Sharp Asymptotics for q-Norms of Random Vectors in High-Dimensional \ell_p^n-Balls
aa r X i v : . [ m a t h . P R ] F e b SHARP ASYMPTOTICS FOR q -NORMS OF RANDOM VECTORSIN HIGH-DIMENSIONAL ℓ np -BALLS TOM KAUFMANN
Abstract.
Sharp large deviation results of Bahadur & Ranga Rao-type are provided for the q -norm of random vectors distributed on the ℓ np -ball B np according to the cone probability measureor the uniform distribution for 1 ≤ q < p < ∞ , thereby furthering previous large deviation resultsby Kabluchko, Prochno and Th¨ale in the same setting. These results are then applied to deducesharp asymptotics for intersection volumes of different ℓ np -balls in the spirit of Schechtman andSchmuckenschl¨ager, and for the length of the projection of an ℓ np -ball onto a line with uniformrandom direction. The sharp large deviation results are proven by providing convenient probabilisticrepresentations of the q -norms, employing local limit theorems to approximate their densities, andthen using geometric results for asymptotic expansions of Laplace integrals of Adriani & Baldi andLiao & Ramanan to integrate over the densities and derive concrete probability estimates. Introduction
The study of convex bodies in high dimensions, known today as asymptotic geometric analysis, hasarisen from the local theory of Banach spaces, which aimed at analyzing infinite-dimensional normedspaces via their finite-dimensional substructures, such as their unit balls. Despite having its originin the realm of functional analysis, the field has since established itself in its own right, consideringproblems also beyond the study of centrally symmetric convex bodies that occur naturally as theunit balls of Banach spaces. In high dimensions convex bodies exhibit certain regularities, such asconcentration of measure phenomena (see e.g. [19]), which make it highly useful to approach themfrom a probabilistic perspective. As pointed out in [5], it might seem counter-intuitive to analyzesomething exhibiting regularities from a probabilistic perspective, as probability concerns itself withstudying the nature of irregularity, i.e. randomness, of given quantities. But as with well knownlimit theorems from probability as the law of large numbers and the central limit theorem, withlarge sample sizes (and analogously - with high dimensionality) random objects exhibit interestingpatterns well characterized in the language of probability and vice versa. Many results analogous tothose from classic probability have been found for high-dimensional convex sets, such as the centrallimit theorem (see, e.g., Anttila, Ball and Perissinaki [4], Klartag [29, 30]). For further backgroundon high-dimensional convexity, see [5, 10, 19, 20].The ℓ np -ball B np , n ∈ N , has been a prominent object of study, as it is the unit ball of the (finite-dimensional) sequence space ℓ np , and has been the subject of a multitude of results. We will nameonly a select few and refer to the survey by Prochno, Th¨ale and Turchi [35] for a comprehensivesummary of classic and contemporary results. Let us denote by U n,p the uniform distribution on theEuclidean ℓ np -ball B np and by C n,p the cone probability measure on the ℓ np -sphere S n − p . Schechtmanand Zinn [40] and Rachev and R¨uschendorf [36] showed a generalization of the Poincar´e-Maxwell-Borel lemma, proving that, for k ∈ N with k < n , the k -dimensional marginal distribution of a Mathematics Subject Classification.
Primary: 52A23, 60F10 Secondary: 46B09, 60D05.
Key words and phrases.
Asymptotic geometric analysis, Bahadur Ranga Rao, high-dimensional convexity, intersec-tion volume, ℓ np -balls, ℓ np -spheres, large deviation principles, precise large deviations, sharp asymptotics, sharp largedeviations, strong large deviations, volume of convex bodies. random vector distributed according to C n,p converges in n to a k -dimensional generalized Gauss-ian distribution. They also provided a probabilistic representation for such random vectors in termsof these generalized Gaussian distributions, which will be a key building block in our main results.The primary quantity of interest of this paper however is the behaviour of the q -norm k Z k q of arandom vector Z in S n − p and B np . This was first studied by Schechtman and Zinn [40], who derivedconcentration inequalities for k Z k q with Z ∼ C n,p and Z ∼ U n,p for q > p . This is closely relatedto the intersection volume of t -multiples of volume-normalized ℓ np -balls D np := vol n ( B np ) − /n B np , i.e.vol n ( D np ∩ t D nq ) with t ∈ [0 , ∞ ), for which Schechtman and Schmuckenschl¨ager [39] gave the asymp-totics for t = 1. Schechtman and Zinn [41] expanded on their previous results in [40], by not onlyconsidering the q -norm, but also images of random vectors under Lipschitz functions in general.Thus, they gave concentration inequalities for f ( Z ), with Z ∼ C n,p and Z ∼ U n,p , p ∈ [1 , f a Lipschitz function with respect to the Euclidean norm. Schmuckenschl¨ager [42] provideda central limit theorem (CLT) for k Z k q with Z ∼ C n,p and Z ∼ U n,p and used it to refine theprevious intersection results in [39] for all t ∈ (0 , ∞ ). Naor [33] gave concentration inequalitiesfor k Z k q with Z ∼ C n,p for q < p , showed that the total variation distance between C n,p and thenormalized surface measure σ n,p on S n − p tends to zero proportional to n − / , and used the previ-ous two results to show a concentration inequality for k Z k q with Z ∼ σ n,p for q < p . Kabluchko,Prochno and Th¨ale [23] gave a multivariate CLT for ( k Z k q , . . . , k Z k q d ) with Z ∼ U n,p in the spiritof [42] and also considered the asymptotics for the intersection volume of multiple ℓ np -balls, i.e.vol n ( D np ∩ t D nq ∩ . . . ∩ t d D nq d ) with t i ∈ [0 , ∞ ). This CLT was furthermore applied by the sameauthors to infer a central limit theorem for the length of B np projected onto a line with uniform ran-dom direction. Moreover, they provided a large deviation principle (LDP) for k Z k q with Z ∼ C n,p and Z ∼ U n,p . In a follow-up paper [25], the same authors showed a CLT for k Z k q , where thedistribution of Z is taken from a wider class of p -radial distributions P n,p, W , introduced by Barthe,Gud´eon, Mendelson and Naor [7], consisting of mixtures of U n,p and C n,p , combined via a measure W on [0 , ∞ ). This class contains both U n,p and C n,p , but also distributions corresponding with ge-ometrically interesting projections (see e.g. [25, Introduction, (iii)]). Finally, they gave a moderateand a large deviation principle for k Z k q with Z ∼ P n,p, W .Generally, studying large deviations within asymptotic geometric analysis has started fairly recentlywith Gantert, Kim and Ramanan [17], who gave an LDP for projections of random points in ℓ np -ballswith distribution C n,p and U n,p onto both random and fixed one-dimensional subspaces. Today,large deviations theory has become a well-established toolbox in high-dimensional convex geometry,giving rise to a plethora of results (see e.g. [1, 2, 23, 24, 25, 27, 28]). Recently, a new tool fromlarge deviations theory was introduced to asymptotic geometric analysis by Liao and Ramanan[32]. They gave sharp large deviation (SLD) results in the spirit of Bahadur and Ranga Rao [6] andPetrov [34] for the projections of random points in ℓ np -balls with distribution C n,p and U n,p onto afixed one-dimensional subspace. While LDPs only give tail asymptotics on a logarithmic scale, thesharp asymptotics provided by sharp large deviations theory can give tail estimates for concretevalues of n ∈ N , which makes them significantly more useful for practical applications. Moreover,a lot of idiosyncrasies of the underlying distributions, that are drowned out on the LDP-scale, arestill visible on the SLD-scale, thus giving a deeper understanding of the geometric interpretation ofthe quantities involved. This paper will follow closely in their footsteps and establish SLD resultsfor the q -norms of random vectors with distribution C n,p and U n,p . Furthermore, we will use theseresults to expand on works of Schechtman and Schmuckenschl¨ager [39], Schmuckenschl¨ager [42], andKabluchko, Prochno and Th¨ale [23] for intersection volumes of ℓ np -balls by giving sharp asymptoticsfor vol n ( D np ∩ t D nq ) at a considerably improved rate for 1 ≤ q < p < ∞ and t > C ( p, q ) bigger thansome constant dependent on p and q only. Additionally, we will also apply our results for ℓ np -spheresto retain sharp asymptotics for the length of the projection of an ℓ np -ball onto the line spanned bya uniform random direction. HARP ASYMPTOTICS FOR q -NORMS OF RANDOM VECTORS IN HIGH-DIMENSIONAL ℓ np -BALLS 3 The paper will proceed as follows: in Section 2 some basic notation and definitions will be providedwhile also giving some appropriate background on the involved large deviations theory. Further-more, we will recapitulate some existing results that are relevant to this paper. In Section 3 wewill present our main results regarding the q -norms of random vectors on ℓ np -spheres and ℓ np -balls.Also, we will present and prove their application to intersections and one-dimensional projectionsof ℓ np -balls, and outline the idea of the two central proofs. In Section 4 we will reformulate thetarget probabilities from the main results in terms of useful probabilistic representations, usingwell-established representations of random vectors in ℓ np -balls of Schechtman and Zinn [40] andRachev and R¨uschendorf [36]. In Section 5 local density approximations of these probabilistic rep-resentations will be provided. In Sections 6 and 7 we will then prove the SLD results for ℓ np -spheresand ℓ np -balls, respectively, by integrating over the density estimates. For that, we will utilize somegeometric results for asymptotic expansions of Laplace integrals from Adriani and Baldi [3] andLiao and Ramanan [32]. 2. Preliminaries
Notation and important distributions.
We denote by vol d the d -dimensional Lebesguemeasure on R d and write B ( R d ) for the σ -field of Borel sets in R d . For a set A ∈ B ( R d ) wewrite A ◦ , A, ∂A , and A c for the interior, closure, boundary and complement of A , respectively.Furthermore, we write h · , · i for the standard scalar product in R d . For g : R d → R d , we denote by J x g ( x ∗ ) the Jacobian of g in x evaluated at x ∗ ∈ R d , and for f : R d → R by ∇ x f ( x ∗ ) and H x f ( x ∗ )the gradient and Hessian of g in x evaluated at x ∗ ∈ R d , respectively, and use the shorthand notation f [ i ,...,i d ] ( x ∗ ) = ∂ i ∂x i . . . ∂ i d ∂x i d f ( x ) (cid:12)(cid:12) x = x ∗ . (1)We write ( x , . . . , x d ) ∈ R d for a standard column vector and for x, y ∈ R d , we write their product x T y as xy , skipping the explicit transpose for brevity of notation. Given a random variable X withdistribution P , we write X ∼ P and denote by E X its expectation. For two random variables X, Y with the same distribution we write X d = Y . For a random vector X in R d and s ∈ R d , denote by ϕ X ( s ) := E [ e h s,X i ] and Λ X ( s ) := log ϕ X ( s ) the moment generating function and cumulant generat-ing function (m.g.f. and c.g.f.), respectively. We call the set of s ∈ R d for which Λ X ( s ) < ∞ theeffective domain D X of Λ X . Moreover, for x ∈ R d we denote by Λ ∗ X ( x ) := sup s ∈ R d [ h x, s i − Λ X ( s )]the Legendre-Fenchel transform of the c.g.f. Λ X , again writing its respective effective domain as J X . When considering sequences in n ∈ N , we denote by o (1) a sequence that tends to zero as n → ∞ .Let us consider the class of distributions at the core of the probabilistic constructions through-out this paper. We say a real-valued random variable X has a generalized Gaussian distribution ifits distribution has Lebesgue density f gen ( x ) := b a Γ (cid:0) b (cid:1) e − (cid:0) | x − m | /a (cid:1) b , x ∈ R , where m ∈ R and a, b >
0, and denote this by X ∼ N gen ( m, a, b ). As mentioned in the introduction,the generalized Gaussian distributions are essential for constructing probabilistically equivalentrepresentations of the quantities of interest, based on results of Schechtman and Zinn [40] andRachev and R¨uschendorf [36]. For these constructions we will be using the specific generalizedGaussian distribution N p := N gen (cid:0) , p /p , p (cid:1) , p ∈ [1 , ∞ ) , with density f p ( x ) := 12 p /p Γ (cid:0) p (cid:1) e −| x | p /p , x ∈ R . TOM KAUFMANN
For X ∼ N p and r >
0, we write E r := E X r for the r -th moment of X , for which it holds that E r := E X r = Γ (cid:18) rp (cid:19) Γ (cid:18) p (cid:19) − . (2)2.2. Background material from (sharp) large deviations theory.
We will give some basicnotions and definitions from large deviations theory. To keep this paper self-contained, we willpresent them here, while referring the reader to [14, 15, 26] for additional background material onlarge deviations. Furthermore, we want to give some insight into the methods of the lesser knowntheory of sharp large deviations.
Definition 2.1.
Let ( P n ) n ∈ N be a sequence of probability measures on R d . We say that ( P n ) n ∈ N satisfies a large deviation principle (LDP) if there are two functions s : N → R and I : R d → [0 , ∞ ] ,such that I is lower semi-continuous and a ) lim sup n →∞ s ( n ) log P n ( C ) ≤ −I ( C ) for all C ⊂ R n closed, b ) lim inf n →∞ s ( n ) log P n ( O ) ≥ −I ( O ) for all O ⊂ R n open,where for B ⊂ R d we define I ( B ) := inf x ∈ B I ( x ) . We call s the speed and I the rate function. Wesay that I is a good rate function, if it has compact sub-level sets. We apply the definition of LDPs to sequences of random variables as well by applying the abovedefinition to the sequence of their distributions. Given a sequence ( X ( n ) ) n ∈ N of i.i.d. random vec-tors in R d , one is frequently interested in the behaviour of the sequence ( S ( n ) ) n ∈ N of their empiricalaverages S ( n ) := n P ni =1 X ( n ) . One of the most well known and most frequently used results in thetheory of large deviations is the theorem of Cram´er, which states that if the c.g.f. Λ X is finite in anopen neighbourhood of the origin, then ( S ( n ) ) n ∈ N satisfies an LDP with speed n and rate functionΛ ∗ X (see e.g. [14, Theorem 2.2.30, Theorem 6.1.3, Corollary 6.1.6]). Hence, under suitable exponen-tial moment assumptions, we can already infer the large deviation behaviour of its empirical average.The classic LDP gives us an idea of the asymptotic deviation behaviour of a sequence of distribu-tions on a logarithmic scale. By doing this however, a lot of subleties of the underlying distributionscan be drowned out. Many small and medium scale properties of a given sequence of distributionsare often missed in the asymptotic analysis of LDPs, since they either disappear for very large n ∈ N or are drowned out by other, more significant phenomena of the distribution. Thus, oneis also interested in considering large deviations on a non-logarithmic scale, which we refer to as“sharp” large deviations (often also “precise” or “strong” large deviations). One of the first andmost prominent results in this regard was shown by Bahadur and Ranga Rao [6]. They showed thatfor a sequence ( X ( n ) ) n ∈ N of i.i.d. random variables and some z > E [ X ( n ) ] with Λ ∗ X ( z ) < ∞ , it holdsthat P (cid:16) S ( n ) > z (cid:17) = 1 √ πn κ ( z ) ξ ( z ) e − n Λ ∗ X ( z ) (1 + o (1)) , where κ ( z ) and ξ ( z ) are only dependent on the distribution of the X ( n ) and the deviation size z .This is proven via a (somewhat implicit) application of the the so-called saddle point method (ormethod of steepest descents), which was established by Debye [13], and brought to the realm ofprobability by Esscher [16] and Daniels [12]. The saddle point method generalizes Laplace’s methodfor integral approximation to the complex plane, and is therefore highly useful when dealing withintegrals over characteristic functions. In general, for appropriate functions f, g and n ∈ N large,the saddle point method gives a way to approximate Laplace-type integrals R P g ( z ) e − nf ( z ) d z alongcomplex paths P , by deforming the integration path using Cauchy’s theorem, into some ˜ P thatpasses through a saddle point of f . The mass of the reformulated integral is then heavily con-centrated around the saddle point and standard integral expansion methods, such as Edgeworth HARP ASYMPTOTICS FOR q -NORMS OF RANDOM VECTORS IN HIGH-DIMENSIONAL ℓ np -BALLS 5 expansion, can be used to great effect. In the realm of probability, this has been used for both tailprobabilities (e.g. Esscher [16], Cram´er [11]) and densities of random variables (e.g. Daniels [12],Richter [37, 38]), by writing them as an integral over their characteristic functions, by using theFourier inversion formula, and then approximating those integrals via the use of a complex saddlepoint. We say that this was used “somewhat implicitly” in certain results, such as those of Esscher[16], Cram´er [11] and Bahadur and Ranga Rao [6], since the technique used therein, which is acertain change of measure, often called exponential tilting or Esscher/Cram´er transform, under thesurface employs saddle points as well. For further background on this method, we refer to the bookof Jensen [22].As mentioned in the introduction, Section 5 will provide density estimates for our probabilisticrepresentations from Section 4, which are derived using the saddle point method. However, sinceour probabilistic representations are given as sums of i.i.d. random vectors, we will refer to previousresults where this was done explicitly, while making sure that the conditions for their applicationare still met in our setting. Generally, the core idea of the saddle point method, which is reformula-ting an integral such that all of its mass heavily concentrates around a critical point, around whichwe can then employ approximation methods, is used in the overall proof of our main results in abroader sense as well. We reformulate our target probabilities via some convenient representations,whose densities we also provide, such that the remaining integrals then heavily concentrate theirmass around a given critical point, such that approximations at that point yield accurate results,as we will see in Sections 6 and 7.2.3. Geometry of ℓ np -balls. For p ∈ [1 , ∞ ], n ∈ N , and x = ( x , . . . , x n ) ∈ R n let us denote by k x k p := (cid:16) n P i =1 | x i | p (cid:17) /p : p < ∞ max {| x | , . . . , | x n |} : p = ∞ (3)the ℓ np -norm of x . Let B np := { x ∈ R n : k x k p ≤ } be the unit ℓ np -ball and S n − p := { x ∈ R n : k x k p =1 } be the unit ℓ np -sphere. We define the uniform distribution on B np and cone probability measureon S n − p as U n,p ( · ) := vol n ( · )vol n ( B np ) and C n,p ( · ) := vol n ( { rx : r ∈ [0 , , x ∈ · } )vol n ( B np ) . The following result is the basis of our probabilistic representations for random vectors with distri-bution C n,p and U n,p and is due to [36] and [40]. Lemma 2.2.
Let p ∈ [1 , ∞ ) , Y = ( Y , . . . , Y n ) be a random vector in R n with Y i ∼ N p i.i.d., and U an independent random variable uniformly distributed on [0 , . Then,i) the random vector Y / k Y k p has distribution C n,p and is independent of k Y k p , ii) the random vector U /n Y / k Y k p has distribution U n,p . LDPs for q -norms in ℓ np -balls. Throughout this paper we assume 1 ≤ q < p < ∞ . The mainvariables of interest will be the q -norms of the random vectors Z ( n ) , Z ( n ) ∈ B np with Z ( n ) ∼ C n,p and Z ( n ) ∼ U n,p . Note, that we will always denote quantities related to Z ( n ) ∼ U n,p cursively. To getnon-trivial results, our target variables also need to be sufficiently rescaled. Thus, for random vectors Z ( n ) , Z ( n ) ∈ B np with Z ( n ) ∼ C n,p and Z ( n ) ∼ U n,p , our target variables will be n /p − /q k Z ( n ) k q and n /p − /q k Z ( n ) k q , respectively. We set k Z k := (cid:16) n /p − /q k Z ( n ) k q (cid:17) n ∈ N and k Z k := (cid:16) n /p − /q k Z ( n ) k q (cid:17) n ∈ N . TOM KAUFMANN
It was shown in [42, Proposition 2.4] and [23, Theorem 1.1] that both k Z k and k Z k converge inlaw to a Gaussian random variable with mean m p,q := E q E p = p q/p q + 1 Γ (cid:16) q +1 p (cid:17) Γ (cid:16) p (cid:17) /q , (4)with E q , E p as in (2), and variances also given with respect to the p -th and q -th moments of N p .Hence, we know that E h n /p − /q k Z ( n ) k q i = m p,q and E h n /p − /q k Z ( n ) k q i = m p,q . Furthermore, LDPs for k Z k and k Z k have been given in previous works, which we want to includehere explicitly. But first, let us look at the following probabilistic representations of k Z k and k Z k , since the LDPs are given with respect to the c.g.f. of these representations: Let ( Y ( n ) ) n ∈ N be a sequence of i.i.d. random vectors Y ( n ) := ( Y ( n )1 , . . . , Y ( n ) n ) with Y ( n ) i ∼ N p , and U a randomvariable, independent of the Y ( n ) i , and uniformly distributed on [0 , n /p − /q k Z ( n ) k q d = n /p − /q k Y ( n ) k q k Y ( n ) k p = (cid:16) n P ni =1 | Y ( n ) i | q (cid:17) /q (cid:16) n P ni =1 | Y ( n ) i | p (cid:17) /p , (5)and n /p − /q k Z ( n ) k q d = n /p − /q U /n k Y ( n ) k q k Y ( n ) k p = U /n (cid:16) n P ni =1 | Y ( n ) i | q (cid:17) /q (cid:16) n P ni =1 | Y ( n ) i | p (cid:17) /p . (6)Define V ( n ) := (cid:16) V ( n )1 , . . . , V ( n ) n (cid:17) ∈ R n with V ( n ) i := (cid:16) | Y ( n ) i | q , | Y ( n ) i | p (cid:17) , (7)and V ( n ) := (cid:16) V ( n )1 , . . . , V ( n ) n (cid:17) ∈ R n with V ( n ) i := (cid:16) | Y ( n ) i | q , | Y ( n ) i | p , U /n (cid:17) . (8)We denote the m.g.f. and c.g.f. of the V ( n ) i as ϕ p ( s ) := Z R e s | y | q + s | y | p f p ( y ) d y and Λ p ( s ) := log Z R e s | y | q + s | y | p f p ( y ) d y, (9)for s = ( s , s ) ∈ R and the Legendre-Fenchel transform of Λ p asΛ ∗ p ( x ) := sup s ∈ R (cid:2) h x, s i − Λ p ( s ) (cid:3) , x ∈ R . Let D p and J p be the effective domains of Λ p and Λ ∗ p , respectively. Since q < p , for the integralin both ϕ p and Λ p to be finite, the sign of the dominant term in the exponent must be negative.Remembering the definition of f p , one can see that this is given for s < /p , thus D p = R × ( −∞ , /p ). For x ∈ J p , let τ ( x ) = ( τ ( x ) , τ ( x ) ) ∈ R be the argument for s ∈ R in Λ ∗ p where thesupremum is attained, i.e. where Λ ∗ p ( x ) = h x, τ ( x ) i − Λ p ( τ ( x )) . (10)We can be sure that such a τ ( x ) exists by the following reasoning: Since x ∈ J p , we have that g x ( s ) := h x, s i − Λ p ( s ) < ∞ for all s ∈ R . Also, since Λ p is strictly convex in s (see standardproperties of the c.g.f.), g x is strictly concave as a sum of a strictly concave and a concave func-tion, and thereby attains its unique supremum. Therefore, there is a unique τ ( x ) ∈ R such thatΛ ∗ p ( x ) = h x, τ ( x ) i − Λ p ( τ ( x )). HARP ASYMPTOTICS FOR q -NORMS OF RANDOM VECTORS IN HIGH-DIMENSIONAL ℓ np -BALLS 7 For the sequence k Z k the following LDP has already been shown by Kabluchko, Prochno andTh¨ale [23, Section 5.1]: Proposition 2.3.
Let ≤ q < p < ∞ and Z ( n ) ∼ C n,p be a random vector in B np . Then thesequence (cid:0) n /p − /q k Z ( n ) k q (cid:1) n ∈ N satisfies an LDP with speed n and good rate function I k Z k ( z ) := inf t , t > t /q t − /p = z Λ ∗ p ( t , t ) : z > ∞ : z ≤ . In [32, Lemma 2.1, Appendix A] Liao and Ramanan established a simplification of a similar ratefunction in a different setting. Their arguments can be analogously applied in our setting to derivethe following result:
Lemma 2.4.
Let z > m p,q such that z ∗ := ( z q , ∈ J p . Then I k Z k ( z ) = inf t , t > t /q t − /p = z Λ ∗ p ( t , t ) = Λ ∗ p ( z ∗ ) , with z ∗ being the unique infimum of Λ ∗ p satisfiying the infimum conditions. To keep this paper self-contained, we will present the analogous proof of this in the Appendix. Forthe sequence k Z k , the following LDP was also provided by Kabluchko, Prochno and Th¨ale in [23,Theorem 1.2]: Proposition 2.5.
Let ≤ q < p < ∞ and Z ( n ) ∼ U n,p be a random vector in B np . Then thesequence (cid:0) n /p − /q k Z ( n ) k q (cid:1) n ∈ N satisfies an LDP with speed n and good rate function I k Z k ( z ) := inf z = z z z , z > (cid:2) I k Z k ( z ) + I U ( z ) (cid:3) : z > ∞ : z ≤ , with I k Z k as in Proposition 2.3 and I U ( z ) := ( − log( z ) : z ∈ (0 , ∞ : otherwise. We again show that the above infimum is attained at a unique point satisfying the infimum condition.
Lemma 2.6.
Assume the same setting as in Proposition 2.5. For z > , we can simplify the ratefunction by combining the two infimum operations to get I k Z k ( z ) = inf z = t /q t − /p t t , t > , t ∈ (0 , (cid:2) Λ ∗ p ( t , t ) − log( t ) (cid:3) . We define I S ( t ) := (cid:2) Λ ∗ p ( t , t ) − log( t ) (cid:3) , t , t ∈ R , t ∈ (0 , , and set z ∗ := ( z q , ∈ R , z ∗∗ := ( z q , , ∈ R . It then holds for z > m p,q with z ∗ ∈ J p that I k Z k ( z ) = I S ( z ∗∗ ) = Λ ∗ p ( z ∗ ) , with z ∗∗ being the unique infimum satisfying the infimum condition. Thus, for z > m p,q both k Z k and k Z k satisfy LDPs with the same speed and rate function. Again,the proof of this is relegated to the Appendix. TOM KAUFMANN
A few remarks on Weingarten maps and curvature.
As outlined in the introduction,we will finish the proof of our first main result in Theorem 3.1 by integrating over a previouslyestablished density estimate via a result of Adriani and Baldi [3] for Laplacian integral expansions.This result has a heavily geometric flavour and relies on the Weingarten maps of certain hypersur-faces, which in our case are simply curves in R . We will therefore just give a brief reminder of theWeingarten map in this setting, recall some of its properties, and refer to the relevant literature(e.g. [21, 31]) or Adriani and Baldi [3] for a more in-depth discussion of the topic.In general, the Weingarten map of a smooth hypersurface M ⊂ R d at a point p ∈ M is an en-domorphism of the tangent space T p M at p , mapping any y ∈ T p M to the directional derivativeof a normal field of M in p in the direction of y . However, as remarked in [3, Example 4.3], for d = 2, hypersurfaces simplify to planar curves and the Weingarten map at a point p simplifies tothe absolute value of the curvature K ( p ) of the curve at p . For implicit curves, i.e. curves given asthe zero set of a function, we have the following formula for its curvature from [18, Proposition 3.1]: Lemma 2.7.
Let F : R → R be a smooth function. For a curve C := { x ∈ R : F ( x ) = 0 } givenas the zero set of F , and a point p ∈ C , where ∇ x F ( p ) = 0 , it then holds that K ( p ) = (cid:0) − F [0 , , F [1 , (cid:1) (cid:18) F [2 , F [1 , F [1 , F [0 , (cid:19) (cid:0) − F [0 , , F [1 , (cid:1)(cid:0) F [1 , + F [0 , (cid:1) / , with derivatives F [ i,j ] = F [ i,j ] ( p ) as in (1) . Remark 2.8. i) Given the set-up of the previous Lemma, straightforward calculation of the above fractiongives that K ( p ) = F [0 , F [2 , − F [0 , F [1 , F [1 , + F [1 , F [0 , (cid:0) F [1 , + F [0 , (cid:1) / . ii) Given that C is the graph of a smooth function f : R → R , i.e. C = { ( x , x ) ∈ R : x = f ( x ) } , and p = ( x, f ( x )), the above reduces to K ( p ) = | f ′′ ( x ) | (cid:0) f ′ ( x ) (cid:1) / . Main Results
Using the concepts and notation established in the previous section, we now proceed to present ourmain results and their applications:3.1.
Sharp asymptotics for q -norms of random vectors in S n − p and B np . For Z ( n ) ∼ C n,p ,we want to give sharp asymptotics for the probability P (cid:0) n /p − /q k Z ( n ) k q > z (cid:1) for z > m p,q suchthat z ∗ ∈ J p , with z ∗ as definded in Lemma 2.4. Before presenting our results, let us define thedeviation-dependent functions ξ ( z ) and κ ( z ), as mentioned also in the sharp large deviation resultsof Bahadur and Ranga Rao [6]. For x ∈ R , we set H x := H τ Λ p ( τ ( x )) (11)to be the Hessian of the c.g.f. Λ p ( τ ) in τ ∈ R , evaluated at τ ( x ). For z > m p,q such that z ∗ ∈ J p ,we then define the deviation-dependent functions as ξ ( z ) := hH z ∗ τ ( z ∗ ) , τ ( z ∗ ) i det H z ∗ , (12) HARP ASYMPTOTICS FOR q -NORMS OF RANDOM VECTORS IN HIGH-DIMENSIONAL ℓ np -BALLS 9 κ ( z ) := 1 − (cid:0) τ ( z ∗ ) + τ ( z ∗ ) (cid:1) / | pq ( p − q ) z q | (cid:12)(cid:12) τ ( z ∗ ) (cid:0) H − z ∗ (cid:1) − τ ( z ∗ ) τ ( z ∗ ) (cid:0) H − z ∗ (cid:1) + τ ( z ∗ ) (cid:0) H − z ∗ (cid:1) (cid:12)(cid:12) ( z q + p q − ) / . (13)Note, that since Λ p is strictly convex, its Hessian is always positive definite and hence invertible,thus the terms above are all well-defined. Now we can formulate our main result for ℓ np -spheres. Theorem 3.1.
Let ≤ q < p < ∞ and Z ( n ) be a random vector in B np with Z ( n ) ∼ C n,p . Then,for some z > m p,q such that z ∗ ∈ J p , it holds that P (cid:16) n /p − /q k Z ( n ) k q > z (cid:17) = 1 √ πn κ ( z ) ξ ( z ) e − n Λ ∗ p ( z ∗ ) (1 + o (1)) . We want to do the same for P (cid:0) n /p − /q k Z ( n ) k q > z (cid:1) with Z ( n ) ∼ U n,p and z > m p,q . Again, westart by defining our deviation-dependent function for z > m p,q γ ( z ) := det H z ∗ τ ( z ∗ ) ( qz q τ ( z ∗ ) + 1) × (cid:12)(cid:12)(cid:12) z q q p (cid:0) H − z ∗ (cid:1) + 2 z q qp (cid:0) H − z ∗ (cid:1) + (cid:0) H − z ∗ (cid:1) + τ ( z ∗ ) z q q ( q − p ) p (cid:12)(cid:12)(cid:12) . (14) Theorem 3.2.
Let ≤ q < p < ∞ and Z ( n ) be a random vector in B np with Z ( n ) ∼ U n,p . Then,for some z > m p,q such that z ∗ ∈ J p , it holds that P (cid:16) n /p − /q k Z ( n ) k q > z (cid:17) = 1 √ πn γ ( z ) e − n Λ ∗ p ( z ∗ ) (1 + o (1)) . We have seen in Section 2.4 that k Z k and k Z k both satisfy LDPs with the same speed and ratefunction for z > m p,q , despite the underlying distributions being different. Comparing Theorem3.1 and Theorem 3.2 now paints a different picture, with the sharp asymptotics for k Z k and k Z k being noticably different. As mentioned in our introductary statements, idiosyncratic phenomenaof underlying distributions, which can be drowned out on the LDP scale, are often still visible onthe scale of sharp large deviations. This is in keeping with what was shown in [32, Theorem 2.4,Theorem 2.6] for one-dimensional projections of ℓ np -spheres and ℓ np -balls. Remark 3.3.
Let us draw a brief comparison between our results and the concentration inequalityderived by Naor in [33] that is closest to our setting. Therein, it is proven in [33, Remark, p. 1062]that for 1 < q ≤ p < ∞ with 1 < p and a random vector Z ( n ) ∼ C n,p , it holds that P (cid:16)(cid:12)(cid:12) n /p − /q k Z ( n ) k q − m p,q (cid:12)(cid:12) ≥ z (cid:17) ≤ C exp (cid:16) − c n z max { ,q } (cid:17) , where C > c > ≤ q < p < ∞ and z > m p,q , and only consider deviations without the absolute value, we can derive from theabove that P (cid:16) n /p − /q k Z ( n ) k q > z (cid:17) ≤ C exp (cid:16) − c n z max { ,q } (cid:17) . Comparing this with our sharp large deviation results from Theorem 3.1, P (cid:16) n /p − /q k Z ( n ) k q > z (cid:17) = 1 √ πn κ ( z ) ξ ( z ) e − n Λ ∗ p ( z ∗ ) (1 + o (1)) , we can see that our results improve on the estimate in terms of n ∈ N by a factor of n − / and giveexplicit and deviation-dependent terms κ ( z ) and ξ ( z ) instead of fixed constants for all z > m p,q . Remark 3.4.
When comparing the SLD results in Theorem 3.1 and Theorem 3.2 to those ofLiao and Ramanan [32, Theorem 2.4, Theorem 2.6], one directly notices the core difference inthe settings. Liao and Ramanan examine projections of random vectors on S n − p and B np withrespective distributions C n,p and U n,p onto fixed one-dimensional subspaces, and therefore have to consider weighted sums of dependent random vectors as probabilistic representations. Thus, alltheir results have to be conditioned on the projection space and include additional terms accountingfor the specifics of the subspace. In our case however, the probabilistic representations are given assums of i.i.d. random variables (see Section 4), which does not necessitate these additional factors.Therefore, when using results from Liao and Ramanan [32], as for example in the proof of Theorem3.2 (see Section 7), we adapt their usage accordingly to the given probabilistic representations in oursetting. Beyond that however, the SLD results share several similarities, especially when comparingthe deviation-dependent terms κ, ξ and γ , which for q = 1 are almost equal.We shall now give an outline of how the above theorems will be proven. Both proofs of Theorem3.1 and Theorem 3.2 contain three essential steps, as already briefly mentioned in the introduction.The first will be rewriting the probabilities in both theorems with respect to convenient probabilisticrepresentations, specifically S ( n ) and S ( n ) as given in (17) of Section 4 as the respective empiricalaverages of the V ( n ) i and V ( n ) i in (7) and (8). The idea is to write the deviation probabilities as anintegral of their distribution over a given “deviation area”. The second step is giving local densityapproximations for these representations. Since the entries of both the V ( n ) and the V ( n ) are highlydependent, no canonical joint densities are available to us to easily do so. However, their Fouriertransforms can be given explicitly, thus, one can use the Fourier inversion theorem to write thedensities of S ( n ) and S ( n ) as integrals over their Fourier transforms. The resulting integrals canthen be approximated using the saddle point method. Since our representations are given as i.i.d.sums of random vectors, for whom this has been done in previous results (see e.g. [9, 12, 37, 38]),we will not prove the density approximations here explicitly. The third and final step then is tocalculate the integrals of these densities over their respective deviation area. For k Z k , this is doneby a result of Adriani and Baldi [3], which construes the boundary of the deviation area and thelevel sets of the rate function in the corresponding LDP as hypersurfaces, which are just planarcurves in our setting, and uses their Weingarten maps to approximate the integral. For k Z k , this isnot applicable, as certain differentiability conditions are no longer met. Thus, a result by Liao andRamanan [32] is used, approximating the integral via the method of resolution of multiple integrals,which as the name suggests, resolves the integral over the 3-dimensional density of S ( n ) into aone-dimensional definite Laplace-type integral, which can be approximated using a generalizationof Watson’s Lemma (see [32, Proof of Lemma 5.1]).3.2. Intersection volumes of ℓ np -balls. We want to use our sharp large deviation results to furtherthe findings of Schechtman and Schmuckenschl¨ager [39] and Schmuckenschl¨ager [42] for intersectionvolumes of t -multiples of different ℓ np -balls. We will first give a brief overview of the original results.For p ∈ [1 , ∞ ), we define D np := vol n ( B np ) − /n B np to be the volume normalized ℓ np -ball and recall thatvol n ( B np ) = 2Γ (cid:16) p (cid:17) n Γ (cid:16) np (cid:17) . We furthermore set c n,p := n /p vol n (cid:0) B np (cid:1) /n and c p := 2 e /p p /p Γ (cid:18) p (cid:19) , and recall that it was shown in [39] that lim n →∞ c n,p = c p . Moreover, for p, q ∈ [1 , ∞ ) , p = q , we set c n,p,q := c n,p c n,q , A p,q,n := c n,p m p,q c n,q , and A p,q := lim n →∞ A p,q,n . HARP ASYMPTOTICS FOR q -NORMS OF RANDOM VECTORS IN HIGH-DIMENSIONAL ℓ np -BALLS 11 Hence, it follows that A p,q = c p m p,q c q = Γ (cid:16) p (cid:17) /p ) Γ (cid:16) q (cid:17) Γ (cid:16) q +1 p (cid:17) /q e /p − /q (cid:16) pq (cid:17) /q . Lastly, for t ≥ n ∈ N , we define t n ≥ t n A p,q A p,q,n = t. Having established the neccessary notation, we shall now recall the result of Schmuckenschl¨ager [42,Theorem 3.3]. Therein, it was shown that for p, q ∈ [1 , ∞ ) , p = q , and t ≥ n (cid:0) D np ∩ t D nq (cid:1) −→ n →∞ A p,q t > : A p,q t = 10 : A p,q t < . (15)To prove this, a central limit theorem for n /p − /q k Z ( n ) k q with Z ( n ) ∼ U n,p and p, q ∈ [1 , ∞ ), p = q , is shown in [42, Proposition 2.4, Proof of Theorem 3.2], since vol n ( D np ∩ t D nq ) can be writtenasvol n ( D np ∩ t D nq ) = vol n (cid:18)(cid:26) z ∈ D np : z ∈ t n A p,q A p,q,n D nq (cid:27)(cid:19) = vol n (cid:18)(cid:26) z ∈ D np : z ∈ t n A p,q m p,q c q,n c p,n D nq (cid:27)(cid:19) = vol n (cid:16)n z ∈ n /p vol n ( B np ) /n B np : z ∈ t n A p,q m p,q n /q − /p vol n ( B np ) − /n B nq o(cid:17) = vol n ( B np ) − vol n (cid:16)n z ∈ B np : z ∈ t n A p,q m p,q n /q − /p B nq o(cid:17) = P (cid:16) n /p − /q k Z ( n ) k q ≤ t n A p,q m p,q (cid:17) . (16)However, we know from the Berry-Esseen Theorem (see [43, Theorem 2.1.3]) that the error of theGaussian approximation given by a central limit theorem decreases with rate n − / . Thus, using(16) and the central limit theorem from [42], we can only infer a rate of convergence of n − / in(15). Using Theorem 3.2, we can considerably refine that rate of convergence in the first of thethree cases in (15) from a sublinear rate to an exponential rate for 1 ≤ q < p < ∞ . Proposition 3.5.
Let ≤ q < p < ∞ . Using the notation established above, it then holds for t > m p,q c n,p,q − that vol n (cid:0) D np ∩ t D nq (cid:1) = 1 − √ πn γ ( t c n,p,q ) e − n Λ ∗ p (( t c n,p,q ) ∗ ) (1 + o (1)) . Proof.
Let 1 ≤ q < p < ∞ , t > m p,q c n,p,q − and assume Z ( n ) is a random vector in B np with Z ( n ) ∼ U n,p . Using (16), we get thatvol n (cid:0) D np ∩ t D nq (cid:1) = P (cid:16) n /p − /q k Z ( n ) k q ≤ t n A p,q m p,q (cid:17) = 1 − P (cid:16) n /p − /q k Z ( n ) k q > t n A p,q m p,q (cid:17) . It now holds that, by t > m p,q c n,p,q − , we have that t m − p,q c n,p,q = t A p,q,n = t n A p,q >
1, and hence t c p,q,n = t n A p,q m p,q > m p,q . Thus, by Theorem 3.2, it follows thatvol n (cid:0) D np ∩ t D nq (cid:1) = 1 − √ πn γ ( t c n,p,q ) e − n Λ ∗ p (( t c n,p,q ) ∗ ) (1 + o (1)) , which finishes our proof. (cid:3) One-dimensional projections of ℓ nq -balls. In Remark 3.4 we have already discussed thedifferences between the setting of the results of Liao and Ramanan [32] and the setting of thispaper. However, a geometrically similar result to those in [32] follows from Theorem 3.1. In [23,Section 2.4 ] Kabluchko, Prochno and Th¨ale derived a central limit theorem for the the length of theprojection of an ℓ np -ball onto the line spanned by a random vector θ ( n ) ∈ S n − with θ ( n ) ∼ C n, asa corollary of their main results. We will proceed similarly and derive sharp large deviation resultsin the same setting. To be specific, in [32] sharp asymptotics where provided for the scalar product of a random vector Z ( n ) ∼ C n,p on S n − p with a random vector θ ( n ) ∼ C n, on S n − , which can benegative. We, on the other hand, consider the absolute value of the scalar prduct of such randomvectors, thereby only considering non-negative values.In the following, for q ∈ [1 , ∞ ], define its conjugate q ∗ via 1 /q + 1 /q ∗ = 1, setting 1 / ∞ = 0by convention. Furthermore, for a vector θ ( n ) ∈ S n − , we write P θ ( n ) B nq for the projection of B nq onto the line spanned by θ ( n ) . Then, our quantity of interest is the projection length vol (cid:0) P θ ( n ) B nq (cid:1) . Corollary 3.6.
Let < q ≤ ∞ and θ ( n ) ∈ S n − be a random vector with θ ( n ) ∼ C n, . Then, forsome z > m ,q ∗ , it holds that P (cid:16) n / − /q vol (cid:0) P θ ( n ) B nq (cid:1) > z (cid:17) = 1 √ πn κ (cid:0) z (cid:1) ξ (cid:0) z (cid:1) e − n Λ ∗ ( z ∗ ) (1 + o (1)) , with Λ as in (9) and ξ, κ as in (12) , (13) , respectively, for q ∗ and p = 2 .Proof. It holds that P (cid:16) n / − /q vol (cid:0) P θ ( n ) B nq (cid:1) > z (cid:17) = P n / − /q x ∈ B nq |h x, θ ( n ) i| > z ! = P (cid:16) n / − /q k θ ( n ) k q ∗ > z (cid:17) . Since 2 < q ≤ ∞ , we have 1 ≤ q ∗ < p , whereby we can apply Theorem 3.1 to the above to getthat P (cid:16) n / − /q vol (cid:0) P θ ( n ) B nq (cid:1) > z (cid:17) = 1 √ πn κ (cid:0) z (cid:1) ξ (cid:0) z (cid:1) e − n Λ ∗ ( z ∗ ) (1 + o (1)) , with Λ , κ, ξ as described above, which concludes our proof. (cid:3) Probabilistic Representation
Recalling the definitions of the random vectors V ( n ) and V ( n ) from (7) and (8), we define S ( n ) := 1 n n X i =1 V ( n ) i and S ( n ) := 1 n n X i =1 V ( n ) i (17)as the empirical averages of their respective coordinates. Furthermore, we define the sets D z := { ( t , t ) ∈ R : t , t > , t /q t − /p > z } , and D z := { ( t , t , t ) ∈ R : t , t > , t ∈ (0 , , t t /q t − /p > z } . It then follows from the reformulations of k Z ( n ) k q and k Z ( n ) k q in (5) and (6) that we can write theprobabilities within Theorem 3.1 and Theorem 3.2 with respect to S ( n ) and S ( n ) , respectively, as P (cid:16) n /p − /q k Z ( n ) k q > z (cid:17) = P n n X i =1 | Y ( n ) i | q > z q n n X i =1 | Y ( n ) i | p ! qp = P (cid:16) S ( n ) ∈ D z (cid:17) , (18) HARP ASYMPTOTICS FOR q -NORMS OF RANDOM VECTORS IN HIGH-DIMENSIONAL ℓ np -BALLS 13 and P (cid:16) n /p − /q k Z ( n ) k q > z (cid:17) = P U qn n n X i =1 | Y ( n ) i | q > z q n n X i =1 | Y ( n ) i | p ! qp = P (cid:16) S ( n ) ∈ D z (cid:17) . (19)We refer to these sets as “deviation areas”, as S ( n ) or S ( n ) lying in D z or D z represents a deviationof k Z ( n ) k q and k Z ( n ) k q . Note that the boundaries of the deviation areas ∂D z = { ( t , t ) ∈ R : t , t > , t /q t − /p = z } and ∂ D z = { ( t , t , t ) ∈ R : t , t > , t ∈ (0 , , t t /q t − /p = z } are the same sets given by the infimum conditions in the respective LDPs for k Z k and k Z k inProposition 2.3 and Proposition 2.5. The fact that for z > m p,q , the rate functions of these LDPsboth assume a unique minimum on ∂D z and ∂ D z , repsectively, as was shown in Lemma 2.4 andLemma 2.6, will be essential to the proof of our main results in Sections 6 and 7. We can expandthis unique infimum property onto the entirety of D z and D z , as the following lemma will show: Lemma 4.1.
Assume the same set-up as in Lemma 2.4 and Lemma 2.6. Let z > m p,q . Theni) z ∗ = ( z q , is the unique infimum of Λ ∗ p on D z ,ii) z ∗∗ = ( z q , , is the unique infimum of I S on D z .Proof. We start off by showing i) . Let t ∈ R such that t ∈ D ◦ z , meaning t /q t − /p > z . Then, for˜ z := t /q t − /p we have that t ∈ ∂D ˜ z , thus, by Lemma 2.4, Λ ∗ p ( t , t ) > Λ ∗ p (˜ z q ,
1) = I k Z k (˜ z ). Weknow that I k Z k ( z ) is strictly increasing in z for z > m p,q , as rate functions are strictly convex, witha unique zero only at the expected value m p,q . Thus, as ˜ z > z > m p,q , we know thatΛ ∗ p ( t , t ) > Λ p (˜ z q ,
1) = I k Z k (˜ z ) > I k Z k ( z ) = Λ ∗ p ( z q ,
1) = Λ ∗ p ( z ∗ ) , showing that z ∗ = ( z q ,
1) minimizes Λ ∗ p over D z . The proof of ii) is analogous, also using the strictmonotonicity of the rate function. (cid:3) Suppose that the distributions of S ( n ) and S ( n ) have respective densities h ( n ) and h ( n ) . Then wecan formulate our probabilities of interest as P (cid:16) n /p − /q k Z ( n ) k q > z (cid:17) = P (cid:16) S ( n ) ∈ D z (cid:17) = Z D z h ( n ) ( x ) d x, (20)and P (cid:16) n /p − /q k Z ( n ) k q > z (cid:17) = P (cid:16) S ( n ) ∈ D z (cid:17) = Z D z h ( n ) ( x ) d x. (21)The following section will be devoted to showing the existence of these densities h ( n ) and h ( n ) andpresenting them explicitly, while Sections 6 and 7 will then approximate their integrals over theirrespective deviation areas D z and D z .5. Joint Density Estimate
Recalling the notation and definitions established in Section 2, we assume the same set-up as inSection 4 and can formulate the following local limit theorems for the densities h ( n ) and h ( n ) of ourprobabilistic representations S ( n ) and S ( n ) . Proposition 5.1.
For S ( n ) = n P ni =1 V ( n ) i with V ( n ) i = ( | Y ( n ) i | q , | Y ( n ) i | p ) , Y ( n ) i ∼ N p i.i.d., and x ∈ J p , it holds that for sufficiently large n ∈ N the distribution of S ( n ) has Lebesgue density h ( n ) ( x ) = n π (det H x ) − / e − n Λ ∗ p ( x ) (1 + o (1)) , where H x = H τ Λ p ( τ ( x )) as in (11) . For the proof of this, we refer to the results of Borovkov and Rogozin [9] or their convenientreformulation in [3, Theorem 3.1]. Therein, a local density estimate is derived for a sum of i.i.d.random vectors in R d via the saddle point method. As discussed in Section 2.2, this means, onewrites the density via the Fourier inversion theorem as a complex integral over its Fourier transformand then uses Cauchy’s theorem to deform the path of integration, such that it passes through acomplex saddle point. For sufficiently large n ∈ N , the mass of the integral then heavily concentratesaround that saddle point and standard integral expansion methods can be used to great effect.Naturally, this requires the conditions of the Fourier inversion theorem to be met, that is, theFourier transform of the density has to be integrable. In [3, Theorem 3.1] this follows from theassumption that all the i.i.d. random vectors have a common bounded density, though it is notedin [3, Remark 3.2], that this can be replaced by any argument ensuring that the Fourier inversiontheorem can be applied. In our setting, the i.i.d. vectors are given by V ( n ) i := ( | Y ( n ) i | q , | Y ( n ) i | p ), whosecoordinates are highly dependent, thus such a density of the V ( n ) i is not available. However, onecan write the Fourier transform of V ( n ) i with respect to the underlying distribution N p of the Y ( n ) i ,and then infer integrability via the properties of its density f p and the Hausdorff-Young inequality,as was done by Liao and Ramanan in [32, Lemma 6.1]. As the considered settings are quite similar,virtually the same arguments can be applied in our case, thereby making sure our referral to [3,Theorem 3.1] is indeed justified. Proposition 5.2.
For S ( n ) = n P ni =1 V ( n ) i with V ( n ) i = ( | Y ( n ) i | q , | Y ( n ) i | p , U /n ) , Y ( n ) i ∼ N p i.i.d., U uniformly distributed on [0 , independently of the Y ( n ) i , and x = ( x , x ) ∈ J p , y ∈ (0 , , itholds that for sufficiently large n ∈ N the distribution of S ( n ) has Lebesgue density h ( n ) ( x , x , y ) = n π y − (det H x ) − / e − n I S ( x ,x ,y ) (1 + o (1)) , where I S ( x , x , y ) := [Λ ∗ p ( x ) − log( y )] and H x := H τ Λ p ( τ ( x )) .Proof. By direct calculation, we can see for y ∈ [0 ,
1] that P (cid:0) U /n ≤ y (cid:1) = P ( U ≤ y n ) = y n , givingthat the density of U /n is given by f U /n ( y ) = n y n − . As U /n is independent of the Y ( n ) i , andthereby also of S ( n ) = ( | Y ( n ) i | q , | Y ( n ) i | p ), the density of S ( n ) = n P ni =1 ( | Y ( n ) i | q , | Y ( n ) i | p , U /n ) is givenby the product of their densities, hence h ( n ) ( x , x , y ) = h ( n ) ( x , x ) f U /n ( y ) = n π y − (det H x ) − / e − n [Λ ∗ p ( x ) − log( y )] (1 + o (1)) . This completes our proof. (cid:3) Proof of the Main Result for ℓ np -spheres In (20) we have reformulated the deviation probability P (cid:0) n /p − /q k Z ( n ) k q > z (cid:1) as an integral ofthe density estimate h ( n ) of the probabilistic representation S ( n ) over the deviation area D z . InProposition 5.1 we have then given h ( n ) explicitly. For the proof of Theorem 3.1 it remains tocalculate that integral. To do so, the integral will be split up into a neighbourhood B z of the point z ∗ , that has been shown in Lemma 4.1 to be the infimum of Λ ∗ p over ¯ D z , and its complement B cz .The LDP from Proposition 2.3 will be used to show the negligibility of the integral outside of the HARP ASYMPTOTICS FOR q -NORMS OF RANDOM VECTORS IN HIGH-DIMENSIONAL ℓ np -BALLS 15 neighbourhood of z ∗ . Within the neighbourhood B z , we use a result from Adriani and Baldi [3],which uses the Weingarten maps of the planar curves given by the boundary of D z ∩ B z and thelevel set of Λ ∗ p at z , to compute the integral. Following that, we will give these Weingarten mapsexplicitly, finishing our proof. Proof of Theorem 3.1.
We assume the set-up of Theorem 3.1 and use the reformulation (20) toproceed by considering P (cid:0) S ( n ) ∈ D z (cid:1) . Let B z ⊂ R be an open neighbourhood around z ∗ . Then itholds that P ( S ( n ) ∈ D z ) = Z D z h ( n ) ( x ) d x = Z D z ∩ B z h ( n ) ( x ) d x + Z D z ∩ B cz h ( n ) ( x ) d x. (22)Since z ∗ / ∈ B cz , by Lemma 4.1 , there exists an η >
0, such thatinf y ∈ D z ∩ B cz Λ ∗ p ( y ) > Λ ∗ p ( z ∗ ) + η, and thus, by the LDP in Proposition 2.3, it holds thatlim sup n →∞ n log P ( S ( n ) ∈ D z ∩ B cz ) ≤ − inf y ∈ D z ∩ B cz Λ ∗ p ( y ) ≤ − Λ ∗ p ( z ∗ ) − η. This gives us that P (cid:16) S ( n ) ∈ D z ∩ B cz (cid:17) ≤ e − n Λ ∗ p ( z ∗ ) − n η (1 + o (1)) = 1 e n η e − n Λ ∗ p ( z ∗ ) (1 + o (1)) . (23)Furthermore, by our density estimate in Proposition 5.1, it holds that Z D z ∩ B z h ( n ) ( x ) d x = n π Z D z ∩ B z (det H x ) − / e − n Λ ∗ p ( x ) d x (1 + o (1)) . (24)To calculate this explicitly, we will rely on a technique established in [3, Proof of Theorem 4.4].Therein, an asymptotic integral expansion of Bleistein and Handelsmann [8, Equation (8.3.63)] forLaplace integrals is reformulated via the Weingarten maps of the integration area and the level setof the exponential function at its minimum, both seen as hypersurfaces. We will present it as oneconcise result, similar to that formulated in [32, Lemma 4.6]. Proposition 6.1.
Let D ⊂ R d be a bounded domain such that ∂D is a differentiable hypersurfacein R d . Furthermore, let g : R d → R be a differentiable function and φ : D → [0 , ∞ ) a nonnegativefunction that is twice differentiable and attains a unique infimum over D at x ∗ ∈ ∂D . Define thehypersurfaces H D = ∂D and H φ = { x ∈ R d : φ ( x ) = φ ( x ∗ ) } , and denote by L D and L φ their respective Weingarten maps at x ∗ . Then, for sufficiently large n ∈ N , it holds that Z D g ( x ) e − n φ ( x ) d x = (2 π ) ( d − / det( L − φ ( L φ − L D )) − / n ( d +1) / hH x φ ( x ∗ ) − ∇ x φ ( x ∗ ) , ∇ x φ ( x ∗ ) i / g ( x ∗ ) e − n φ ( x ∗ ) (1 + o (1)) . The proof of this can be found in the proof of [3, Equations (4.5), (4.6)]. Let us now check thatthese conditions hold for the integral in (24). We have that D z ∩ B z is bounded, and for z > m p,q ,we can write ∂D z as the graph of the infinitely differentiable function f : (0 , ∞ ) → (0 , ∞ ) with f ( t ) = z − p t p/q , thus both ∂D z and ∂D z ∩ B z are differentiable planar curves. Furthermore, we knowthat Λ p is infinitely differentiable in its effective domain by virtue of being a c.g.f., which, by standardproperties of the Legendre-Fenchel transform, yields that Λ ∗ p is infinitely differentiable on its effectivedomain as well. Hence, for z > m p,q such that Λ ∗ p ( z ∗ ) < ∞ and B z small enough that Λ ∗ p ( x ) < ∞ forall x ∈ B z , we have infinite differentiability of Λ ∗ p ( x ), τ ( x ) and Λ p ( τ ( x )) in x on D z ∩ B z . Lemma4.1 gives us the uniqueness of z ∗ = ( z q ,
1) as an infimum on D z and D z ∩ B z . Nonnegativity ofΛ ∗ p follows directly by the standard properties of rate functions. By the infinite differentiability of Λ p ( τ ( x )) in x , we get the differentiability of g ( x ) := (det H x ) − / = (det H τ Λ p ( τ ( x )) − / in x . Thus,in view of the above, we can use Proposition 6.1 for D = D z ∩ B z ⊂ R with g ( x ) = (det H x ) − / , φ ( x ) = Λ ∗ p ( x ), and x ∗ = z ∗ , and get that Z D z ∩ B z h ( n ) ( x ) d x = n π (2 π ) / det( L − ( L Λ − L D )) − / (det H z ∗ ) − / e − n Λ ∗ p ( z ∗ ) n / hH x Λ ∗ p ( z ∗ ) − ∇ x Λ ∗ p ( z ∗ ) , ∇ x Λ ∗ p ( z ∗ ) i / (1 + o (1)) , (25)for the respective Weingarten maps at z ∗ of the curves H D = ∂ ( D z ∩ B z ) and H Λ = { x ∈ R : Λ ∗ p ( x ) = Λ ∗ p ( z ∗ ) } . Let us present the following identities for some of the terms in the fraction above, resulting fromthe definition of τ ( x ) and the properties of the Legendre-Fenchel transform: Lemma 6.2.
It holds thati) ∇ x Λ ∗ p ( x ) = τ ( x ) , ii) H x Λ ∗ p ( x ) = H x − . Proof.
We start by showing that ∇ x Λ ∗ p ( x ) = τ ( x ). We have defined τ ( x ) as the supremum of[ h x, τ i − Λ p ( τ )] in τ ∈ R (see (10)), thus it follows that ∇ τ (cid:2) h x, τ i − Λ p ( τ ) (cid:3)(cid:12)(cid:12) τ = τ ( x ) = x − ∇ τ Λ p ( τ ( x )) = 0 . With this, it follows that ∇ x Λ ∗ p ( x ) = ∇ x (cid:2) h x, τ ( x ) i − Λ p ( τ ( x )) (cid:3) = τ ( x ) + J x τ ( x ) x − ∇ x Λ p ( τ ( x ))= τ ( x ) + J x τ ( x ) x − J x τ ( x ) ∇ τ Λ p ( τ ( x ))= τ ( x ) + J x τ ( x ) (cid:2) x − ∇ τ Λ p ( τ ( x )) (cid:3) = τ ( x ) . Let us now prove that H x (Λ ∗ p ( x )) = H x − . On the one hand, it follows from the above that H x Λ ∗ p ( x ) = J x τ ( x ) , (26)while on the other hand, it holds that H x Λ ∗ p ( x ) = H x (cid:2) h x, τ ( x ) i − Λ p ( τ ( x )) (cid:3) = H x (cid:2) h x, τ ( x ) i (cid:3) − H x (cid:2) Λ p ( τ ( x )) (cid:3) = J x (cid:2) ∇ x h x, τ ( x ) i (cid:3) − J x (cid:2) ∇ x Λ p ( τ ( x )) (cid:3) = J x (cid:2) τ ( x ) + J x τ ( x ) x (cid:3) − J x (cid:2) J x τ ( x ) ∇ τ Λ p ( τ ( x )) (cid:3) = J x τ ( x ) + J x (cid:2) J x τ ( x ) x (cid:3) − H x τ ( x ) ∇ τ Λ p ( τ ( x )) − J x τ ( x ) J x (cid:2) ∇ τ Λ p ( τ ( x )) (cid:3) = 2 J x τ ( x ) + H x τ ( x ) (cid:2) x − ∇ τ Λ p ( τ ( x )) (cid:3) − J x τ ( x ) J x τ ( x ) H τ Λ p ( τ ( x ))= 2 J x τ ( x ) − J x τ ( x ) J x τ ( x ) H x Λ p ( τ ( x )) , (27)Equating the terms (26) and (27) yields HARP ASYMPTOTICS FOR q -NORMS OF RANDOM VECTORS IN HIGH-DIMENSIONAL ℓ np -BALLS 17 J x τ ( x ) = 2 J x τ ( x ) − J x τ ( x ) J x τ ( x ) H x Λ p ( τ ( x )) ⇔ J x τ ( x ) − J x τ ( x ) J x τ ( x ) H x Λ p ( τ ( x )) ⇔ I − J x τ ( x ) H x Λ p ( τ ( x )) ⇔ J x τ ( x ) = H x Λ p ( τ ( x )) − , where I denotes the identity matrix in R . Again using (26) on the above yields H x Λ ∗ p ( x ) = J x τ ( x ) = H x Λ p ( τ ( x )) − = H − x , and thereby finishes the proof. (cid:3) Via Lemma 6.2, we get D H x Λ ∗ p ( z ∗ ) − ∇ x Λ ∗ p ( z ∗ ) , ∇ x Λ ∗ p ( z ∗ ) E = D H z ∗ τ ( z ∗ ) , τ ( z ∗ ) E . With the definition of ξ ( z ) in (12) the integral in (25) hence simplifies as follows: Z D z ∩ B z h ( n ) ( x ) d x = 1 √ πn ξ ( z ) (cid:0) det( L − ( L Λ − L D ) (cid:1) − / e − n Λ ∗ p ( z ∗ ) (1 + o (1)) . (28)We see that it only remains to prove that det( L − ( L Λ − L D )) = κ ( z ) . We proceed to calculate theWeingarten maps of the curves H D and H Λ explicitly. As discussed in Section 2.5, the Weingartenmap of a planar curve at a point x reduces to the absolute value of its curvature in x . As previouslymentioned, ∂D z is the graph of a function f : (0 , ∞ ) → (0 , ∞ ) with f ( t ) = z − p t p/q . Thus, thesame holds locally for H D = ∂ ( D z ∩ B z ) in a neighbourhood of z ∗ , so by the curvature formula forgraphs of functions, as seen in Remark 2.8 ii), it holds that L D = | f ′′ ( z q ) | (1 + f ′ ( z q ) ) / , where f ′ ( t ) = (cid:16) pq − z − p t p/q ) − (cid:17) ⇒ f ′ ( z q ) = p q − z − q , and f ′′ ( t ) = pq − (cid:0) pq − − (cid:1) z − p t p/q ) − ⇒ f ′′ ( a q ) = ( p − pq ) q − z − q . This yields L D = | ( p − pq ) q − z − q | (1 + p q − z − q ) / = | pq ( p − q ) z q | ( z q + p q − ) / . (29)The curve H Λ is the zero set of the function F ( x ) := Λ ∗ p ( x ) − Λ ∗ p ( z ∗ ). From Lemma 6.2 we knowthat ( F [1 , , F [0 , ) = (cid:16) ∂∂x Λ ∗ p ( z ∗ ) , ∂∂x Λ ∗ p ( z ∗ ) (cid:17) = τ ( z ∗ )and (cid:18) F [2 , F [1 , F [1 , F [0 , (cid:19) = ∂ ∂ x Λ ∗ p ( z ∗ ) ∂ ∂x ∂x Λ ∗ p ( z ∗ ) ∂ ∂x ∂x Λ ∗ p ( z ∗ ) ∂ ∂ x Λ ∗ p ( z ∗ ) = H − z ∗ , for derivatives F [ i,j ] = F [ i,j ] ( z ∗ ) as in (1). Hence, by the curvature formula for implicit curves fromLemma 2.7 and Remark 2.8 i), we get L Λ = (cid:12)(cid:12) τ ( z ∗ ) (cid:0) H − z ∗ (cid:1) − τ ( z ∗ ) τ ( z ∗ ) (cid:0) H − z ∗ (cid:1) + τ ( z ∗ ) (cid:0) H − z ∗ (cid:1) (cid:12)(cid:12)(cid:0) τ ( z ∗ ) + τ ( z ∗ ) (cid:1) / . (30) Since both L D and L Λ are one-dimensional, it follows from (29) and (30) thatdet( L − ( L Λ − L D )) = L − ( L Λ − L D ) = 1 − L D L Λ = κ ( z ) . for κ ( z ) as in (13). It now follows with (28) that Z D z ∩ B z h ( n ) ( x ) d x = 1 √ πn ξ ( z ) κ ( z ) e − n Λ ∗ p ( z ∗ ) (1 + o (1)) . (31)Comparing (31) with the upper bound of the integral outside of B z in (23), we can see that theintegral over B cz is negligible for large n ∈ N . Thus, combining (22), (23) and (31) finishes the proofof Theorem 3.1. (cid:3) Proof of the Main Result for ℓ np -balls We use the notation and definitions established in Sections 2 through 4. Let 1 ≤ q < p < ∞ and z >m p,q such that I k Z k ( z ) < ∞ . We proceed similarly to the previous proof, using the reformulation of P (cid:0) n /p − /q k Z ( n ) k q > z (cid:1) from (21) in conjunction with the density approximation from Proposition5.2. The resulting integral over D z is again split into a neighbourhood of the minimum of I S over D z and its complement, which, according to Lemma 4.1, is attained in z ∗∗ = ( z q , , ∂ D z stillheavily dictates the value of the overall approximation. However, since this result is formulated for acertain neighbourhood of the origin, we first need to construct a sufficient transformation, mappingour deviation area into such a neighbourhood. After that, we calculate the specific approximationin our setting. Proof of Theorem 3.2.
We assume the set-up of Theorem 3.2 and use the reformulation (21) toproceed by considering P (cid:0) S ( n ) ∈ D z (cid:1) . Let B z ⊂ R be an open neighbourhood around z ∗∗ =( z q , , P (cid:16) S ( n ) ∈ D z (cid:17) = Z D z ∩ B z h ( n ) ( x , x , y ) d x d x d y + Z D z ∩ B cz h ( n ) ( x , x , y ) d x d x d y. (32)As in the proof of Theorem 3.1, we can follow from Lemma 4.1 ii) and the LDP in Proposition 2.5that there is an η >
0, such that P (cid:16) S ( n ) ∈ D z ∩ B cz (cid:17) ≤ e − n I S ( z ∗∗ ) − nη (1 + o (1)) = 1 e n η e − n Λ ∗ p ( z ∗ ) (1 + o (1)) . (33)Let us now consider the first integral in (32). Since z ∗ ∈ J p , we know that I S ( z ∗∗ ) < ∞ , hence, forsufficiently small B z , we have that x = ( x , x ) ∈ J p and y ∈ (0 , Z D z ∩ B z h ( n ) ( x , x , y ) d x d x d y = n π Z D z ∩ B z y − (det H x ) − / e − n I S ( x ,x ,y ) d x d x d y (1 + o (1)) . As we have seen in Lemma 4.1, I S attains its infimum on D z at z ∗∗ . However, we cannot use theresult of Adriani and Baldi from Proposition 6.1 here, since at z ∗∗ the boundary of D z ∩ B z is notdifferentiable, and thereby not smooth. Hence, we use the following result of Liao and Ramanan[32] and apply it to our setting. HARP ASYMPTOTICS FOR q -NORMS OF RANDOM VECTORS IN HIGH-DIMENSIONAL ℓ np -BALLS 19 Proposition 7.1.
Let D be a bounded subset in R such that D ⊂ { x = ( x , x , x ) ∈ R : x ≥ , x ≥ } and contains a neighbourhood of (0 , , in { x = ( x , x , x ) ∈ R : x ≥ , x ≥ } . Let h ( n ) : R → R , n ∈ N , be a sequence of functions that take the form h ( n ) ( x ) = g ( n ) ( x ) e − n f ( x ) , where f is a nonnegative function that is smooth in a neighbourhood of D and its minimum on D is attained uniquely at x ∗ = (0 , , , and each g ( n ) is smooth with g ( n ) ( x ) having order o ( e nx ) in aneighbourhood of D . Then, we have the following asymptotic expansion: Z D h ( n ) d x = √ πn / g ( n ) ( x ∗ ) f [1 , , f [0 , , q | f [0 , , | e − nf ( x ∗ ) (1 + o (1)) , with derivatives f [ i,j,k ] = f [ i,j,k ] (0 , , as in (1) . To apply this, we use a transformation of D z ∩ B z , mapping z ∗∗ = ( z q , ,
1) to := (0 , , I : R → R with I ( x , x , y ) = ( y q x − z q x q/p , − y, x −
1) = ( t , t , t ) . It then holds that I ( z ∗∗ ) = and I ( D z ) = ˜ D z := { t ∈ R : t > , t ∈ [0 , , t > − } .Furthermore, in a neighbourhood of z ∗∗ small enough such that t < I is invertible with I − ( t , t , t ) = t + z q ( t + 1) q/p (1 − t ) q , t + 1 , − t ! . Let us calculate the Jacobian of I − : J t I − ( t ) = − t ) q q ( t + z q ( t +1) q/p )(1 − t ) q +1 z q qp ( t +1) ( q/p ) − (1 − t ) q − . (34)Thus, we have that | det J t I − ( t ) | = (1 − t ) − q . We set g ( x , x , y ) = y − (det H x ) − / , as well as˜ B z := I ( B z ), and transform the area of integration via I − , yielding P (cid:16) S ( n ) ∈ D z ∩ B z (cid:17) = Z D z ∩ B z h ( n ) ( x , x , y ) d x d x d y = n π Z D z ∩ B z y − (det H x ) − / e − n [Λ ∗ p ( x ,x ) − log( y )] d x d x d y (1 + o (1))= n π Z D z ∩ B z g ( x , x , y ) e − n I S ( x ,x ,y ) d x d x d y (1 + o (1))= n π Z ˜ D z ∩ ˜ B z g ◦ I − ( t ) e − n I S ◦ I − ( t ) (1 − t ) − q d t (1 + o (1)) . We now set ˜ g ( t ) := (1 − t ) − q g ◦ I − ( t ) and ˜ f ( t ) := I S ◦ I − ( t ), then P (cid:16) S ( n ) ∈ D z ∩ B z (cid:17) = n π Z ˜ D z ∩ ˜ B z ˜ g ( t ) e − n ˜ f ( t ) d t (1 + o (1)) . (35)It now holds that ˜ D z ∩ ˜ B z is bounded and ˜ D z ∩ ˜ B z ⊂ ˜ D z = { t ∈ R : t > , t ∈ [0 , , t > − } . Itholds further that I S ◦ I − is nonnegative, since I S is a nonnegative function. We can see from (34)that all partial derivatives of I − are again differentiable in a neighbourhood of ˜ D z ∩ ˜ B z for ˜ B z smallenough such that ( t , , t ) / ∈ ˜ D z ∩ ˜ B z . Thereby, I − is twice differentiable in such a neighbourhood.By the same arguments as in the proof of Theorem 3.1, Λ ∗ p is twice differentiable, thereby also I S , and hence, I S ◦ I − is twice differentiable as well. Furthermore, I has been constructed thusly,that I ( z ∗∗ ) = , so I S ◦ I − ( ) = I S ( z ∗∗ ). It holds that ∈ ∂ ˜ D z and we know from Lemma 4.1,that I S is uniquely minimized on D z at z ∗∗ . Hence, minimizes I S ◦ I − uniquely on ˜ D z . We set˜ g ( n ) ( t ) = ˜ g ( t ) for all n ∈ N , as in our setting variability of ˜ g in n ∈ N is not required. For sufficientlysmall ˜ B z we know that ˜ g ( t ) = (1 − t ) − q h (1 − t ) − (cid:0) det H ( I − ( t ) , I − ( t ) ) (cid:1) − / i is differentiable on˜ D z ∩ ˜ B z as a composition of differentiable functions. Thus, ˜ D z ∩ ˜ B z , ˜ f and ˜ g meet the requirementsof Proposition 7.1 and we get the following reformulation of our integral in (35): n π Z ˜ D z ∩ ˜ B z ˜ g ( t ) e − n ˜ f ( t ) d t (1 + o (1)) = n π √ πn / ˜ g ( )˜ f [1 , , ˜ f [0 , , q | ˜ f [0 , , | e − n ˜ f ( ) (1 + o (1)) . (36)It remains to calculate the quantities in the above explicitly. First off, it holds that˜ g ( ) = (det H z ∗ ) − / and ˜ f ( ) = I S ( z ∗∗ ) = Λ ∗ p ( z ∗ ) . (37)Let us now consider the derivatives of ˜ f in the above fraction:˜ f [1 , , = ∂∂t ˜ f ( ) = ∂∂t I S ◦ I − ( ) = ∇ ( x ,x ,y ) I S ( z ∗∗ ) ∂∂t I − ( ) . With Lemma 6.2 i) we can follow that ∇ ( x ,x ,y ) I S ( z ∗∗ ) = (cid:18) ∂∂x Λ ∗ p ( x , x ) , ∂∂x Λ ∗ p ( x , x ) , − y (cid:19) (cid:12)(cid:12)(cid:12) ( x ,x ,y )= z ∗∗ = (cid:18) τ ( x ) , τ ( x ) , − y (cid:19) (cid:12)(cid:12)(cid:12) ( x ,x ,y )= z ∗∗ = ( τ ( z ∗ ) , τ ( z ∗ ) , − , and we know from (34) that ∂∂t I − ( ) = (cid:0) (1 − t ) − q , , (cid:1) (cid:12)(cid:12)(cid:12) t = = (1 , , , which gives ˜ f [1 , , = ( τ ( z ∗ ) , τ ( z ∗ ) , −
1) (1 , ,
0) = τ ( z ∗ ) . (38)Similarly, we get˜ f [0 , , = ∇ ( x ,x ,y ) I S ( z ∗∗ ) ∂∂t I − ( ) = ( τ ( z ∗ ) , τ ( z ∗ ) , − ∂∂t I − ( ) , where ∂∂t I − ( ) = q ( t + z q ( t + 1) q/p )(1 − t ) q +1 , , − ! (cid:12)(cid:12)(cid:12) t = = ( qz q , , − , yielding ˜ f [0 , , = ( τ ( z ∗ ) , τ ( z ∗ ) , −
1) ( qz q , , −
1) = qz q τ ( z ∗ ) + 1 . (39)And lastly, we calculate ˜ f [0 , , . We start by noting that ∂∂t I − ( t ) (cid:12)(cid:12)(cid:12) t = = z q qp ( t + 1) ( q/p ) − (1 − t ) q , , ! (cid:12)(cid:12)(cid:12) t = = (cid:18) z q pq , , (cid:19) , HARP ASYMPTOTICS FOR q -NORMS OF RANDOM VECTORS IN HIGH-DIMENSIONAL ℓ np -BALLS 21 and ∂ ∂ t I − ( t ) (cid:12)(cid:12) t = = ∂∂t z q qp ( t + 1) ( q/p ) − (1 − t ) q , , ! (cid:12)(cid:12)(cid:12) t = = z q qp (cid:0) qp − (cid:1) ( t + 1) ( q/p ) − (1 − t ) q , , ! (cid:12)(cid:12)(cid:12) t = = (cid:18) z q q p − z q qp , , (cid:19) . Again, by Lemma 6.2 ii), we get that H ( x ,x ,y ) I S ( z ∗∗ ) = (cid:0) H − z ∗ (cid:1) (cid:0) H − z ∗ (cid:1) (cid:0) H − z ∗ (cid:1) (cid:0) H − z ∗ (cid:1)
00 0 y − . It thereby follows that˜ f [0 , , = ∂ ∂ t I S ◦ I − ( )= ∂∂t (cid:20) ∇ ( x ,x ,y ) I S ( I − ( t )) ∂∂t I − ( t ) (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = = ∂∂t (cid:2) ∇ ( x ,x ,y ) I S ( I − ( t )) (cid:3) (cid:12)(cid:12)(cid:12) t = ∂∂t I − ( ) + ∇ ( x ,x ,y ) I S ( z ∗∗ ) ∂ ∂ t I − ( )= (cid:18) z q qp , , (cid:19) H ( x ,x ,y ) I S ( z ∗∗ ) (cid:18) z q qp , , (cid:19) + (cid:16) τ ( z ∗ ) , τ ( z ∗ ) , − (cid:17) (cid:18) z q q p − z q qp , , (cid:19) = (cid:18) z q qp , , (cid:19) (cid:18) z q qp (cid:0) H − z ∗ (cid:1) + (cid:0) H − z ∗ (cid:1) , z q qp (cid:0) H − z ∗ (cid:1) + (cid:0) H − z ∗ (cid:1) , (cid:19) + τ ( z ∗ ) (cid:16) z q q p − z q qp (cid:17) = z q q p (cid:0) H − z ∗ (cid:1) + 2 z q qp (cid:0) H − z ∗ (cid:1) + (cid:0) H − z ∗ (cid:1) + τ ( z ∗ ) (cid:16) z q q p − z q qp (cid:17) . (40)Plugging the terms from (37), (38), (39) and (40) into the fraction in (36), we get that˜ g ( )˜ f [1 , , ˜ f [0 , , q | ˜ f [0 , , | = (det H z ∗ ) − / ( τ ( z ∗ ) ) − ( qz q τ ( z ∗ ) + 1) − × (cid:12)(cid:12)(cid:12) z q q p (cid:0) H − z ∗ (cid:1) + 2 z q qp (cid:0) H − z ∗ (cid:1) + (cid:0) H − z ∗ (cid:1) + τ ( z ∗ ) (cid:16) z q q p − z q qp (cid:17)(cid:12)(cid:12)(cid:12) − / = " det H z ∗ ( τ ( z ∗ ) ) ( qz q τ ( z ∗ ) + 1) × (cid:12)(cid:12)(cid:12) z q q p (cid:0) H − z ∗ (cid:1) + 2 z q qp (cid:0) H − z ∗ (cid:1) + (cid:0) H − z ∗ (cid:1) + τ ( z ∗ ) z q q ( q − p ) p (cid:12)(cid:12)(cid:12) − / = γ ( z ) − , with γ ( z ) as in (14). Hence, it follows for the integral in (36) that n π Z ˜ D z ∩ ˜ B z ˜ g ( t ) e − n ˜ f ( t ) d t (1 + o (1)) = 1 √ πn γ ( z ) e − n Λ ∗ p ( z ∗ ) (1 + o (1)) . (41)Combining the representation from (32) with the two integral estimates from (33) and (41) showsthat the integral in the complement of B z can be neglected and we have that P (cid:16) n /p − /q k Z ( n ) k q > z (cid:17) = P (cid:16) S ( n ) ∈ D z ∩ B z (cid:17) = 1 √ πn γ ( z ) e − n Λ ∗ p ( z ∗ ) (1 + o (1)) , which proves our second main result for ℓ np -balls. (cid:3) Appendix
Proof of Lemma 2.4.
Let z > m p,q and z ∗ := ( z q , ∈ R be such that z ∗ ∈ J p . Then it holds that I k Z k ( z ) = inf t , t > t /q t − /p = z Λ ∗ p ( t , t ) = inf ˜ t , ˜ t >
0: ˜ t = z ˜ t Λ ∗ p (˜ t q , ˜ t p ) = inf ˜ t > Λ ∗ p ( z q ˜ t q , ˜ t p ) . We set t z := ( z q ˜ t q , ˜ t p ), then with (10) it follows that I k Z k ( z ) = inf ˜ t > sup s ∈ R ( h s, t z i − Λ p ( s )) = inf ˜ t > h h τ ( t z ) , t z i − Λ p ( τ ( t z )) i . Our goal is to show that the infimum is attained at t ∗ z := z ∗ , i.e. at ˜ t = 1. Recall the definition g t ( s ) := h s, t i − Λ p ( s ) for t ∈ J p from Section 2.4. By the definition of τ ( t z ) it holds that g t z ( s )attains its supremum at τ ( t z ), thus it holds that ∇ s g t z ( s ) (cid:12)(cid:12) s = τ ( t z ) = t z − ∇ s Λ p ( s ) (cid:12)(cid:12) s = τ ( t z ) = 0 , whichgives t z = (cid:0) z q ˜ t q , ˜ t p (cid:1) = (cid:18) ∂∂s Λ p ( s ) (cid:12)(cid:12) s = τ ( t z ) , ∂∂s Λ p ( s ) (cid:12)(cid:12) s = τ ( t z ) (cid:19) . (42)We now aim to write ∂∂s Λ p ( s ) with respect to ∂∂s Λ p ( s ) and then use the above equations. To doso, we firstly want to reformulate Λ p along the lines of [17, Lemma 5.7]. It holds thatΛ p ( s ) := log Z R e s | y | q + s | y | p f p ( y ) d y = log p /p Γ (cid:0) p (cid:1) Z R e s | y | q − p (1 − ps ) | y | p d y ! . The change of variables x = (1 − ps ) /p y then givesΛ p ( s ) = log (cid:18) (1 − ps ) − /p Z R e s − ps q/p | x | q f p ( x ) d x (cid:19) = − p log(1 − ps ) + log ϕ | X | q (cid:18) s (1 − ps ) q/p (cid:19) , where ϕ | X | q is the m.g.f. of | X | q . Hence, ∂∂s Λ p ( s ) = ∂∂s (cid:20) log ϕ | X | q (cid:18) s (1 − ps ) q/p (cid:19)(cid:21) = ϕ | X | q (cid:18) s (1 − ps ) q/p (cid:19) − ∂∂s (cid:20) ϕ | X | q (cid:18) s (1 − ps ) q/p (cid:19)(cid:21) = ϕ | X | q (cid:18) s (1 − ps ) q/p (cid:19) − Z R (1 − ps ) − q/p | x | q e s − ps q/p | x | q f p ( x ) d x = (1 − ps ) − q/p ϕ | X | q (cid:18) s (1 − ps ) q/p (cid:19) − ϕ ′| X | q (cid:18) s (1 − ps ) q/p (cid:19) , HARP ASYMPTOTICS FOR q -NORMS OF RANDOM VECTORS IN HIGH-DIMENSIONAL ℓ np -BALLS 23 where ϕ ′| X | q (cid:16) s (1 − ps ) q/p (cid:17) = ϕ ′| X | q ( t ) (cid:12)(cid:12)(cid:12) t = s − ps q/p . Moreover, with the above we get that ∂∂s Λ p ( s ) = (1 − ps ) − + ∂∂s (cid:20) log ϕ | X | q (cid:18) s (1 − ps ) q/p (cid:19)(cid:21) = (1 − ps ) − + ϕ | X | q (cid:18) s (1 − ps ) q/p (cid:19) − ∂∂s (cid:20) ϕ | X | q (cid:18) s (1 − ps ) q/p (cid:19)(cid:21) = (1 − ps ) − + ϕ | X | q (cid:18) s (1 − ps ) q/p (cid:19) − Z R qs (1 − ps ) ( q + p ) /p | x | q e s − ps q/p | x | q f p ( x ) d x = (1 − ps ) − + qs (1 − ps ) ( q + p ) /p ϕ | X | q (cid:18) s (1 − ps ) q/p (cid:19) − ϕ ′| X | q (cid:18) s (1 − ps ) q/p (cid:19) = (1 − ps ) − + qs − ps (1 − ps ) − q/p ϕ | X | q (cid:18) s (1 − ps ) q/p (cid:19) − ϕ ′| X | q (cid:18) s (1 − ps ) q/p (cid:19) = (1 − ps ) − + qs (1 − ps ) − ∂∂s Λ p ( s ) . (43)Plugging in the identities from (42) into (43) it follows for ( s , s ) = (cid:0) τ ( t z ) , τ ( t z ) (cid:1) :˜ t p = (1 − pτ ( t z ) ) − + qτ ( t z ) (1 − pτ ( t z ) ) − z q ˜ t q . (44)Using this, we can calculate the derivative of Λ ∗ p ( t z ) in t (we write t instead of ˜ t for notationalbrevity), where τ ( t z ) is considered as a function in t as well. It holds that ∂∂t Λ ∗ p ( t z ) = ∂∂t Λ ∗ p ( z q t q , t p )= ∂∂t (cid:2) h t z , τ ( t z ) i − Λ p ( τ ( t z )) (cid:3) = ∂∂t (cid:2) z q t q τ ( t z ) + t p τ ( t z ) − Λ p ( τ ( t z )) (cid:3) = z q qt q − τ ( t z ) + z q t q ∂∂t τ ( t z ) + pt p − τ ( t z ) + t p ∂∂t τ ( t z ) − ∂∂t Λ p ( τ ( t z ))= z q qt q − τ ( t z ) + z q t q ∂∂t τ ( t z ) + pt p − τ ( t z ) + t p ∂∂t τ ( t z ) − J t ( τ ( t z )) ∇ s Λ p ( s ) (cid:12)(cid:12) s = τ ( t z ) = z q qt q − τ ( t z ) + z q t q ∂∂t τ ( t z ) + pt p − τ ( t z ) + t p ∂∂t τ ( t z ) − ∂∂t τ ( t z ) ∂∂s Λ p ( s ) (cid:12)(cid:12) s = τ ( t z ) − ∂∂t τ ( t z ) ∂∂s Λ p ( s ) (cid:12)(cid:12) s = τ ( t z ) . We now use the identity from (42), which yields ∂∂t Λ ∗ p ( t z ) = z q qt q − τ ( t z ) + z q t q ∂∂t τ ( t z ) + pt p − τ ( t z ) + t p ∂∂t τ ( t z ) − ∂∂t τ ( t z ) z q t q − ∂∂t τ ( t z ) t p = z q qt q − τ ( t z ) + pt p − τ ( t z ) . (45) Reformulating the identity in (44) yields t p = (1 − pτ ( t z ) ) − + qτ ( t z ) (1 − pτ ( t z ) ) − z q t q ⇔ (1 − pτ ( t z ) ) t p − − t − = z q t q − qτ ( t z ) . (46)Thus, if we set ∂∂t Λ ∗ p ( t a ) = 0, we get from (45) and (46) that ∂∂t Λ ∗ p ( t z ) = 0 ⇔ z q qt q − τ ( t z ) + pt p − τ ( t z ) ⇔ − pτ ( t z ) ) t p − − t − + pt p − τ ( t z ) ⇔ t = 1 . Hence, the infimum of Λ ∗ p over ∂D z is attained at t ∗ z = ( z q ,
1) = z ∗ Since Λ ∗ p is strictly convex(see properties of the Legendre-Fenchel transform), this minimum is unique. Thereby, our claim isproven. (cid:3) Proof of Lemma 2.6.
Let z > m p,q , such that z ∗ = ( z q , ∈ J p . Furthermore, set z ∗∗ := ( z q , , I S ( t ) := [Λ ∗ p ( t , t ) − log( t )], t ∈ R . We use the definitions of I k Z k and I U , together withLemma 2.4, to get that I k Z k ( z ) = inf z = t /q t − /p t t , t > , t ∈ (0 , I S ( t )= inf z = z z z > , z ∈ (0 , inf t , t > t /q t − /p = z Λ ∗ p ( t , t ) + I U ( z ) = inf z = z z z > , z ∈ (0 , [Λ p ( z q , − log( z )] . Due to [42], we know that E [ n /p − /q k Z ( n ) k q ] = m p,q , which implies that I k Z k ( m p,q ) = 0. Since I k Z k is strictly convex, by the standard properties of rate functions, it follows that for z > m p,q itholds that I k Z k ( z ) = Λ ∗ p ( z q ,
1) is strictly increasing in z . Since z ≤ z = z z , and 1 < q , we have z q ≥ z > m p,q , meaning that Λ ∗ p ( z q ,
1) is strictly increasing in z . Furthermore, we can see that − log( z ) is strictly decreasing in z . Hence, rewriting z with respect to z then gives I k Z k ( z ) = inf z = z/z z ∈ (0 , (cid:20) Λ p (cid:16)(cid:16) zz (cid:17) q , (cid:17) − log( z ) (cid:21) , which is strictly decreasing in z . Thus, choosing z = 1 gives z = z and I k Z k ( z ) = I S ( z ∗∗ ) = Λ ∗ p ( z ∗ ) , finishing the proof. (cid:3) Acknowledgments
The author would like to thank Kavita Ramanan for the insightful exchanges on the topic ofsharp large deviations in asymptotic geometric analysis. Furthermore, the author would like tothank his supervisor Christoph Th¨ale for the helpful discussions, feedback and constructive criticismthroughout the writing of this paper.
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Tom Kaufmann: Faculty of Mathematics, Ruhr University Bochum, Germany
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