝐋, where Ξ›=diag(πœ†1,…,πœ†π‘‘). We will initially consider the Gaussian 𝐙 and define π™βˆ—=𝐋𝐙; since 𝐋 is orthonormal, it follows that π™βˆ—βˆΌπ‘(𝟎,πˆπ‘‘).
Define π™π‘Š=𝑑𝐓𝑑𝐓𝐙𝐡=π™βˆ—π‘‘πš²π™βˆ—π΅.(4.55) Then, for fixed 𝑑, 𝔼𝐙[π‘Š]=1𝐡𝔼𝐙𝑑𝑖=1πœ†π‘–π‘π‘–βˆ—2ξƒ­=1,Var𝐙[π‘Š]=1𝐡2Var𝐙𝑑𝑖=1πœ†π‘–π‘π‘–βˆ—2ξƒ­βˆ‘=2𝑑𝑖=1πœ†2π‘–ξ‚€βˆ‘π‘‘π‘–=1πœ†π‘–ξ‚2πœ†β‰€21βˆ‘π‘‘π‘–=1πœ†π‘–<2𝛿3.(4.56) Chebyshev’s inequality gives ℙ𝐙||π‘Šβˆ’π”Όπ™[π‘Š]||ξ€Έ<>𝛿Var𝐙[π‘Š]𝛿2,thatis,ℙ𝐙||||ξ€Έπ‘Šβˆ’1>𝛿<2𝛿.(4.57) By (2.5) there is a 𝑑0 such that, for all 𝑑>𝑑0, β„™(π“π‘‘βˆ‰π΄π‘‘)<𝛿. Thus, for all 𝑑>𝑑0, ℙ𝐓,𝐙||||ξ€Έξ€·||||ξ€Έπ‘Šβˆ’1>π›Ώβ‰€β„™π‘Šβˆ’1>π›Ώβˆ£π“βˆˆπ΄+β„™(π“βˆ‰π΄)<3𝛿.(4.58) Hence, π‘Šπ‘‘π‘β†’1. Now 𝑑𝐡𝑑𝐕𝑑𝑑𝐓𝑑𝑑𝐓𝑑𝐕𝑑=𝑑‖‖𝐙𝑑‖‖2‖‖𝐕𝑑‖‖2π‘Šπ‘‘,(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}, {πœ†π‘‘,𝑖}, and {𝐡𝑑} to be as defined in (4.52) and the statement of Theorem 2.2, 𝔼‖‖𝐕𝑑‖‖2ξ‚„=𝑑𝐡𝑑𝔼‖‖𝐓𝑑𝐕𝑑‖‖2ξ‚„.(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 𝔼‖𝐓𝐕‖2=𝔼𝑑𝑖=1πœ†2𝑖𝑉2𝑖=𝑑𝑖=1πœ†2𝑖𝔼𝑉2𝑖.(4.61) But, again, 𝐕 is spherically symmetric so this is 𝑑𝑖=1πœ†2𝑖𝔼𝑉21ξ€»=1𝑑𝑑𝑖=1πœ†2𝑖𝑑𝑗=1𝔼𝑉2𝑗=𝐡𝑑𝑑𝔼‖𝐕‖2ξ€».(4.62) Turning now to convergence in mean square itself, note that, by Proposition 4.12, 𝔼‖‖‖𝑑1/2π΅π‘‘π“π‘‘π•π‘‘β€–β€–β€–ξ‚Άβˆ’12ξƒ­ξ‚ƒξ€·β€–β€–π•βˆ’π”Όπ‘‘β€–β€–ξ€Έβˆ’12𝔼‖‖‖𝑑=βˆ’21/2π΅π‘‘π“π‘‘π•π‘‘β€–β€–β€–ξ‚Ήξ€Ίβ€–β€–π•βˆ’π”Όπ‘‘β€–β€–ξ€»ξ‚Ά.(4.63) But (1.2) implies that ‖𝐕𝑑‖m.s.β†’1, and hence it is sufficient to show that 𝔼‖‖‖𝑑1/2π΅π‘‘π“π‘‘π•π‘‘β€–β€–β€–ξ‚Ήξ€Ίβ€–β€–π•βˆ’π”Όπ‘‘β€–β€–ξ€»βŸΆ0.(4.64) Now, by Lemma 4.11 and Proposition 4.12, ‖‖‖𝑑1/2π΅π‘‘π“π‘‘π•π‘‘β€–β€–β€–π‘ξƒ¬β€–β€–β€–π‘‘βŸΆ1,𝔼1/2𝐡𝑑𝐓𝑑𝐕𝑑‖‖‖2‖‖𝐕=𝔼𝑑‖‖2ξ‚„βŸΆ1.(4.65) We now require Scheffe’s Lemma, which states that, for any sequence of random variables {π‘Œπ‘‘}, if 𝔼[π‘Œ2𝑑]β†’1 and π‘Œπ‘‘π‘β†’1, then 𝔼[π‘Œπ‘‘]β†’1. Hence 𝔼[β€–(𝑑1/2/𝐡𝑑)𝐓𝑑𝐕𝑑‖]β†’1. Now (1.2) also implies that 𝔼[‖𝐕𝑑‖]β†’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||||β€–β€–π™βˆΆ=𝑑‖‖𝑑1/21‖‖𝐕𝑑‖‖||||βˆ’1<𝛿.(4.66)

Now, for πžπ‘‘ independent of 𝐕𝑑 (and 𝐙𝑑),ℙ𝑑1/2π•π‘‘β‹…πžπ‘‘ξ€Έξ‚΅π™β‰€π‘Ž=β„™π‘‘β‹…πžπ‘‘β€–β€–π™β‰€π‘Žπ‘‘β€–β€–π‘‘1/21β€–β€–π•π‘‘β€–β€–ξ‚Άξ‚΅π‘Žβ€–β€–π™=Φ𝑑‖‖𝑑1/21‖‖𝐕𝑑‖‖,(4.67) and soΦ𝑑(π‘Ž(1βˆ’π›Ώ))<β„™1/2π•π‘‘β‹…πžπ‘‘β‰€π‘Žβˆ£π΄(𝑑)ξ€Έ<Ξ¦(π‘Ž(1+𝛿)).(4.68)

For any event 𝐸,||||β„™(𝐸)βˆ’β„™(𝐸∣𝐴)=β„™(𝐴𝑐)||β„™(πΈβˆ£π΄π‘||)βˆ’β„™(𝐸∣𝐴)≀ℙ(𝐴𝑐)(4.69) and, in particular,||ℙ𝑑1/2π•π‘‘β‹…πžπ‘‘ξ€Έξ€·π‘‘β‰€π‘Žβˆ’β„™1/2π•π‘‘β‹…πžπ‘‘ξ€Έ||ξ€·π΄β‰€π‘Žβˆ£π΄β‰€β„™π‘π‘‘ξ€Έ.(4.70)

Given πœ–>0, by (1.1) we may define 𝑑0 such that, for all 𝑑>𝑑0, β„™(𝐴𝑐𝑑)<πœ–. Thus, for all 𝑑>𝑑0,Ξ¦(π‘Ž(1βˆ’π›Ώ))βˆ’πœ–<ℙ𝐕𝑑𝑑1/2π•π‘‘β‹…πž(𝑑)ξ€Έβ‰€π‘Ž<Ξ¦(π‘Ž(1+𝛿))+πœ–.(4.71)

By taking 𝑑 large enough we can make 𝛿 and πœ– as small as desired. Moreover, since Ξ¦(β‹…) is bounded and monotonic, βˆƒπ›Ώβˆ—>0 such that |Ξ¦(π‘Ž(1+𝛿))βˆ’Ξ¦(π‘Ž)|<πœ–forall𝛿 with |𝛿|<π›Ώβˆ—, and hencelimπ‘‘β†’βˆžβ„™ξ€·π‘‘1/2π•π‘‘β‹…πžπ‘‘ξ€Έβ‰€π‘Ž=Ξ¦(π‘Ž).(4.72)

To prove Part 2, first note that, whereas π™π‘‘β‹…πžπ‘‘βˆΌπ‘(0,1), (𝐓𝑑𝐙𝑑)β‹…πžπ‘‘βˆΌπ‘(0,β€–π“π‘‘πžπ‘‘β€–2), and soℙ𝐕𝑑𝑑𝐡𝑑1/2ξ€·π“π‘‘π•π‘‘ξ€Έβ‹…πžπ‘‘ξƒͺβ‰€π‘Ž=ℙ𝐙𝑑1β€–β€–π“π‘‘πžπ‘‘β€–β€–ξ€·π“π‘‘π™π‘‘ξ€Έβ‹…πžπ‘‘π΅β‰€π‘Žπ‘‘1/2𝑑1/2β€–β€–π“π‘‘πžπ‘‘β€–β€–β€–β€–π™π‘‘β€–β€–π‘‘1/21‖‖𝐕𝑑‖‖ξƒͺξƒ©π‘Žπ΅=Φ𝑑1/2𝑑1/2β€–β€–π“π‘‘πžπ‘‘β€–β€–β€–β€–π™π‘‘β€–β€–π‘‘1/21‖‖𝐕𝑑‖‖ξƒͺ.(4.73)

But a unit vector πžπ‘‘ chosen uniformly at random can be written as π™βˆ—π‘‘/||π™βˆ—π‘‘|| for some standard 𝑑-dimensional Gaussian π™βˆ—π‘‘. Hence, by Theorem 2.2,𝑑1/2𝐡𝑑1/2β€–β€–π“π‘‘πžπ‘‘β€–β€–=𝑑1/2β€–β€–π™βˆ—π‘‘β€–β€–β€–β€–π“π‘‘π™βˆ—π‘‘β€–β€–π΅π‘‘π‘1/2⟢1.(4.74)

We now define the eventπ΄βˆ—π‘‘||||𝐡∢=𝑑1/2𝑑1/2β€–β€–π“π‘‘πžπ‘‘β€–β€–β€–β€–π™π‘‘β€–β€–π‘‘1/21‖‖𝐕𝑑‖‖||||βˆ’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]). Setπ‘Žπ‘‘βˆΆ=(2log𝑑)βˆ’1/2,π‘π‘‘βˆΆ=(2log𝑑)1/2βˆ’12π‘Žπ‘‘ξ€Ίξ€»loglog𝑑+log(4πœ‹).(4.76)

Also let 𝐺(β‹…) be the distribution function of a Gumbel random variable, and let 𝑍1,…,𝑍𝑑 be independent and identically distributed 𝑁(0,1) random variables. Then𝐺𝑑1(𝑐)∢=β„™π‘Žπ‘‘ξ‚΅max𝑖=1,…,π‘‘π‘π‘–βˆ’π‘π‘‘ξ‚Άξ‚Άβ‰€π‘βŸΆπΊ(𝑐).(4.77)

Replacing 𝑐 with π‘π‘‘βˆΆ=[loglog𝑑+log(4πœ‹)]/2=((2log𝑑)1/2βˆ’π‘π‘‘)/π‘Žπ‘‘ or with π‘βˆ—π‘‘βˆΆ=βˆ’π›Ό(2log𝑑)1/2+[loglog𝑑+log(4πœ‹)]/2=((1βˆ’π›Ό)(2log𝑑)1/2βˆ’π‘π‘‘)/π‘Žπ‘‘ (𝛼>0) givesβ„™ξ‚΅max𝑖=1,…,𝑑𝑍𝑖≀(2log𝑑)1/2ξ‚Άξ€·π‘βˆΌπΊπ‘‘ξ€Έβ„™ξ‚΅βŸΆ1,max𝑖=1,…,𝑑𝑍𝑖≀(1βˆ’π›Ό)(2log𝑑)1/2ξ‚Άξ€·π‘βˆΌπΊβˆ—π‘‘ξ€ΈβŸΆ0.(4.78)

Choose 𝛿 in (4.66) small enough that (1βˆ’π›Ώ)(2+πœ–)1/2>21/2. Thenℙ𝑑1/2max𝑖=1,…,𝑑𝑉𝑑,𝑖≀((2+πœ–)log𝑑)1/2βˆ£π΄π‘‘ξ‚Άξ‚΅=β„™max𝑖=1,…,𝑑𝑍𝑑,𝑖≀‖‖𝐙𝑑‖‖𝑑1/21‖‖𝐕𝑑‖‖((2+πœ–)log𝑑)1/2βˆ£π΄π‘‘ξ‚Άξ‚΅>β„™max𝑖=1,…,𝑑𝑍𝑑,𝑖≀(1βˆ’π›Ώ)((2+πœ–)log𝑑)1/2βˆ£π΄π‘‘ξ‚Άξ‚΅>β„™max𝑖=1,…,𝑑𝑍𝑑,𝑖≀(2log𝑑)1/2βˆ£π΄π‘‘ξ‚Άξ‚΅>β„™max𝑖=1,…,𝑑𝑍𝑑,𝑖≀(2log𝑑)1/2ξ‚Άξ€·π΄βˆ’β„™π‘π‘‘ξ€Έ.(4.79)

Similarly by choosing 𝛿 in (4.66) small enough that (1+𝛿)(2βˆ’πœ–)1/2>21/2(1βˆ’π›Ό) for some small 𝛼>0,ℙ𝑑1/2max𝑖=1,…,𝑑𝑉𝑑,𝑖β‰₯((2βˆ’πœ–)log𝑑)1/2βˆ£π΄π‘‘ξ‚Άξ‚΅>𝑃max𝑖=1,…,𝑑𝑍𝑑,𝑖β‰₯(1βˆ’π›Ό)(2log𝑑)1/2ξ‚Άξ€·π΄βˆ’β„™π‘π‘‘ξ€Έ.(4.80)

In each case the first term tends to 1 and β„™(𝐴𝑐𝑑)β†’0, proving the desired result.

References

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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.


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