On the Concentration of Measure Phenomenon for Stable and Related Random Vectors

Transcription

1 The Annals of Probability 2003, Vol. 00, No. 00, On the Concentration of Measure Phenomenon for Stable and elated andom Vectors By Christian Houdré 2 and Philippe Marchal 3 Université Paris XII and Georgia Institute of Technology and Ecole Normale Supérieure Concentration of measure is studied, and obtained, for stable and related random vectors. Let X be a standard normal vector in d and let f : d be Lipschitz with constant one with respect to the Euclidean distance. A seminal result of Borell [B] and of Sudakov and Tsirel son [ST] asserts that for all x>0, /2 P fx mfx x 1 Φx e x2, 1 2 where mfx is a median of fx and where Φ is the one dimensional standard normal distribution function. The inequality 1 has seen many extensions and to date, most of the conditions under which these developments hold require the existence of finite exponential moments for the underlying vector X. Itisthus natural to explore the robustness of this concentration phenomenon and to study the corresponding results for stable vectors. It is the purpose of these notes to initiate this study and to present a few concentration results for stable and related vectors, freeing us from the exponential moment requirement. Our main result will imply that if X is an α stable random vector in d, then for all x>0, Cα, d P fx mfx x 1 x α, 2 where the constant Cα, d will be explicit. Let X IDb, 0,ν, i.e., let X be d-dimensional infinitely divisible vector without Gaussian component. For all u d, its characteristic function ϕ X is given, by ϕ X u =e ψu,with ψu=i u, b + d e i u,y 1 i u, y 1 y 1 νdy, 3 where b d and where ν 0 the Lévy measure is a positive Borel measure without atom at the origin and such that d y 2 1νdy < + throughout,, and are respectively the Euclidean inner product and norm in d. 1 eceived AMS 2000 subject classifications. 60G70, 62G07, 62C20, 41A25 Key words and phrases. concentration of measure phenomenon, stable random vectors, infinite divisibility. 2 Many thanks to the Laboratoire de Probabilités et Modèles Aléatoires, Université deparis VI for its hospitality while part of this research was done. 3 I would like to thank the School of Mathematics at the Georgia Institute of Technology where part of this research was done. 1

2 2 C. HOUDÉ and P. MACHAL It is well known that there is a one-to-one relationship between Lévy processes and infinitely divisible laws. More precisely, if {Xt : t 0}is a Lévy process without Gaussian component on d, then for all t 0, u d, ϕ Xt u =Ee i u,xt = e tψu, 4 where ψ is as in 3. Hence, an infinitely divisible vector X canbeviewedasx1, the Lévy process {Xt :t 0}at time 1. ecall also that X is α-stable, 0 <α<2, if the measure ν is given, for any Borel set B B d, by + νb = λdξ 1 B rξ dr, 5 S d 1 0 r1+α where λ is a finite positive measure on S d 1 the unit sphere of d called the spherical component of the Lévy measure. X is symmetric α-stable SαS if and only if λ is symmetric in which case, ϕ X u =e S cα d 1 u,ξ α λdξ, πγ 2 α/2 where c α = α2 α. Moreover, X is rotationally invariant if and only if Γ 1 + α/2 λ is uniform on S d 1 and then ϕ X u =e c α,d u α, where c α,d = c α u/ u,ξ S α λdξ does not depend on u d. In particular, d 1 if λ is the uniform probability measure on S d 1 Γd/2Γ 2 α/2, c α,d = α2 α Γd+α/2. We refer the reader to Sato s book [S] for a good introduction to Lévy processes and infinitely divisible laws. In order to prove our first theorem, we need the lemma below. For the mean rather than a median and x rather than x/2 the result is obtained in [H]. However, it is standard that applying this result to the function gy = mindy, A,x, y d,wherea={f m}and m is a median of f, leads to deviation from a median. Indeed, Eg x/2, and therefore g Eg x/2 whenever f m x. Lemma 1 Let X IDb, 0,νwith ν boundedly supported, let V 2 = d x 2 νdx, and let = inf {ρ >0:ν{x: x >ρ}= 0}. Then for any Lipschitz function with constant 1 f : d, P fx mfx x e x 2 x 2 + V 2 2 for all x>0, and where mfx is a median of fx. log1+ x 2V 2, 6 Above and below, the Lipschitz property is usually taken with respect to the fx fy Euclidean norm, i.e., f is Lipschitz if f Lip =sup <+ ; however x y x y other norms could be considered, e.g., see emark 2 ii. We can now state:

3 CONCENTATION OF MEASUE 3 Theorem 1 Let X be an α-stable vector with Lévy measure ν given by 5. Let f: d be such that f Lip 1. Then P fx mfx x K 1λS d 1 α2 αx α, 7 whenever K2 λs d 1 1/α x, α2 α and where mfx is a median of fx while K 1, K 2 are two absolute constants. Proof For any >0, we have the identity in distribution X = d Y +Z,where Y and Z are mutually independent infinitely divisible vectors with respective characteristic function ϕ Y = e ψ Y and ϕ Z = e ψ Z.Foru d, the exponents are given by ψ Z u = e i u,y 1 ν X dy, y > with ψ Y u =i u, b + e i u,y 1 i u, y 1 y 1 ν X dy, y b = b y > y1 y 1 ν X dy, where the last integral is understood coordinatewise and so is the above difference and where ν X is the Lévy measure of X. Next, P fx mfx x P fy mfx x+pz 0. 8 Let us first estimate the second probability in 8 involving the compound Poisson random vector Z. P Z 0=1 PZ =0 1 exp x > ν X dx =1 exp λdξ S d 1 =1 exp λsd 1 α α =1 exp C 2 α, λ α rξ > dr r 1+α C 2 α, 9

4 4 C. HOUDÉ and P. MACHAL where C 2 := C 2 α, λ = λsd 1. α Turning our attention to Y, we first compute the quantities involved in Lemma 1: where P fy mfy x e x V 2 = x x 2 ν X dx = λdξ S d 1 = λdξ S d 1 rξ r 2 dr 0 r 1+α 2 x 2 + V 2 2 rξ 2 dr r 1+α log1+ x 2V 2, 10 = C 1 α, λ 2 α, 11 with C 1 α, λ = λsd 1. Hence 10 becomes 2 α P fy mfy x e x x C 1 α log 1+ α x 2C 1 := H x 12 where C 1 := C 1 α, λ = λsd 1 2 α. Now rewrite the first probability in 8 as e x 2 x 1+ α 2 x 2C 1 P fy mfx x =PfY mfy, 13 x + mfx mfy. 14 We want to bound mfx mfy and, as it will become clear from the proof, only the case mfx <mfy which we assume presents some interest and needs some work. To this end, remark that for any x 0andany function f, wehave PfX x PfY x PX Y =PZ 0. Set P m = P fx mfx 1/2. Then, P m P fy mfx = P fx mfx P fy mfx P Z 0. Moreover, if f is Lipschitz with f Lip 1, P m P Z 0 PfY mfx = P fy mfy mfx mfy H mfy mfx,

5 CONCENTATION OF MEASUE 5 where the second inequality follows from Lemma 1 applied to f, andwithh given in 12. Since H is decreasing, set I y =sup{z 0,H z y}. Thus, provided P m >PZ 0, mfy mfx I P m P Z 0. Choose δ 0, 1/2. Then for every such that C 2 /δ 1/α 15 we have, using the same estimates as in 9, P m P Z 0 1/2 δ. Moreover, for every positive A, 13 entails A/2 P fy mfy A H A e A/2 2C1 A α. Thus if 2C1 A/2 2/αA e A/2, 16 A 1/2 δ then H A 1/2 δ, or, equivalently, I 1/2 δ A. As a consequence, if satisfies both conditions 15 and 16, we have mfy mfx I P m PZ 0 I 1/2 δ A. Using this together with 12 and 14 yields P fy mfx 2 + A P fy mfy 2 ec 1 α. Setting x =2+A,weobtain PfY mfx x ec 12 + A α x α 17 Finally, combining 17 and 9, we conclude that for any A>0, P fx mfx x ec 1 + C A α x α 18 whenever x is large enough so that there exists δ 0, 1/2 satisfying and x 2+A x 2+A C2 δ 1/α 19 1/α 2C1 e A/2 2/αA. 20 A 1/2 δ

6 6 C. HOUDÉ and P. MACHAL ForagivenA, the domain of validity of 18 can be found by optimizing δ in 19 and 20. Taking for instance A = 2 leads by equating the right hand sides of 19 and 20 to δ = C 2 /2eC 1 + C 2 =2 α/22 α + eα, and so P fx mfx x 4α ec 1 + C 2 x α = 4α 2 α + eαλs d 1 α2 αx α, whenever 22 α + eαλs d 1 1/α x 4. α2 α emark 1 i The estimate in 7 is sharp in x as can be seen by taking X SαS a one dimensional symmetric α-stable random variable with parameter σ>0andcharacteristic function ϕ X u =e σα u α. In that case e.g., see Proposition in [ST1] where lim x + xα P X x =σ α A α, { 1 α 2Γ2 αcosπα/2 A α =, α 1 1 π α=1 and σ α =2λ1c α = πγ2 α/2 2 α αγ1+α/2 2λ1. For d =1,andXsymmetric, our constants are C 1 = λ1+λ 1 2 α = 2λ1 2 α and C 2 = 2λ1 α. Thus, the dependency in α in the constants of 7 is sharp as α 0, but explodes as α 2in contrast to σ α A α. This problem will be addressed in the sequel. We also note that the dependency on the dimension d is sharp. Indeed, if X is a stable vector in d, then see Theorem in [ST1] lim x + xα P X x=c α λs d 1 A α. 21 ii As usual left tails inequalities also follow from 7 by applying the result to f, and as is also classical, the estimates can equivalently be given in terms of enlargements of sets. For α>1, a median can be replaced by the mean changing the range of x too by properly modifying the above proof or by using K2 λs d 1 1/α E fx mfx 2 α2 α + 2 K 1 λs d 1 α 1 α2 α which follows from integrating the tail inequality 7. K2 λs d 1 1 α/α, α2 α iii The methodology of proof presented above works as well for any infinitely divisible vector X. However, it requires estimates on x > ν Xdx andon

7 CONCENTATION OF MEASUE 7 x x 2 ν X dx which, in general, are unavailable in the absence of further knowledge of the Lévy measure. If the Lévy measure of X has the form + ν B = λdξ 1 B rξ Lrdr, 22 S d 1 0 r1+α for some slowly varying function L on [0,, in which case X is in the domain of attraction of a stable random vector with Lévy measure given by 5, the proof of Theorem 1 and standard estimates on regularly varying functions see for instance [BGT], Theorem give the following bound: P fx mfx x K 1λS d 1 Lx α2 αx α, 23 for every x such that L is locally bounded on [x, andsuchthat x α Lx Kα 2, where the constants K 1, K 2 are the same as in Theorem 1. A similar result can also be obtained when X in the domain of attraction of a stable vector with Lévy measure ν. Inthatcase,Lr in 22 should be replaced by Lr, ξ, thus if L 1 r Lr, ξ L 2 r, in 23, Lx should be replaced by L 2 x. When α is close to 2, the upper bound in Theorem 1 has the form KλS d 1 /2 αx α as soon as x α >K λs d 1 /2 α. We would like to obtain a better bound, namely of the form K λs d 1 /x α, at the price of strengthening the condition on x. To do so, we begin by a result improving Lemma 1. The setting and the notation below are as in Lemma 1, in addition, let W 3 = d x 3 νdx. Lemma 2 If V 2 /W 3 > 2/, lets 0 be the unique positive solution of e s 1 s = V 2 1, s > 0, W 3 and let x 0 =2 V 2 W3 s 0. Let f : d be such that f Lip 1. Then, if x x 0, x 2 P fx EfX x exp 4, V 2 W 3 while for x x 0, x x P fx EfX x K exp + 2W 3 3 log 1+ 2 x 2W 3, with x 2 0 K =exp 4 exp x 0 V 2 W3 + x0 + 2W 3 3 log 1+ 2 x 0 2W 3. 24

8 8 C. HOUDÉ and P. MACHAL Proof ecall that Theorem 1 in [H] asserts that P fx EfX x e x 0 h 1 sds, 25 for all 0 <x<hn, where N =sup { t 0:Ee t X < + } and where h 1 is the inverse of hs = u e s u 1νdu, 0 <s<n. d When the Lévy measure has its support in the Euclidean ball of center 0 and radius inwhichcase,n =+ Lemma 1 follows by bounding the function g s x =e sx 1, between 0 and by a linear interpolation, using also the convexity of the exponential. It is easily seen that for every x [0,], the following improved inequality holds: g s x sx + es 1 s 2 x 2, for all s 0. Therefore, hs V 2 W 3 s + W 3 2 es 1 2max V 2 W 3 s, W 3 2 es 1. So if V 2 /W 3 > 2/, there exists a unique positive s 0 such that e s0 1 s 0 = V 2 W 3 1, and for s s 0, while for s s 0, Let x 0 =2 V 2 W3 hs 2 V 2 W 3 s, hs 2 W 3 2 es 1. s 0,wehavefort x 0, h 1 t t 2, V 2 W3 while for t x 0, h 1 t 1 log 1+ 2 t 2W 3. The lemma follows. We are now ready to state our second theorem. We will express the deviation from the expectation here since it exists. As already mentioned, a result in terms of the median can be easily derived. Theorem 2 Using the notation of Theorem 1, assume α>3/2and let M =1/2 α. Then there exists an absolute constant K 3 such that P fx EfX x K 3λS d 1 x α, 26

9 CONCENTATION OF MEASUE 9 for every x satisfying x α 4λS d 1 M log M log1 + 2M log M. Proof We use the notation of the proof of Theorem 1. The second and third moment of the Lévy measure ν Y of Y are given by V 2 = λsd 1 2 α 2 α, and W 3 = λsd 1 3 α 3 α, which entails V 2 W 3 1=M. As α 3/2, M 2and and So for x x 0, Lemma 2 gives log M s 0 2 log M, 2λS d 1 M log M 3 α α 1 x 0 4λSd 1 M log M 3 α α P fy EfY x K exp From 24, 27 and since 2W 3 Suppose next that 3 x x log 1+ 2 x 0 2W 3 + 2W 3 3 log 1+ 2 x 2W 3. x 0, it follows that 4λS d 1 M log M K exp 3 α α log1 + 2M log M. 28 x α 4λS d 1 M log M log1 + 2M log M. 29 Then setting = x, it first follows since α<2 that K<e, and second by 27 and since M>2, that x>x 0. Therefore since W 3 = λs d 1 3 α /3 α, P fy EfY x eexp λSd 1 3 αxα 3 αx α log 1+ 2λS d 1 e 2 3 αxα exp log 1+ 2λS d 1 2e2 λs d 1 x α, 30

10 10 C. HOUDÉ and P. MACHAL for all x satisfying 29. Let us now estimate the difference between EfX and EfY. EfX EfY = E fy + Z fy 1 {Z 0} E Z x ν X dx x > = λsd 1 α 1 1 α x 4 log 2 log1 + 4 log 2, 31 where we used the compound Poisson structure of Z to get the next to last inequality. and our choice of = x, M>2, α 1 > 1/2, and the range of x given by 29, to get the last one. The estimate 31 as well as 30 and 9, finally give the result by proceeding as in the proof of Theorem 1 K 3 =1+8e 2 will do. emark 2 i Unless λs d 1 is bounded above independently of d, Theorem 1 and 2 are not dimension free, even for X with independent components. This is to be expected in view of 21, and is in sharp contrast to the Gaussian case. ii X has independent components if and only if λ is discrete and concentrated on the intersections of the axes of d with S d 1. In that case, the natural Lipschitz property is with respect to the l 1 norm. Thus taking, in the Lévy Khintchine representation 3 S d 1 1 the l 1 unit ball of d instead of S d 1, and correspondingly changing ν to ν 1, 7 continues to hold replacing λs d 1 byλ 1 S d 1 1, where λ 1 is the spherical component of ν 1.In fact, for any norm of d, a result similar to 7 continues to hold with the type of changes just described. As already mentioned, the above theorems are not dimension free, in particular, when the components of X are independent and for simplicity of notation identically distributed. However using Corollary 3 in [H] improved versions of Theorem 1 and 2 can be obtained with mixture of Lipschitz norms. More precisely, while we are not able to improve the constant in the upper bound of 26, the additional conditions we assume on the function f enable us to extend for c below is small the range on which our inequality holds. Again, for X with iid components the measure ν is concentrated on the axes of d see [S] p. 67, i.e., νdx 1,...,dx d = d δ 0 dx 1 δ 0 dx k 1 νdx k δ 0 dx k+1 δ 0 dx d. 32 k=1 Denoting by e 1,...,e d the canonical basis of d, Theorem 2 now becomes:

11 CONCENTATION OF MEASUE 11 Theorem 3 Let X be a stable vector with index α>3/2and Lévy measure ν given by 32. Letf: d be such that sup d x d k=1 fx + ue k fx 2 νdu a 2, 33 and sup fx + ue k fx c u, 34 x d for all k =1,...,d, u. Then, P fx EfX x K 4λS d 1 c α + K 5 a α x α, 35 whenever x α 44α α 1 c α 1 λs d 1, where K 4 and K 5 are absolute constants. Proof We only briefly describe the changes to the previous proofs to get the result. First, and as previously done, decompose X as X = d Y + Z.Next,use Corollary 3 in [H]: under the assumptions on f stated above, P fy EfY x exp x 4c log 1+ cx 2a 2. Then, proceed as in the proof of Theorem 1, using also the comparison between EfX andefy given in the proof of Theorem 2: EfX EfY λs d 1 1 α /α 1. Hence, taking = Kx/2αc, for some constant 0 <K<1, chosen below, leads to: P fx EfX x 2αα λs d 1 c α + α4α α/2 a α αk α x α, whenever x α 2αα 1 α 1 K 1 α 1 K cα 1 λs d 1, with α>1. The choices K =1/2, α>3/2 yield the result. A slightly improved result holds if in the integral in 33, is replaced by u where is chosen as above. EFEENCES [BGT] N.H. Bingham, C.M. Goldie and J.L. Teugels, egular Variation. Cambridge University Press, Cambridge, [B] C. Borell, The Brunn Minkowski inequality in Gauss space. Invent. Math , [H] C. Houdré, emarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Proba , [ST1] G. Samorodnitsky and M. Taqqu, Stable non-gaussian random processes. Stochastic models with infinite variance. Stochastic Modeling. Chapman and Hall, New York, 1994

12 12 C. HOUDÉ and P. MACHAL [S] K-I. Sato, Lévy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. evised by the author. Cambridge Studies in Advanced Mathematics, 68 Cambridge University Press, Cambridge, [ST] V.N. Sudakov and B.S. Tsirel son, Extremal properties of half spaces for spherically invariant measures. Zap. Nauch. Sem. LOMI , English translation in: J. Soviet Math , Université Paris XII Laboratoire d Analyse et de Mathématiques Appliquées CNS UM 8050 Créteil 94010, France and School of Mathematics Georgia Institute of Technology Atlanta, GA 30332, USA houdre@math.gatech.edu DMA Ecole Normale Supérieure Paris France Philippe.Marchal@ens.fr