A note on deviation inequalities on {0, 1} n. by Julio Bernués*

Transcription

1 A oe o deviaio iequaliies o {0, 1}. by Julio Berués* Deparameo de Maemáicas. Faculad de Ciecias Uiversidad de Zaragoza Zaragoza (Spai) I. Iroducio. Le f: (Ω, Σ, ) IR be a radom variable. Roughly speakig, a deviaio iequaliy is a upper esimae of he probabiliy of he eve { ω Ω f(w) A f > }, where A f is a cerai fixed real umber oly depedig o f. (Tha is we measure how he fucio f deviaes from a fixed value A f ). Paricularly impora are he deviaio iequaliies for fucios defied o {0, 1}. Cosider he aural iclusio {0, 1} l p, 1 p ad he meric iheried i {0, 1} from he usual orm i l p ; (for p = 1 he disace is called Hammig disace). Such meric space is called l p -cube. The cardial of a subse of I {0, 1} is deoed by I. Give f: {0, 1} IR, deoe by σ p is Lipschiz cosa wih respec o he meioed disace. The firs resul o his opic is he followig oe. (See [A-M]): Theorem. Le {0, 1} be provided wih he couig probabiliy ad he Hammig disace. For every Lipschiz fucio f: {0, 1} IR wih Lipschiz cosa σ 1 ad every > 0 { f M f > } exp σ 1 (1) where M f is he media of he fucio f. Remark. I [M-S] he auhors re-prove he iequaliy above, obaiig somewha worse umerical cosas, by usig marigale eciques. For 1 < p <, he same argume as i [M-S] proves he iequaliy, { f M f > } 4 exp 4σp () for f: {0, 1} IR. I [T] ad [J-S], he auhors obai sharper deviaio iequaliies by resricig he class of fucios o {0, 1}. I he secod referece he auhors cosider {0, 1} equipped wih ay produc probabiliy, i.e. = (sδ 0 + (1 s)δ 1 ) where δ 0, δ 1 are Dirac measures ad 0 s 1. Their resul is he followig: Theorem. Le {0, 1} be provided wih he probabiliy described above ad he disace ρ p. Le f: {0, 1} IR. For every p <, ad > 0 { f M f > } 4 exp p 4 σ p p (3) where σ p deoes he Lipschiz cosa of a (ay) covex exesio of f o [0, 1]. * Parially suppored by Gra DGICYT PS , PS ad D.G.A. 1

2 Remark. (i) The Theorem was firs proved by Talagrad, [T], for p = ad he couig probabiliy, ad i was exeded o a eve more geeral siuaio ha he oe cosidered here i [J-S]. (ii) Usig he relaio ( a i p ) 1/p ( 1/p 1/ a i ) 1/, 1 p, (ai ) IR ad he argumes i [T] i is o difficul o prove ha for every 1 p <, ad > 0 { f M f > } 4 exp 4 1 σ p p (4) (iii) Noe ha he eve { f M f > } depeds oly o he values ake by f o {0, 1}, bu he upper boud depeds o he covex exesio. This is why, as oed i [J-S], we ca subsiue σ p by he ifimu of he Lipschiz cosas of every covex exesio of f o [0, 1]. The aim of he prese work is o improve hese iequaliies for a leas cerai class of fucios. II. Resuls. Proposiio 1. Le ( [0, 1], Σ, ) be a probabiliy measure over he ierval [0, 1]. Le ψ: [0, 1] [0, 1] a o-decreasig ad cocave fucio, ad le x 0 [0, 1]. The for every 0 1 { ψ ψ(x 0 ) > } { x x 0 > x 0 } Proof: Sice ψ is cocave he lef derivaive ψ () exiss for every ad ψ is a posiive odecreasig fucio. For every 0 1 we have, ψ (y) dy Now, for ay 0 x 1 ad by he Mea Value Theorem, 0 ψ (y) dy ψ () ψ(x) ψ(x 0 ) ψ (ξ) x x 0 x x 0 1 ξ where ξ is bewee x ad x 0. Noe ha for every 0 1, x x 0 > ξ = x x 0 > x 0. Ideed, if x x 0 x ( 0 he ξ x 0 ad so x x 0 ξ. Therefore, he calculaios made above yield o x 0 ) ad so, ψ(x) ψ(x 0 ) > = x x 0 > ξ = x x 0 > x 0 { ψ ψ(x 0 ) > } { x x 0 > x 0 } /// Remark. (i) If he probabiliy is symmeric respec o, say, x 0 = 1 he ψ( 1 ) is a media of he fucio ψ ad { ψ M ψ > } { x 1 > 4 } Corollary 1. Uder codiios of Proposiio 1, if is he biomial probabiliy, ha is ( k ) = ( k) k = 0,...,, he for every 0 1 { ψ M ψ > } exp 8

3 Corollary. Le {0, 1} be equipped wih he couig probabiliy ad le f: {0, 1} IR be a fucio such ha (i) For every η, η {0, 1}, η 1 η 1 = f(η) f(η ) (ii) For every η, η, η {0, 1} such ha η 1 = η = η 1 + we have The, > 0, f(η ) f(η ) f(η ) f(η) { f M f > } exp 8 f where f is he variaio of f i.e. f = f(1,..., 1) f(0,..., 0) Proof: Codiio (i) implies ha η 1 = η 1 = f(η) = f(η ) ha is, f ca be cosidered as a fucio of η 1 defied o {0,..., }. Exed by liear ierpolaio f o he ierval [0, ] ad deoe his ew fucio wih he same leer f. Noe ha codiio (ii) says ha fucio f is cocave o [0, ]. Normalize ad defie ψ: [0, 1] [0, 1] as ψ(x) = f(x) f(0) f The fucio ψ o-icreasig cocave ad by (i) we have [0, 1] equipped wih he biomial probabiliy. Also oe ha, sice f is o-decreasig f( ) is a media of f ad so, ψ( 1 ) is a media of ψ. Now Corollary 1 implies ha { f M f > } = { η ψ( ) ψ(1 ) } > exp f 8 f (5) wheever f. If > f he eve { f M f > } is empy ad he iequaliy rivial. /// Remark. Le us compare (5) wih he iequaliies (1),(),(3) ad (4). I is imediae o check ha for ay covex exesio of f o [0, 1] we have, (1) If 1 p f σ σ p 1 p 1 σp 1 p 1 ad so, f σp p 1 σ p p 1 Excep for he umerical cosas, (5) is sharper ha (1), (), (4). ( ) A example where he improveme is subsaial is: f(η) = log 1 + η i For such a fucio f = log( + 1) ad (5) gives he esimae { f M f > } exp 8 log ( + 1) O he oher had is imediae o check ha for every 1 p, σ p = log. Hece, for every 1 p ad > 0, log ( + 1) (log ) p 1 σp p 1 σ p p 1 3

4 () If < p < f f 1 1/p p 1 σp 1 p 1 σp 1 p 1 σp This makes (5) o be sharper ha (). Also, he depedece respec o i (5) is beer ha he oe i (3). I our las resul we examie he siuaio whe we drop he cocaviy codiio (ii) of Corollary. Before saig i we eed o iroduce some oaio. Noaio. For every k deoe i k he iclusio i k : {0, 1} k {0, 1} give by i k ( (η1,..., η k ) ) = (η 1,..., η k, 0..., 0) Lemma 1. Le ψ: [0, 1] [0, 1] be a o-decreasig fucio. a = a(ε, ψ) [ 1 4, ] 1 such ha For every 0 < ε < 1 here exiss x, y [ a ε 8, a + ε 8] = ψ(x) ψ(y) ε Proof: Wrie δ = ε. Sice ψ is a o-decreasig posiive fucio ad ψ(1) 1 we have, 8 1/ 1/4 [ψ( + δ) ψ( δ)] d = = 1/+δ ψ() d 1/+δ 1/4+δ 1/4+δ ψ() d ψ() d 1/ δ ψ() d 1/4 δ 1/+δ 1/ δ 1/4 δ 1/ δ ψ() d δ Therefore here is a umber a ( 1 4, 1 ) such ha ψ(a + δ) ψ(a δ) 8 δ = ε; (ideed, if i was such, we would have 1/ 1/4 [ψ( + δ) ψ( δ)] d > 1/ 1/4 8δ d = δ which is a coradicio). Now, sice ψ is o-decreasig we coclude ha x, y [a δ, a + δ] = ψ(x) ψ(y) 8 δ = ε Proposiio. Le {0, 1} be equipped wih he couig measure, ad le F : {0, 1} IR be a fucio such ha for every η, η {0, 1}, η 1 η 1 = F (η) F (η ). The for every > 0 here exiss k = k() [ ] such ha he fucio f = F ik verifies { f M f > } exp 18 F Proof: Exed F o he ierval [0, ] by liear ierpolaio ad deoe his ew fucio wih he same leer F. Normalize, ad defie ψ: [0, 1] [0, 1] as ψ(x) = F (x) F (0) F /// Wrie ε = F. Noe ha if F ( f), he iequaliy is rivial. Assume 0 < ε = F a [ 1 4, 1 ] be he umber associaed o ψ ad ε i Lemma 1 ad wrie k = [a] [ ]. < 1. Le 4

5 k By hypohesis F (resp f) is of he form F ( η i ) (resp f( η i )); herefore we ca ake as a ( [ k ] ) media of f, M f = f ad ( [ k ] ) ( ψ k ) { f M f > } = { f f > } = { ψ > F } I is clear ha, k a k a = 1, ad so if 1 ε 16 (or equivalely if 8 ), we have ε ha k [a ε 16, a + ε ]. We will rea his resricio laer. 16 If 8 ε Lemma 1 assers ha ad so, ψ ( k η i ) ( k ) k ψ > ε = η i a > ε 8 k η i a > ε k 8 = η i k + a k > ε k 8 = η i k > ε 16 Therefore if 8 ε we have ( k ) { ψ ψ > k F } { η i k > 16 F } exp 18k F exp 18 F exp 18 F I he secod iequaliy we have used Amir-Milma s iequaliy (1) for he fucio f(η) = ad he rivial relaio k 1. We coclude by observig ha he resricio 8 F sice also < F, we have is o such. Ideed, if < 8 F η i he, exp 18 F exp 18 F > exp 1 16 > 1 ad he iequaliy is rivial. /// As a applicaio we obai some resuls abou he problem of embeddig l p -cubes io l q -cubes, 1 p, q. We firs recall a defiiio. Defiiio. Two fiie meric spaces (M, d) ad (M, d ) are called c-isomorphic if here is a biyecive map f: M M such ha f Lip f 1 Lip c where f Lip is he Lipschiz cosa of f. Proposiio 3. Le 0 < ε < 1, 1 p, q ad IN. Wrie c = ε log e. There exiss a se 3 I {0, 1} wih I [c] such ha he meric spaces (I, p ) ad (I, q ) are (1 + ε)-isomorphic. Proof: Le X = η ie i be a radom variable defied o some probabiliy space where η i are idepede ideically disribued radom variables (i.i.d.) akig values 0 ad 1 wih probabiliy 1. Le X 1,... X N i.i.d. copies of X wih N = [c]. Suppose firs ha 0 < 1 p 1 so ha he fucio q f() = 1/p 1/q is icreasig ad cocave. We esimae he probabiliy of he followig eve 5

6 { 1/p X i X j p { { ( } 1/q X i X j q 1/q > X i X j q ε1/q = 1/q ) 1/p ( 1/q η 1/q ( i) ε1/q 1/p 1/q > η 1/q } i) = 1/q 1/p 1/p( η i I is easy o see ha { 1/p X i X j p 1/p η i ) 1/p 1/q ( ) 1/p 1/q > ε ( } ) 1/p 1/q (by Corollary ) exp ε 3 1/q X i X j q 1/q X i X j q ε1/q 1/q }, i j 1 ( ) N exp ε 3 If we make his las expresio be posiive we assure he exisace of a eleme ω i he probabiliy space such ha 1/q X i (ω) X j (ω) q (1 ε) 1/p X i (ω) X j (ω) p (1 + ε) 1/p X i (ω) X j (ω) p 1/q 1/p 1/p ad he resul ow follows wih I = {X 1 (ω),... X N (ω)}. If 0 < 1 q 1 p ad he resuls follows. jus chage he role of p ad q /// Remark. I [B-M-W] he auhors obaied (1+ε)-embeddigs of he whole l k p-cube io he l q -cube, 1 p < q, wih c(ε, p, q)k 3. Our resuls says ha over a large subse I of {0, 1} he orms p ad q are (1 + ε)-equivale. Uforuaely, we do o kow i geeral wha he se I looks like. I he case q =, he se I ca be ideified wih he l [c] -cube, sice η η = 1 for every η η {0, 1} [c], ad we obai a embeddig of he l k -cube io he l p -cube wih k proporioal o (his fac was already poied ou i [B-B]). Remark. The same resul ca be give if he meric i {0, 1} is give by orms i IR ha make (e i ) be a 1-symmeric basis. I his case, codiios o he growh of e i are eeded. Akowledgemes. This work is par of my docoral hesis wrie uder he supervisio of Dr. Jesús Basero whose advice ad ecouragme have bee uvaluable. I am also graeful o Nigel J. Kalo for useful ideas i Proposiio. Refereces. [A-M] [B-B] [B-M-W] [J-S] [M-S] [T] Amir, D. ad Milma, V.D.: Ucodiioal ad symmeric ses i -dimesioal ormed spaces. Israel J. Mah. 37 (1980), 3-0. Basero, J. ad Berués, J.: Applicaios of deviaio iequaliies o fiie meric ses. Mah. Nachr.153 (1991), Bourgai, J., Milma, V.D., ad Wolfso, H.: O he ype of meric spaces. Tras. of he A.M.S. 94 (1986), Johso, W.B. ad Schechma, G.: Remarks o Talagrad s deviaio iequaliy for Rademacher fucios. Lec. Noes i Mah Pg Spriger-Verlag Milma, V.D. ad Shechma, G.: Asympoic heory of fiie dimesioal ormed spaces. Lec. Noes i Mah Spriger-Verlag Talagrad, M.: A isoperimeric heorem o he cube ad he Kichie - Kahae iequaliies. Proc. AMS, 104 (1988),