Order statistics and concentration of l r norms for log-concave vectors
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1 Jounal of Funcional Analysis Ode saisics and concenaion of l noms fo log-concave vecos Rafał Laała a,b, a Insiue of Mahemaics, Univesiy of Wasaw, Banacha, Waszawa, Poland b Insiue of Mahemaics, Polish Academy of Sciences, ul. Śniadecich 8, Waszawa, Poland Received 30 Novembe 010; acceped 17 Febuay 011 Available online Mach 011 Communicaed by K. Ball Absac We esablish uppe bounds fo ails of ode saisics of isoopic log-concave vecos and apply hem o deive a concenaion of l noms of such vecos. 011 Elsevie Inc. All ighs eseved. Keywods: Log-concave measues; Ode saisics; Concenaion of volume 1. Inoducion and noaion An n-dimensional andom veco is called log-concave if i has a log-concave disibuion, i.e. fo any compac nonempy ses A,B R n and λ 0, 1, P X λa + 1 λb PX A λ PX B 1 λ, whee λa + 1 λb ={λx + 1 λy: x A, y B}. By he esul of Boell [3] a veco X wih full dimensional suppo is log-concave if and only if i has a densiy of he fom e f, whee f : R n, ] is a convex funcion. Log-concave vecos ae fequenly sudied in convex geomey, since by he Bunn Minowsi inequaliy unifom disibuions on convex ses as well as hei lowe dimensional maginals ae log-concave. Reseach paially suppoed by MNiSW Gan no. N N and he Foundaion fo Polish Science. * Addess fo coespondence: Insiue of Mahemaics, Univesiy of Wasaw, Banacha, Waszawa, Poland. addess: laala@mimuw.edu.pl /$ see fon mae 011 Elsevie Inc. All ighs eseved. doi: /j.jfa
2 68 R. Laała / Jounal of Funcional Analysis A andom veco X = X 1,...,X n is isoopic if EX i = 0 and CovX i,x j = δ i,j fo all i, j n. Equivalenly, an n-dimensional andom veco wih mean zeo is isoopic if E,X = fo any R n. Fo any nondegeneae log-concave veco X hee exiss an affine ansfomaion T such ha TXis isoopic. In ecen yeas hee wee deived numeous impoan popeies of log-concave vecos. One of such esuls is he Paouis concenaion of mass [10] ha saes ha fo any isoopic log-concave veco X in R n, P X C n exp n fo 1. 1 One of puposes of his pape is he exension of he Paouis esul o l noms, ha is deiving uppe bounds fo P X, whee x = n x i 1/.Fo [1, his is an easy consequence of 1 and Hölde s inequaliy, howeve he case > equies in ou opinion new ideas. We show ha P X Cn 1/ exp n 1/ fo 1, >, whee C is a consan depending only on see Theoem 8. Ou mehod is based on suiable ail esimaes fo ode saisics of X. Fo an n-dimensional andom veco X by X1 X X n we denoe he noninceasing eaangemen of X 1,..., X n in paicula X1 = max{ X 1,..., X n } and Xn = min{ X 1,..., X n }. Random vaiables X,1 n, ae called ode saisics of X. By 1 we immediaely ge fo isoopic log-concave vecos X, P X exp 1 C fo C n/. The main esul of he pape is Theoem 3 which saes ha he above inequaliy holds fo C logen/ as shows he example of exponenial disibuion his ange of is fo n/ opimal up o a univesal consan. Tail esimaes fo ode saisics can be also applied o povide opimal esimaes fo sup #I=m P I X, whee he supemum is aen ove all subses of {1,...,n} of cadinaliy m [1,n] and P I denoes he coodinaewise pojecion. The deails will be pesened in he fohcoming pape [1]. The oganizaion of he aicle is as follows. In Secion we discuss uppe bounds fo ails of ode saisics and hei connecions wih exponenial concenaion and Paouis esul. Secion 3 is devoed o he deivaion of ail esimaes of l noms fo log-concave vecos. Finally Secion 4 conains a poof of Theoem 4, which is a cucial ool used o deive ou main esul. Thoughou he aicle by C,C 1,... we denoe univesal consans. Values of a consan C may diffe a each occuence. Fo x R n we pu x = x = n x i 1/.. Tail esimaes fo ode saisics If he coodinaes of X ae independen symmeic exponenial andom vaiables wih vaiance one hen i is no had o see ha MedX C 1 logen/ fo any 1 n/. So we may obain a easonable bound fo PX, n/ in he case of isoopic log-concave vecos only fo 1 C logen/. Using he idea ha exponenial andom vecos ae exemal in he
3 R. Laała / Jounal of Funcional Analysis class of uncondiional log-concave vecos i.e. such vecos ha η 1 X 1,...,η n X n has he same disibuion as X fo any choice of signs η i { 1, 1} one may easily deive he following fac. Poposiion 1. If X is a log-concave and uncondiional n-dimensional isoopic andom veco hen P X exp 1C en fo C log. Poof. The esul of Bobov and Nazaov [] implies ha fo any i 1 <i < <i and >0, P X i1,..., X i = PX i1,...,x i exp 1C. Hence P X 1 i 1 < <i n P X i1,..., X i en exp 1C exp 1 C n exp 1C if C logen/. Howeve fo a geneal isoopic log-concave veco wihou uncondiionaliy assumpion we may bound PX i1,...,x i only by exp /C fo C. This suggess ha we should ahe expec bound exp /C han exp /C. If we y o apply he union bound as in he poof of Poposiion 1 i will wo only fo C logen/. Anohe appoach may be based on he exponenial concenaion. We say ha a veco X saisfies exponenial concenaion inequaliy wih a consan α if fo any Boel se A, P X A + αb n 1 1 exp if PX A and >0. Poposiion. If he coodinaes of an n-dimensional veco X have mean zeo and vaiance one and X saisfies exponenial concenaion inequaliy wih a consan α 1 hen P X exp 1 3α en fo 8α log. Poof. Since VaX i = 1wehaveP X i 1/soP X i + exp /αfo >0. Le μ be he disibuion of X. Then he se { A = x R n :# { i: x i } < } has measue μ a leas 1/ fo 4α logen/ indeed we have fo such
4 684 R. Laała / Jounal of Funcional Analysis μ A n n = P 1 { Xi } E 1 { Xi } n exp α n en 1. Le A = A4α logen/. Ifz = x + y A + sb n hen less han / of x i s ae bigge han 4α logen/ and less han / of y i s ae bigge han s, so en P X 4α log + s 1 μ A + sb n exp 1 s. α Fo log-concave vecos i is nown ha exponenial inequaliy is equivalen o seveal ohe funcional inequaliies such as Cheege s and specal gap see [9] fo a deailed discussion and ecen esuls. The song conjecue due o Kannan, Lovász and Simonovis [6] saes ha evey isoopic log-concave veco saisfies Cheege s and heefoe also exponenial inequaliy wih a unifom consan. The conjecue howeve is wide open a ecen esul of Klaag [7] shows ha in he uncondiional case KLS conjecue holds up o log n consan see also [5] fo examples of nonpoduc disibuions ha saisfy specal gap inequaliy wih unifom consans. Bes nown uppe bound fo Cheege s consan fo geneal isoopic log-concave measue is n α fo some α 1/4, 1/ see [9,4]. The main esul of his pape saes ha despie he appoach via he union bound o exponenial concenaion fails he naual esimae fo ode saisics is valid. Namely we have Theoem 3. Le X be an n-dimensional log-concave isoopic veco. Then P X exp 1 en fo C log. C Ou appoach is based on he suiable esimae of momens of he pocess N X, whee N X := n 1 {Xi }, 0. Theoem 4. Fo any isoopic log-concave veco X and p 1 we have E N X p Cp p n fo C log. We pospone a long and bi echnical poof ill he las secion of he pape. Le us only menion a his poin ha i is based on wo ideas. One is he Paouis lage deviaion inequaliy 1 and anohe is an obsevaion ha if we esic a log-concave disibuion o a convex se i is sill log-concave. Poof of Theoem 3. Obseve ha X implies ha N X / on X / and veco X is also isoopic and log-concave. So by Theoem 4 and Chebyshev s inequaliy we ge p
5 R. Laała / Jounal of Funcional Analysis P X p ENX p + EN X p Cp p povided ha C logn /p. So i is enough o ae p = ec 1 and noice ha he esicion on follows by he assumpion ha C logen/. As we aleady noiced one of he main ools in he poof of Theoem 4 is he Paouis concenaion of mass. One may howeve also do he opposie and deive lage deviaions fo he Euclidean nom of X fom ou esimae of momens of N X and he obsevaion ha he disibuion of UX is again log-concave and isoopic fo any oaion U. Moe pecisely he following saemen holds. Poposiion 5. Suppose ha X is a andom veco in R n A 1,A < and any U On, such ha fo some consans E N UX l A1 l l fo A,l n. Then P X n exp 1 n CA 1 fo max{ca 1,A }. Poof. Le us fix A. Hölde s inequaliy gives ha fo any U 1,...,U n On, E l l 1/l N Ui X EN Ui X l A1 l l fo l n. Now le U 1,...,U l be independen andom oaions in On disibued accoding o he Haa measue hen fo l n, A1 l l l E X E U N Ui X = E X EU1 N U1 X l l = EX npy X, Y = n l E X PY X Y1 l, whee Y is a andom veco unifomly disibued on S n 1. Since Y 1 is symmeic, EY 1 = 1/n and EY 4 1 C/n we ge by he Paley Zygmund inequaliy ha PY 1 1 4n 1/C 1 which gives P X n E X C1 P Y X Y1 l C1 A l 1 l. n To conclude he poof i is enough o ae l = ec1 A n. 1 1
6 686 R. Laała / Jounal of Funcional Analysis Concenaion of l noms The aim of his secion is o deive Paouis-ype esimaes fo concenaion of X = n X i 1/. We sa wih pesening wo simple examples. Example 1. Le he coodinaes of X be independen symmeic exponenial.v. s wih vaiance one. Then 1/ = ne X1 1/ 1 C n1/ fo [1,, 1 C log n and p 1/p E X1 p 1/p p C fo p, 1. I is also nown ha in he independen exponenial case wea and song momens ae compaable [8], hence fo, 1/ 1/ = E sup a i X i Theefoe we ge a 1 i 1/ 1/ + C sup E a i X i a 1 i 1/ 1/ + C sup E a i X i 1/ + C. a 1 p 1/p 1/ 1 C n1/ fo p and n C. Example. Fo 1 lex be an isoopic andom veco such ha Y = X X n / n has he exponenial disibuion wih vaiance one. Then by Hölde s inequaliy X n 1/ 1/ Y and i p 1/p n 1/ 1/ Y p 1 C n1/ 1/ p fo p, 1. The examples above show ha he bes we can hope is p 1/p C n 1/ + n 1/ 1/ p fo p, 1, p 1/p C n 1/ + p fo p, [, 3
7 R. Laała / Jounal of Funcional Analysis and p 1/p Clog n + p fo p. 4 O in ems of ails, P X exp 1C n1/ 1/ P X exp 1C fo Cn 1/, [1, ], 5 fo Cn 1/, [, 6 and P X exp 1C fo C log n. 7 Case [1, ] is a simple consequence of he Paouis heoem. Poposiion 6. Esimaes and 5 hold fo all isoopic log-concave vecos X. Poof. We have X n 1/ 1/ X by Hölde s inequaliy, hence and heefoe also 5 immediaely follows by he Paouis esul. Case = is also vey simple. Poposiion 7. Esimaes 4 and 7 hold fo all isoopic log-concave vecos X. Poof. We have P X n P X i n exp /C. Wha is lef is he case << we would lie o obain 6 and 3. We almos ge i excep ha consans explode when appoaches. Theoem 8. Fo any δ>0 hee exis consans C 1 δ, C δ C1 + δ 1/ such ha fo any + δ, P X exp 1 C 1 δ fo C 1 δn 1/ and p 1/p C δ n 1/ + p fo p. The poof of Theoem 8 is based on he following slighly moe pecise esimae.
8 688 R. Laała / Jounal of Funcional Analysis Poposiion 9. Fo > we have P X exp 1 1/ C 1/ fo C n 1/ + log n o in ems of momens p 1/p 1/ C n 1/ + log n + p fo p. Poof. Le s = log n.wehave n X = X i =0 s X. Theoem 3 yields P X en C3 log + exp 1 C fo >0. 8 Obseve ha s log en Cn j j C n. =0 Thus fo 1,..., 0 we ge whee Hence by 8 j=1 s 1/ s P X C n 1/ + P Y =0 =0 Y := X C 3 log en. s 1/ P X C n 1/ + =0 s PY =0 s =0 Fix >0and choose such ha = / / 1/. Then s, =0 exp 1 C 1/.
9 R. Laała / Jounal of Funcional Analysis s = =0 s =0 1 1 C, so we ge 1/ P X C n 1/ + log n + 1 exp 1C. Poof of Theoem 8. Obseve ha 1/ C1 + δ 1/ fo + δ and log n n 1/ and apply Poposiion Poof of Theoem 4 Ou cucial ool will be he following esul. Poposiion 10. Le X be an isoopic log-concave n-dimensional andom veco, A ={X K}, whee K is a convex se in R n such ha 0 < PA 1/e. Then n P A {X i } C 1 PA log PA + ne /C 1 fo C 1. 9 Moeove fo 1 u C, # { i n: P A {X i } e u PA } C u log PA. 10 Poof. Le Y be a andom veco disibued as he veco X condiioned on he se A ha is PY B = PA {X B} PA = PX B K. PX K Noice ha in paicula fo any se B, PX B PAPY B. The veco Y is log-concave, bu no longe isoopic. Since his is only a mae of pemuaion of coodinaes we may assume ha EY 1 EY EY n. Fo α>0le m = mα = # { i: EY i α }. We have EY1 EY m α. Hence by he Paley Zygmund inequaliy, P m Yi 1 αm m P Y i 1 E m Y i 1 4 E m Y i E m Y i 1 C.
10 690 R. Laała / Jounal of Funcional Analysis This implies ha m P Xi 1 αm 1 C PA. Howeve by he esul of Paouis, m P Xi 1 αm exp 1C3 mα fo α C 3. So fo α C 3,exp 1 C 3 mα PA/C and we ge ha We have mα = # { i: EY i α } C 4 α log PA fo α C PA {X i } PA = PY i exp 1 CEYi 1/ and 10 follows by 11. Tae C 3 and le 0 be a nonnegaive inege such ha 0 C Define I 0 = { i: EYi }, I 0 +1 = { i: EYi < 4 0 } and By 11 we ge I j = { i:4 j EY i < 4 1 j }, j = 1,,..., 0. #I j C 4 4 j log PA fo j = 0, 1,..., 0 and obviously #I 0 +1 n. Moeove fo i I j, j 0, Thus Y i PY i P EYi j 1 1/ n 0 +1 PY i = PY i #I 0 + e j=0 i I j 0 C 4 log PA 1 + e j=1 C 1 log PA + ne /C 1. exp 1 1C j j=1 #I j exp 1C j j exp 1C j + ene /C
11 R. Laała / Jounal of Funcional Analysis To finish he poof of 9 i is enough o obseve ha n P A {X i } n = PA PY i. The following wo examples show ha esimae 9 is close o be opimal. Example 1. Tae X 1,X,...,X n o be independen symmeic exponenial andom vaiables wih vaiance 1 and A ={X 1 }. Then PA = 1 e and n P A {X i } n = PA PX i = n 1PA exp /, i= i= heefoe he faco ne /C in 9 is necessay. Example. Tae A ={X 1,...,X } hen n P A {X i } PA. So impovemen of he faco PA log PA in 9 would imply in paicula a bee esimae of PX 1,...,X han exp 1 C and we do no now if such bound is possible o obain. Poof of Theoem 4. We have N X n, so he saemen is obvious if n Cp, in he sequel we will assume ha n 10p. Le C 1 and C be as in Poposiion 10 inceasing C i if necessay we may assume ha PX 1 e /C i fo C i and i = 1,. Le us fix p 1 and C log n, hen p max{c 1, 4C } and ne /C 1 p if C is lage enough. Le l be a posiive inege such ha p l p and l = fo some inege. Since EN X p 1/p EN X l 1/l i is enough o show ha E N X l Cl l. Recall ha by ou assumpion on p,wehave n 5l. To shoen he noaion le B i1,...,i s ={X i1,...,x is } and B = Ω. Define
12 69 R. Laała / Jounal of Funcional Analysis we need o show ha n l ml := EN X l = E 1 {Xi } = n i 1,...,i l =1 PB i1,...,i l, Cl l ml. 1 We divide he sum in ml ino seveal pas. Le j 1 be such inege ha We se and n j1 < log j1 1. I 0 = { i 1,...,i l {1,...,n} l : PB i1,...,i l >e l}, I j = { i 1,...,i l {1,...,n} l : PB i1,...,i l e j l,e j 1 l ]}, 0 <j<j 1 I j1 = { i 1,...,i l {1,...,n} l : PB i1,...,i l e j 1 1 l }. Since {1,...,n} l = j 1 j=0 I j we ge ml = j 1 j=0 m j l, whee m j l := l i 1,...,i l I j PB i1,...,i l fo 0 j j 1. I is easy o bound m j1 l namely since #I j1 n l we have i 1,...,i l I j1 PB i1,...,i l n l e j1 1l l l. To esimae m 0 l we define fis fo I {1,...,n} l and 1 s l, P s I = { i 1,...,i s : i 1,...,i l I fo some i s+1,...,i l }. By Poposiion 10 we ge fo s = 1,...,l 1 PB i1,...,i s+1 i 1,...,i s+1 P s+1 I 0 n i 1,...,i s P s I 0 i s+1 =1 P B i1,...,i s {X is+1 } C 1 PB i1,...,i s log PB i1,...,i s + ne /C 1. i 1,...,i s P s I 0
13 R. Laała / Jounal of Funcional Analysis Obseve ha we have PB i1,...,i s >e l fo i 1,...,i s P s I 0 and ecall ha ne /C 1 p 4l, hence PB i1,...,i s+1 5C 1 l PB i1,...,i s. i 1,...,i s+1 P s+1 I 0 i 1,...,i s P s I 0 So, by easy inducion we obain m 0 l = PB i1,...,i l 5C1 l l 1 PB i1 i 1,...,i l I 0 i 1 P 1 I 0 5C 1 l l 1 ne /C 1 Cl l. Now comes he mos involved pa of he poof esimaing m j l fo 0 <j<j 1. I is based on suiable bounds fo #I j. We will need he following simple combinaoial lemma. Lemma 11. Le l 0 l 1 l s be a fixed sequence of posiive ineges and Then F = { f :{1,,...,l 0 } {0, 1,,...,s}: 1 i s # { : f i } l i }. #F s eli 1 Poof of Lemma 11. Noice ha any funcion f :{1,,...,l 0 } {0, 1,,...,s} is deemined by he ses A i ={: f i} fo i = 0, 1,...,s.Taef F, obviously A 0 ={1,...,l 0 }.If he se A i 1 of cadinaliy a i 1 l i 1 is aleady chosen hen he se A i A i 1 of cadinaliy a mos l i may be chosen in ai ai ai 1 l i li 1 0 l i + li. li li 1 l i li eli 1 l i ways. We come bac o he poof of Theoem 4. Fix 0 <j<j 1,le 1 be a posiive inege such ha 1 < C 1+1. Fo i 1,...,i l I j we define a funcion f i1,...,i l :{1,...,l} {j,j + 1,..., 1 } by he fomula j if PB i1,...,i s exp j+1 PB i1,...,i s 1, f i1,...,i l s = if exp +1 PB i 1,...,is PB i1,...,i s 1 < exp, j < < 1, 1 if PB i1,...,i s <exp 1PB i1,...,i s 1.
14 694 R. Laała / Jounal of Funcional Analysis Noice ha fo all i 1, PX i1 e /C < exp 1PB, so f i1,...,i l 1 = 1 fo all i 1,...,i l. Pu Fo f = f i1,...,i l F j and >j,wehave so F j := { f i1,...,i l : i 1,...,i l I j }. exp j l < PB i1,...,i l <exp # { s: fs }, # { s: fs } j l =: l. 13 Obseve ha he above inequaliy holds also fo = j. Wehavel 1 /l = and 1 =j+1 l l so by Lemma 11 we ge Pu #F j 1 =j+1 el 1 l l e l. Now fix f F j we will esimae he cadinaliy of he se We have I j f := { i 1,...,i l I j : f i1,...,i l = f }. n := # { s {1,...,l}: fs= }, = j,j + 1,..., 1. n j + n j+1 + +n 1 = l, moeove if i 1,...,i s 1 ae fixed and fs= < 1 hen s and by he second pa of Poposiion 10 wih u = +1 /C i s may ae a mos values. Thus 4C log PB i1,...,i s 1 4C +j l 4C l 1 1 #I j f n n 1 =j m n = n n 4C l l n exp Obseve ha by peviously deived esimae 13 we ge n l = j l, exp + j =: m =j + jn.
15 R. Laała / Jounal of Funcional Analysis hence jn j+ l C + j l. =j =j We also have n 1 j 1 l C j 1 l logn /4l j 3 l, whee he las inequaliy holds since C logn /4l and C may be aen abiaily lage. So we ge ha fo any f F j, This shows ha Hence Theefoe Cl l n n1 #I j f exp j 4l l Cl l 3 exp 8 j l. Cl l 3 #I j #F j exp 8 j l Cl l exp + 38 j l. m j l = PB i1,...,i l #I j exp j 1 l Cl l exp j 3 l. i 1,...,i l I j j 1 1 ml = m 0 l + m j1 l + Cl l j=1 m j l l l C l C l exp j 3 l j=1 and 1 holds. Acnowledgmens This wo was done while he auho was aing pa in he Themaic Pogam on Asympoic Geomeic Analysis a he Fields Insiue in Toono. The auho would lie o han Radosław Adamcza and Nicole Tomcza-Jaegemann fo hei helpful commens.
16 696 R. Laała / Jounal of Funcional Analysis Refeences [1] R. Adamcza, R. Laała, A. Liva, A. Pajo, N. Tomcza-Jaegemann, Geomey of log-concave ensembles of andom maices and appoximae econsucion, pepin. [] S.G. Bobov, F.L. Nazaov, On convex bodies and log-concave pobabiliy measues wih uncondiional basis, in: Geomeic Aspecs of Funcional Analysis, in: Lecue Noes in Mah., vol. 1807, Spinge, Belin, 003, pp [3] C. Boell, Convex measues on locally convex spaces, A. Ma [4] O. Guedon, E. Milman, Inepolaing hin-shell and shap lage-deviaion esimaes fo isoopic log-concave measues, pepin, hp://axiv.og/abs/ [5] N. Hue, Specal gap fo some invaian log-concave pobabiliy measues, pepin, hp://axiv.og/abs/ [6] R. Kannan, L. Lovász, M. Simonovis, Isopeimeic poblems fo convex bodies and a localizaion lemma, Discee Compu. Geom [7] B. Klaag, A Bey Esseen ype inequaliy fo convex bodies wih an uncondiional basis, Pobab. Theoy Relaed Fields [8] R. Laała, Tail and momen esimaes fo sums of independen andom vecos wih logaihmically concave ails, Sudia Mah [9] E. Milman, On he ole of convexiy in isopeimey, specal gap and concenaion, Inven. Mah [10] G. Paouis, Concenaion of mass on convex bodies, Geom. Func. Anal
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