The Inequalities of Khintchine and Expanding Sphere of Their Action

Transcription

1 Usekhi Mat Nauk Vol 5 No (3-6) (Russia) Russia Math Surveys Vol 5 No (849-94) (glish) Prerit Ser No Math Ist Aarhus Research Reort No 34 (995) Det Theoret Statist Aarhus To the ceteary of the birth of Aleksadr Yakovlevitch Khitchie ( ) The Iequalities of Khitchie ad xadig Shere of Their Actio G PSKIR 3 ad A N SHIRYAV 33 Cotets: Itroductio The strog law of large umbers of Borel ad its refiemets The iequalities of Khitchie 3 Martigale extesios of Khitchie s iequalities I (relimiary cosideratios) 4 Martigale extesios of Khitchie s iequalities II ( the iequalities of Burkholder: > ) 5 The maximal iequalities of Khitchie 6 Martigale extesios of maximal Khitchie s iequalities ( the iequalities of Davis: = ; the iequalities of Burkholder-Davis: ) 7 Comariso iequalities for martigale trasforms 8 O the best costats i the Khitchie s iequalities 9 The iequalities of Khitchie i the (exoetial) Orlicz saces O the best costats i the iequalities of Burkholder ( > ) ad Davis ( = ) O some refiemets of the iequalities of Khitchie ( the iequalities of Rosethal their modificatios ad extesios to the martigale case ) 3 Research suorted by Daish Natural Sciece Research Coucil ad by Daish Research Academy 33 Research suorted by Deartmet of Theoretical Statistics (Uiversity of Aarhus) ad by Steklov Mathematical Istitute (Russia Academy of Scieces) AMS 98 subject classificatios Primary 65 6G4 6G4 6G5 Secodary 4C5 4C 6F 6G46 Key words ad hrases: The iequalities of Khitchie Burkholder-Davis-Gudy (BDG-iequalities) Rosethal; the strog law of large umbers ad its refiemets the law of iterated logarithm; the subgaussia iequalities; the systems of Haar ad Walsh; the iequalities of Paley ad Marcikiewicz-Zygmud; martigale local martigale suermartigale submartigale martigale differeces martigale trasforms stoig times the quadratic variatio the redictable quadratic variatio (quadratic characteristic) Doob s decomositio; maximal iequalities; the method of symmetrizatio; the method of radomizatio; comariso iequalities for martigale trasforms; the decomositio of Davis; the domiatio roerty of Leglart; the best costats (i iequalities) Schur-covexity (-cocavity) the Schur coditio; the iequalities of Khitchie i exoetial Orlicz saces; the iequalities of Khitchie for (comlex) Steihaus variables; good -iequality; martigale extesio of the subgaussia iequality (Secod editio) gora@imfaudk

2 Itroductio I year 93 A: Ya: Khitchie ublished the aer Über dyadische Brüche [74] i which he was tryig to fid the right rate of covergece i the strog law of large umbers of : Borel ad roved the followig statemet: Let " = ("; "; ) be a sequece of ideedet ad idetically distributed radom variables defied o some robability sace (; F; P ) ad takig values + ad with robability = The for ay iteger = m m there exists a uiversal costat B such that for each sequece of umbers a = (a; a; ) ad every the followig iequality is valid: (I) a k " k B ja k j = This statemet formulated i terms of robability theory ( i (I) the symbol deotes the mathematical exectatio with resect to the measure P ) admits the followig formulatio which is accustomed i the metrical theory of fuctios Let r = (r; r; ) be a system of Rademacher fuctios r k = r k (x) k defied o the set [; [ with Lebesgue measure ad determied by the equalities: + ; x < r(x) = ; x ; r k (x) = r k x k The for ay iteger = m m there exists a uiversal costat B = B = such that for each sequece of umbers a = (a; a; ) ad every the followig iequality is valid: (I ) a k r k B ja k j = where k k deotes the L [; [ -orm: kk = jj = It is well-kow (ad will be show below) that the iequalities (I) (I ) remai valid for ay < < Together with the estimate from below ( which also holds for all > ): (II) or equivaletly: (II ) A ja k j = A ja k j = a k " k a k r k with a uiversal costat A > ad A = A = we arrive to the iequalities:

3 (III) or equivaletly: (III ) A ja k j = a k " k B ja k j = A ja k j = a k r k B ja k j = which are widely called the iequalities of Khitchie It is the aim of the reset aer to give a exositio of the basic results which relate to these iequalities from differet oits of view their roofs alicatios clarificatios values of the best costats A B exadig shere of actio of the iequalities (III) (III ) by relacig P a k" k or the system of Rademacher fuctios r = (r k ) k with a the sequece radom sequece f = (f ) or systems of fuctios with a more geeral structure As we shall see below the iequalities (III) exted i a atural way to radom sequeces f = (f ) which aear to be martigales or sequeces related to them (for examle local martigales) The ext atural ste should be the extesio to martigale or local martigale f = (f t ) t for the case of cotiuous time The results exosed below will cocer oly the case of discrete time From a formal oit of view may of them could be deduced as cosequeces of more geeral results for cotiuous time but it should be remarked that the corresodig (cotiuous time) theory eeds to attach rather comlicated otios ad results from stochastic calculus At the same time i the case of discrete time arameter oe succeeds to obtai rather simle ad trasaret formulatios ad roofs by miimal tools This exlais that from the oit of view of exadig shere of actio of the iequalities of Khitchie the mai attetio is devoted to the case of discrete time whereas the corresodig case of cotiuous time is much ivestigated i the theory of stochastic rocesses The reset exositio will be give i a chroological order which i our oiio gives a better uderstadig of those motives ad aims which led A: Ya: Khitchie ad subsequet researchers i coectio with Khitchie s iequalities to their refiemets extesios etc The strog law of large umbers of Borel ad its refiemets To uderstad why A: Ya: Khitchie eeded the iequality (I) let us tur back to the history related to the strog law of large umbers of Borel which i the umber-theoretical cotext might be formulated i the followig way Let = [; [ let F be the family of Borel subsets ad let P be Lebesgue measure Cosider the biary decomositio x = : xx of a umber x I other words let () x = x k k For uiqueess of such a reresetatio we shall cosider oly those decomositios which cotai ifiite umber of zeros For examle i betwee the two reresetatios = ad =

4 uder cosideratio we allow oly the first oe Puttig k (x) = x k k we see that for a i = or : P f x : (x) = a ; ; (x) = a g 8 = P x : a + a + + a x < a + a + + a a = P x : x + a + + a ; a + a + + a + 9 = From this it follows that = (; ; ) with k = k (x) k is a Beroulli sequece of ideedet ad idetically distributed radom variables takig values ad with robability = Puttig r k (x) = k (x) k we get a system r = (r; r; ) of Rademacher fuctios r k = r k (x) k Clearly i this way the radom quatities r ; r ; form a sequece of ideedet radom variables with P f r k = g = P f r k = g = = k The system of Rademacher fuctios is obviously orthoormal r k r l = kl but ot comlete because r k = By erestroika of this system oe gets two well-kow systems of Haar ad Walsh havig ot oly the roerty of orthogoality but comleteess as well; see below for their defiitios ad formulatios of a martigale character The above metioed result of Borel (99 [3]) states that almost every umber x from [; [ is ormal i the sese that with robability oe the fractio of zeros ad uits i the biary decomositio () coverges to = that is for! : () or i terms of Rademacher fuctios: k (x)! ( P -ae ) r k (x)! ( P -ae ) Deotig i accordace to the traditios of robability theory S = r + r + + r we fid S = DS = ad for! : (3) S! ( P -as ) The roof of the statemet () or equivaletly (3) which is called the strog law of large umbers of Borel is rather simle: By the Chebyshev-Markov iequality ad simle estimate S 4 3 it follows that for ay > : ad thus: P su m P S! > S m >! P m m S S m >! 3 m 4 m! m! 4

5 which roves the validity of the roerty (3) The ext ste i the clarificatio of the rate of covergece i (3) has bee made by Hausdorff (93 [6]) who oted that for each " > : (4) S +"! ( P -as ) ie: that +" S = o with robability oe The roof of this statemet is also rather simle ad is based uo the fact that S r = O(r ) from which for a iteger r > =" we fid that: P! > m +" S m r m +" su S m m r m m P C r m! S m > m +" because r" > ( where C is a costat deedig o r ) m r m r+r"!! Remark As oted i [6] (Theorem 43) already i this way it was ossible to obtai the followig result; with robability oe for! : (5) lim su js j log we get: Ideed because: (6) ex e t + e t e ts = = S It follows that for ay " > : ad so by (6): P S m m log m +"! e =!! = P ex Sm m m +"! P su m = P ex Sm! ex (+") log m m m ex S m m +" S m! ex +" m log m m Sm m m +"!! 5

6 By symmetry: P su m! js m j +" m log m!! which is by the freedom of choice of " > equivalet to the roerty (5) Next refiemets of the rate of covergece i (4) have bee obtaied by Hardy ad Littlewood (94 [59]) who showed that with robability oe: (7) js j = O log ad by Steihaus (9 [4]) who established that: (8) lim su js j 8 log 3 I the above metioed aer from year 93 [74] A: Ya: Khitchie made a ext ste towards the establishmet of the rate of covergece i () by showig that ( P -as ): (9) lim su js j log log I other words he was the first who foud a statemet i which the iterated logarithm aears that figures as we ow kow i the fial formulatio of Khitchie s law of the iterated logarithm (94 [78]); with robability oe: () lim su js j log log = From the above stated historical excursio we see that the refiemets of the rate of covergece i (3) were coected with fidig a good estimate of the robability: P js j t P = P ( js j t() ) ad where t = t() which was rovided covergece of the series fially (by the Borel-Catelli lemma) gave ossibility to obtai statemets (4) (5) (7) ad (8) For this i the cases cosidered above: t() = (Borel) t() = +" (Hausdorff) t() = = log ((5); Révész) t() = = log = (Hardy-Littlewood Steihaus) It was the aim of obtaiig a good estimate of robability more geeral form robability validity of iequalities (I) ad with the hel of this to get the exoetial (subgaussia) estimate: P P js j t or i a little bit P j a k" k j t that led A: Ya: Khitchie to rove the 6

7 () P with some absolute costat a k " k > t! C ( = e = 3:84 ) C e t = P ja kj This estimate is give by A: Ya: Khitchie i year 93 [74] whe he roved the validity of a weak variat of the law of the iterated logarithm (9) I the ext year (94) i the aer [75] A: Ya: Khitchie obtaied the fial formulatio () i which for the estimate of robability P js j t he alied a direct aalysis of the biomial distributio P (S = k) Our exositio i the sequel will be essetially cocetrated aroud the iequalities of Khitchie (I) i themselves ad their extesios Cocerig the law of iterated logarithm itself o the way to which have bee derived the iequalities of Khitchie we shall here otice oly that the ext ricile fact about the ossibility of extedig the law of iterated logarithm to the radom sequeces of a more geeral structure has bee obtaied i 99 by A: N: Kolmogorov i [78] His result may be formulated i the followig way Let ; ; be a sequece of ideedet radom variables with zero mea = B = P k!! ad suose that there exist costats M such that: () The (3) lim su! M = o =! B log log B S B log log B = (P -as) ( I 937 Marcikiewicz ad Zygmud established that i () o-small caot be relaced by O-caital ) If additioally to the ideedece of the quatities ; ; we suose also idetical distributio the as show by Hartma ad Witer i 94 for the validity of the law of the iterated logarithm it is sufficiet oly that = < The iequalities of Khitchie Let " = ("; "; ) be a sequece of ideedet ad idetically distributed radom variables defied o the robability sace (; F; P ) ad takig values 6 with robability = Let us deote f = (f; f; ) with: () f = a k " k where a = (a; a; ) is a sequece of umbers Put: () S (f ) = fk! = where f k = f k f k f = Sice " k = 6 the: 7

8 (3) S (f ) = ja k j!= I this otatio the iequalities (III) get the followig form: (4) A S (f ) jf j B S (f ) > To rove the right-had iequality i the case = m m A: Ya: Khitchie [74] writes that: jf j m = k ++k=m; ki = k ++k=m ; ki (m)! (k )! (k )! (m)! (k )! (k )! It is evidet that for m = k + + k k i : Hece: (5) jf j m (m)! m k! k! (k )! (k )! m m! = (m)! m m! k ++k=m ; ki ja k j m = (m)! ja j k ja j k j" j k j" j k ja j k ja j k m! k! k! m m! ja j k ja j k m S (f ) which roves the right-had iequality i (4) for = m (oly such a case did A: Ya: Khitchie cosider ) with the costat: B m = (m)! m m! which exactly coicides with the momet ( m ) of order m of a stadard ormal radom variable ( = ; D = ) From the give cosideratio oe ca also see that the right-had side of the iequality (5) is exactly equal to P a k k m where ; ; are ideedet stadard ormal radom variables because: (m)! a k k m= ja j k ja k j j j k j k j = k ++k=m; ki k ++k=m ; ki = (m)! m m! k ++k=m ; ki (k )! (k )! (m)! (k )! (k )! m! k! k! (k )! (k )! k k k! k! ja j k ja j k ja j k ja j k 8

9 I this way oe ca give to the iequalities of Khitchie the followig form: (6) a k " k m By Stirlig s formula! = (m)! a k k m e e R =(+) < R < =() m m m m m! D e with D = Thus from (5) by use of the Chebyshev-Markov iequality we get that for: t m = S (f ) oe has: P jf j > t t m jf j m (m)! D m S (f ) e t m tm m S (f ) m!! m De m De t =(S (f )) = De e t =(S (f )) I this way from the iequality (5) the estimate is obtaied: (7) P jf j > t C e t =(S (f )) with C = De = e = 3:84 which was recisely what gave to A: Ya: Khitchie a ossibility to derive the statemet (9) (Below we will see that the iequality (7) holds with the costat C = ) Beig cocered with the establishmet of Khitchie s iequality: (8) jf j m m B m S (f ) with the costat B m = (m)!! it aturally aears a questio about is that costat the best ossible oe It is reasoable that the aswer to this questio deeds o are we iterested i that the iequality holds true with give ad fixed costat for all or just for fixed ( whe the otimal costat i (8) deeds o it ) I Sectio 8 below we will reset such tye of results Here we shall oly ote that i 96 S: B: Stechki [39] showed that i (8) beig cosidered for all the costat B m = (m)!! aears ideed to be the best ossible 3 The above exosed method of A: Ya: Khitchie which gives the right-had iequality i (4) works oly for = m m Let us show that the validity of these iequalities for all > follows from the iequality (7) I this cotext it might be useful to rove ideedetly (from the roof of iequality (4) for = m ) the validity of iequality (7) ad the shortly afterwards to derive from it the iequalities (4) for all > For this otice that sice cosh x (ex +e x ) e x = the for all > : 9 :

10 P f jf j t g = P f f t g = P8 9 ex(f ) ex(t) Y Y e t e f = e t = ex (S (f )) t By the freedom of choice of > hece: cosh(a k ) e t (S P f jf j t g mi > ex (f )) t ex jak j t = ex (S (f )) because the miimum is attaied for = t=(s (f )) So we have just roved that the iequality (7) holds ideed with C = Tur ow to the derivatio of iequalities of Khitchie (4) for > The case < for the right-had iequality i (4) ad the case < for the left-had iequality i (4) are trivial sice: jf j = S (f ) ad the orm kfk = jfj = is o-decreasig i > This moreover shows that i (4) oe may take B = for ad A = for Let ow > The from (7) ( with C = ) we fid that: (9) f S (f ) Z Z P = t f ) > t S (f t e t = dt = = (=) which roves the right-had iequality i (4) with the costat: B = = (=) > dt I the case < < by the Cauchy-Buyakovskii iequality: whece: = f S (f ) = f S (f )4!= B 4 f S (f ) f S (f )= f S (f )=! f S (f )!= B 4= ie the left-had iequality i (4) holds true with the costat: f S (f )!=

11 A = B 4 This comletes the roof of the iequalities of Khitchie (4) with the costats A ad > give by: B ; () B = () A = B 4 ; < ; : = (=) ; > I Sectio 8 the best values for the costats A ad B i the iequalities of Khitchie will be cosidered (comare () ad () with (88) ad (89)) 4 I the ext two sectios it will be cosidered a geeralizatio of the iequalities of Khitchie (4) to the case of martigale sequece (f ) But before comig to that geeralizatio we shall cosider oe more iterestig exadig shere of actio of Khitchie s iequalities coected with a observatio that the iequality (5) ca take form (6) where o the right-had side of that iequality ; ; are ideedet ormally distributed radom variables with k = ad D k = k Havig the iequality (6) it is atural to ose a questio if it remais valid if oe assumes that ; ; belog to the class 6 cosistig of ideedet idetically distributed radom variables with symmetric distributio ad k = D k = k Such a settig of questio was cosidered by Utev [48] 984 Pielis [9] 994 ad the authors (T: Figiel P: Hitczeko W: B: Johso G: Schechtma J: Zi) i the article [4] 994 From the results of these works it follows that for = ad 3 ad for ( ; ; ) 6 : ad cosequetly: i= i= a i " i a i " i i= a i i = if (;;)6 which shows a defiite extremal role of Rademacher fuctios i= a i i 3 Martigale extesios of Khitchie s iequalities I ( relimiary cosideratios) Let us begi with two results from the classic theory of orthogoal series It is well-kow that with the system of Rademacher fuctios r = (r ; r ; ) oe coects two remarkable orthogoal ad comlete systems of fuctios amely of Haar h = (h ; h ; ) ad of Walsh w = (w ; w ; ) which are defied i the followig way: For x [; ] : (3) h (x) = h (x) = r (x) j= r h (x) = j+ if k x < k ; = j + k ; k j ; j j j otherwise

12 ad: (3) w(x) = w (x) = r (x) r k (x) if = + + k + where > > k Let the fuctio ' L [; ] or ' L [; ] with ' = Deote: (33) ' (h) = ('; h k )h k ' (w) = ('; w k )w k where ('; h k ) ad ('; w k ) are coefficiets of Fourier-Haar ad Fourier-Walsh resectively I year 93 Paley [4] showed that if: f = ' (w) the for all < < ad there are such uiversal costats A ad B that: (34) A S (f ) f B S (f ) where: S (f ) = fk = f k = f k f k I year 937 Marcikiewicz [96] oted that the result of Paley (34) (for Walsh fuctios) follows from the validity of (34) for Haar fuctios ( with f = '(h) ) ad the fact (recorded also by Paley i [4] ad iitially roved by Walsh [49] i year 93) that ' (h) = ' (w) I year 938 Marcikiewicz ad Zygmud [97] showed that if = (; ; ) is a sequece of ideedet radom variables with i = i the for all oe fids such uiversal costats A ad B that agai the iequality (34) holds with f = + + ad: S = (f ) = k Aaretly D: Burkholder ad R: Gudy were the first who realized ([4] [5] [5]) that sequeces f = (f; f; ) for which oe obtais iequalities of tye (34) both i the case of Khitchie f = P a k" k ad i the cases of Paley f = ' (w) Marcikiewicz f = ' (h) Marcikiewicz ad Zygmud f = P k osses oe imortat ad remarkable roerty all of them aear to be martigales So the two-sided iequalities cosidered above aear to be iequalities for secial classes of martigales It was this circumstace that defied martigale directio of ivestigatio of the validity of the iequalities of tye (34) where basic ad fudametal results were obtaied by D: Burkholder R: Gudy B: Davis with whose ames are coected the so-called BDG-iequalities (see Sectio 6 below) first roots of which were the iequalities of Khitchie 3 Let us recall eeded defiitios of martigales ad related otios Let (; F ; (F ) ; P ) be a filtred robability sace (; F ; P ) equied with a o-decreasig family (filtratio) of

13 -algebras (F ) such that F m F F m Defiitio A stochastic sequece = ( ) of radom variables = (!) is called a martigale if: (i) is F -measurable ; (ii) j j < ; (iii) the martigale roerty is satisfied: P -as (35) ( + j F ) = where ( + j F ) is the coditioal exectatio of + with resect to F Defiitio A stochastic sequece = ( ) of radom variables = (!) is called a local martigale if: (i) is F -measurable ; (ii) the coditioal exectatio ( + j F ) is well-defied for all ; (iii) the martigale roerty (35) is satisfied (We say that the coditioal exectatio ( + j F ) is well-defied if the set f! : ( + + j F ) < g [ f! : ( + j F ) < g coicides with u to a set of P - robability zero I this case we set ( + j F ) = ( + + j F ) ( + j F ) ; for more details see [38] VII ) Defiitio 3 A radom variable = (!) a Markov time if for each : f! : g F with values i the set f; ; ; +g is called If moreover Pf < g = the we say that = (!) is a stoig time With the hel of stoig times oe obtais the followig criterium for whe a stochastic sequece forms a local martigale: A give sequece = ( ) of F -measurable radom variables forms a local martigale if ad oly if there exists a sequece of stoig times ( k ) k such that k " ad for each k the stoed sequece (k) = ( ^k ) is a martigale (see [38] VII Theorem ) If M deotes the class of all martigales ad M loc the class of all local martigales the evidetly: M M loc The above give criterium makes it ossible that the roof of some or the other roerty of local martigales is reduced (by corresodig localizatio) to the case of martigales This fact clarifies why most of the results i the sequel will be formulated for martigales although they exted to the case of local martigales as well If i (35) the equality is relaced by the iequality ( ) the oe says that the sequece forms a submartigale (suermartigale) 3

14 As i the case P P of Khitchie f = k"k a as well as i the case of Marcikiewicz ad f = Zygmud k the martigale roerty of the sequece f = (f ) is evidet from the ideedece of its comoets ad roerties jak"kj = jakj < jkj < ad (ak"k) = (k) = This is less evidet i the case of the system of Haar where f = '(h) = P ('; h k)hk However oe could observe that if F = (h ; ; h) the (' j F ) coicides ( P -as) with f : f = (' j F ) Hece immediately it follows that f = (f) forms ( with resect to the family (F ) ) a (Lévy) martigale sice: (f+ j F ) = (' j F + ) j F = (' j F ) = f The case of the Walsh system reduces to the case of the Haar system sice as already oted above ' (h) = ' (w) ad (as it is easily see) the sequece ' (h) is a martigale (with resect to F ) 4 Martigale extesios of Khitchie s iequalities II ( the iequalities of Burkholder: > ) Let us suose that f = (f) is a martigale defied o a filtred robability sace ; F ; (F ); P with f = F = f;; g So f = Puttig d = ad: d = f f ( f ) we obtai a sequece d = (d) associated with the martigale f = (f) which is called a martigale differece sequece; d is F -measurable jdj < ad: (d+ j F ) = Put: (4) S(f ) = I the theory of martigales the quatity: (4) [f ] = f k= f k = is widely called the quadratic variatio of the sequece (f ; ; f) I the square-itegrable case ( jfj < ) a sigificat role is layed by the quadratic characteristic: (43) f = (fk) j F k for which we have the followig imortat roerties: d k = 4

15 (i) (ii) (iii) f f f [f ] f is F -measurable; is a martigale; is a martigale Proerty (i) which is evidetfrom defiitio (43) is also exressed by sayig that the sequece f is redictable ( ie f is F -measurable ) Proerty (ii) is checked directly ad its equivalet formulatio is the fact that the submartigale (f ) admits Doob s decomositio: (44) f = f + M with redictable sequece f ad martigale (M ) Proerty (iii) is established by a straightforward verificatio It is clear from (4) ad (4) that: (45) [f ] = S (f ) All of the above metioed iequalities have the followig form (comare with (34)): (46) A S (f ) jf j B S (f ) herewith i the case of Khitchie they hold for > i the case of Marcikiewicz-Zygmud for ad i the case of Haar ad Walsh for > For which values of the result (46) remai valid for a arbitrary martigale f = (f )? Basic results i this directio were obtaied by Burkholder [4] who foud that the iequality (46) holds ( with some uiversal costats A ad B ) for all > From the examles i Sectio 6 below it will be clear that it is imossible to exted the validity of the left-had iequality i (46) for = However as established by B: Davis [3] for = the roer form corresodig to the aalogue of the iequality (46) is the followig oe: (47) K 3 S (f ) max jf k j L 3 S (f ) k By the well-kow iequality of Doob for submartigale jf j ) for ay > : (48) jf j max jf j k jf j ( see [38] VII 3 Theorem so the result of Burkholder ( for > ) ad the result of Davis ( for = ) admit the coulig formulatio: For all there exist such uiversal costats K 3 ad L 3 ( ot deedet either o or o the martigale f ) that: (49) K 3 S (f ) max jf j L 3 S(f ) k or i the equivalet forms: 5

16 (4) K 3 (4) K 3 S (f ) [f ] max k jf j max jf j k L 3 S (f ) L 3 [f ] where K 3 = K3= ad L 3 = L3= Let us tur to the basic stes i the roof of the iequality (46) established by Burkholder for > From the techical oit of view the roof of Burkholder is rather comlicated The scheme of the roof roosed below differs from the traditioal oe ad it seems to us better clarifies the key role of the iitial iequalities of Khitchie I the roof oe establishes the validity of the iequality (46) for arbitrary martigales Let " = (" k ) k be a Rademacher system Accordig to the iequalities of Khitchie for > : (4) = fg A ja k j a k " k fg = B ja k j Suose i additio that we ca rove that for ay sequece of umbers b = (b k ) k with b k = 6 ad for the give martigale f = (f ) the followig iequality is valid: (43) F f f3g b k d k f4g G f with some ( uiversal ie: ot deedet o ad f ) costats F ad G Note that the sequece: with the form: (44) F f f (b) = f (b) f (b) = P b kd k = P b kf k is also a martigale ad (43) ca be writte i f (b) G f ad might be viewed as a comariso iequality for martigales f ad f (b) It will be see from the sequel that iequalities (44) are valid oly for > (I the case = we will have maximal iequalities (6)) Let us ow show how the iequalities of Khitchie (4) + comariso iequalities (44) together with the ideas of radomizatio lead ( for > ) to the iequality of Burkholder Havig the robability sace (; F ; P ) ad martigale f = f (!) defied o it cosider a ew robability sace ( " ; F " ; P " ) ad a sequece of Rademacher fuctios " = " k (! " ) k defied o it O " ; F F " ; P P " defie radom variables "k (!;! " ) ad d k (!;! " ) by uttig: " k (!;! " ) = " k (! " ) ad d k (!;! " ) = d k (!) 6

17 It is clear that the sequeces: " k (!;!") k ad d k (!;!") k are ideedet with resect to the measure P P" Usig this fact ad alyig the iequalities of Khitchie with a k = d k (!) " k = " k (!") ad comariso iequalities with b k = " k (!") d k = d k (!) we fid that: (45) A = " by f4g d = by fg k (!) G" = G " = G F " I this way it is showed that: A G " d k (!)" k (!") (F ubii) d k (!;!")" k (!;!") = " d k (!) d k (!;!") d k (!)" k (!") d k = (!) = G by f3g by fg d k (!) G F d k (!) " G B F B F d k = (!) " k (!")d k (!) d k (!;!")" k (!;!") d = k (!) ( for > sice the comariso iequalities are valid exactly for such values of ; see Sectio 7 below) 3 Let us here idicate a articular case (Marcikiewicz-Zygmud) of the validity of comariso iequalities (43) ( for all ) Namely suose that d = (d) is a sequece of ideedet radom variables with d = It is clear that f = (f) is a martigale I tis case the iequalities (43) are easily roved by alyig the method of symmetrizatio which is as follows Together with (; F ; P ) cosider a ew robability sace ( ; F ; P ) ad o it defied a sequece of ideedet radom variables d = (d ) havig the same distributio as d = (d) : Law d j P = Law d j P The defiig o ; F F ; P P ) variables d k (!;! ) = d k (!) ad d k (!;! ) = d k (! ) ad takig ito accout that jx+yj jxj +jyj x; y R we fid ( b k = 6 ): 7

18 = fag b k d k (!) = = = b k d k (!;! )d (!; k! ) d k b k d k (!) b k d k (J ese) d k (!;! )d (!; k! ) b k d k (!;! )d (!; k! ) d k + where i {a} we use the symmetry of the distributio of the sequece d d which leads to the fact that: Law b(dd ); b (dd ); jp P = Law dd ; d d ; jp P I this way the right-had iequality i (43) is roved with G = To rove the left-had iequality it is sufficiet to observe that if b k = 6 the by the above roved: f j = d k = b k b k d k b k d k from where the left-had iequality i (43) follows with F = Ufortuately such a simle roof of comariso iequalities (43) relies heavily uo the assumtio of ideedece of the radom variables d k k ad does ot work i the case whe radom variables d k k form oly a martigale differece sequece d k! 5 The maximal iequalities of Khitchie I the scheme cosidered by A: Ya: Khitchie besides the iequalities (III) ( for > ) the validity of the followig maximal iequalities (which are ot cosidered by Khitchie himself) ca be established: For > there exist such uiversal costats A 3 ad B 3 that: (5) A 3 If we deote = f 3 g ja k j max f k = P i= a i" i ad: f 3 = m max jf kj k the (5) ca be writte i the followig form: (5) A 3 S (f ) f 3 m! f 3 g a k " k B 3 B 3 S (f ) = ja k j 8

19 where S (f ) is defied by (3) It is clear that by the (ordiary) left-had iequality of Khitchie i (4) we get the left-had iequality i (5) with A 3 = A sice jf j f 3 For the roof of the right-had iequality i (5) we shall use the well-kow maximal iequality of Lévy: (53) P max k js k j > t o P8 js j > t9 which holds for sums S = + + of ideedet ad symmetrically distributed ( P ( k B) = P ( k B) B B(R) ) radom variables (see for istace [38] IV 4) Puttig k = a k " k k = ; ; ad hece usig the fact that for : Z Z f 3 = t P ff 3 > tg dt t P fjf j > tg dt = jf j from the right-had iequality i (4) we get ( for ) the right-had iequality i (5) with the costat B 3 = B Cocerig the case < < we should ote that for such s by Jese s iequality: f 3 (f 3 ) = ad so this case is reduced to the case = I this: B 3 = B = = = sice B = The questio about the best costats A 3 ad B3 i the maximal iequalities of Khitchie (5) (still) remais oe Some asects i this directio may be foud i the works [37] [38] [5] [5] [68] [7] 6 Martigale extesios of maximal Khitchie s iequalities ( the iequalities of Davis: =; the iequalities of Burkholder-Davis: ) I the case > the iequalities of Khitchie ad comariso iequalities (44) gave a ossibility to obtai the iequalities of Burkholder (46) for martigales f = (f ) As already oted above i the case = the iequalities (46) geerally fail to hold Here is the corresodig examle Let d = (d ) be a Beroulli sequece of ideedet ad idetically distributed radom variables takig values 6 with robability = Put f = d + + d ad: = mi f : f = g It is well-kow that P f < g = but = = Usig the martigale ad stoig time we shall ow costruct a ew martigale g = (g ) by uttig: g = f ^ ^ = d k = I(k )d k f = (f ) 9

20 The: (6) g = g + g = g +!! sice g + ad g +! ( P -as ) O the other had: S (g) = ^!! Comarig this with (6) we see that the left-had iequality i (46) caot be satisfied with a sigle costat A which does ot deed o As already remarked i Sectio 4 Davis [3] have discovered that the roer shae of the corresodig aalogue of the iequality (46) has the form (47) ad the for all the maximal iequalities must have the form (49) Let us ow show that the roof of these iequalities ca be obtaied ( uo the same scheme as the iequalities (46) with > ) from: (i) maximal iequalities of Khitchie (5) for the system of Rademacher fuctios ( layig the role of radomizatio of the martigale f ); ad (ii) maximal comariso iequalities for martigale trasforms (comare with (44)): (6) F f 3 3 f 3 3 g f 3 (b) f 4 3 g G 3 f 3 where of umbers with b i = 6 Ideed by usig the same otatio as i the chai of iequalities (45) we fid that for ( f 3 g f 3 g f3 3 g ad f4 3 g are defied i (5) ad (6) ): f 3 = max m jf m j f m (b) = P m b kd k ad b = (b i ) i is a arbitrary sequece 3 (63) A 3 by f g S (f ) = " (F ubii) " max m by f4 3 g G 3 " " m max m max m m d k (!;! " )" k (!;! " ) m max m m " k (! " )d k (!) m d k (!) d k (!)" k (! " ) = G 3 f 3 = G 3 " max d k (!;! " ) m by f3 3 g G3 F 3 " max m d k (!;! " )" k (!;! " ) m

21 = G3 F 3 by f 3 g " G3 F 3 I this way we get for all : max m m m B 3 d = k (!) d k (!)" k (! " ) = G3 B 3 F 3 S (f ) with: K 3 S (f ) f 3 L 3 S (f ) K 3 = A3 G 3 L 3 = B3 F 3 where (A 3 ; B3 ) ad (G3 ; F 3 ) are the costats from the maximal iequalities of Khitchie (5) ad maximal comariso iequalities (6) 7 Comariso iequalities for martigale trasforms The above exosed scheme of the roof of iequalities of the form (46) or (5) clarifies the key role of the two igrediets Khitchie s iequalities ((4) (5)) ad comariso iequalities for martigale trasforms ( 44) (6)) Sice the situatio with Khitchie s iequalities is clear let us address the questio of comariso iequalities I the roofs give above ( see (45) ad (63) ) we have used the martigale trasform: of a rather secial form: (7) f (b) = f (b) = f (b) b k d k where the umbers b k take the two values 6 Let us ow aturally formulate a questio of the validity of comariso iequalities i a more geeral form admittig for b = (b k ) k a arbitrary redictable sequece ( ie such oe that b k is F k-measurable with jb k j for all k ; F = f;; g b = cost ) I this way it becomes of iterest to clarify the validity of iequalities of the form: (7) (73) max f (b) G f f m (b) m G 3 max f m m where f (b) = f k (b) is a martigale trasform with a redictable sequece k b = (b k) k or i a more geeral form the iequalities of the tye:

22 (74) (75) max g G f gm 3 G m max m fm for two martigales f = (f ) ad g = (g ) such that ( P -as ) for all : (76) g f I the above give iequalities oe assumes that G ad G 3 are uiversal ( ie ot deedet either o the martigales or o ) costats Naturally it is of cosiderable iterest to address the questio about the best costats ( Uder the best costat say G i (74) we uderstad that value that if G < G the oe ca fid a robability sace ( ; F ; P ) filtratio (F ) ad martigales g ad f such that g f but g > G f ) The validity of iequalities (7) for > ad (73) for ca be established for examle by usig techiques develoed i works of Burkholder ad Davis Below we give a roof of the iequality (73) for the case = The corresodig roof for > from the oit of view of idea is rather similar ad ca be obtaied by the very same method like the roof of Theorem 7 i 9 of Chater [89] followig the scheme of the roof give below for the case = Let us uderlie that the formulatio ad roof of Theorem 7 i 9 of Chater i [89] have bee doe at oce for local martigales i the case of cotiuous time (the case of discrete time ca be imbedded ito it) ad i this maer it is iterestig as a roof which works i a rather geeral situatio I the case of discrete time a very iterestig result about the validity ( for ) of the iequality (74) uder assumtio (76) was obtaied by Burkholder [] [3] The formulatio of this will be give below i art 4 of this sectio 3 So suose that o the filtred robability sace ; F ; (F ) ; P with F = f ;; g a martigale f = (f ; F ) with f = is give Let d = f f (= f ) such that: (77) f = d k d = Let b = (b ; F ) ( with F = F ) be a redictable sequece with jb j Set: (78) f (b) = b k d k f 3 = max k jf kj f 3 (b) = max jf k(b)j k Our aim is to rove the followig comariso iequality; for ay stoig time : (79) f 3 (b) G3 f 3 or i a more detailed otatio: (7) max b k d k G 3 max d k where G 3 is a uiversal costat ( from the estimates give below it follows G3 Itroduce the followig otatio: = 85 )

23 d 3 = g = max jd kj k I = I jd j > d 3 ~g = I k d k g (b) = I k b k d k I k d k jf k ~g (b) = I k b k d k jf k f = g ~g f (b) = g (b) ~g (b) Varg = ji k d k j Var~g = At oce we ote sice jb k j that: (7) Varg(b) = ji k b k d k j ad: Var~g(b) = Ik b k d k jf k = Lemma It takes lace the followig iequality: (7) max with the costat C = 8 Proof We have: (73) max f m (b) m ^ Varg ^ fmg = ^ f (b) Varf (b) C max jf j ^ ^ + Var~g ^ ji k d k j ji k d k j Ik d k jf k Varg jb k j Ik d k jf k Var~g Varg(b) + Var~g(b) ^ ^ ^ ji k d k j + ji k d k j Fk P k jik d k j(ji k d k j Fk) where the equality fmg follows from the fact that is a martigale ad stoig time ^ From (73) i a evidet way by assig to the limit whe! we get the iequality: (74) max m f m (b) ji k d k j 3

24 Let us begi to estimate the right-had side of that iequality Sice jdkj d 3 k ad o f! : jdkj > d 3 k g we have jd kj + d 3 k jd kj d 3 k the jd kj d 3 k Therefore: (75) jikdkj But: d 3 k = d3 (76) d 3 = k max jd kj = max k jf k fkj max k jf kj = f 3 The desired iequality (7) follows evidetly from (74)-(76) The lemma is roved To formulate the ext lemma deote: f = f f f (b) = f (b)f (b) Note that: ad: Also ut: It is evidet that: sice jbkj f = f (b) = dkik + dkikjf k bk f 3 = dkik + dkikjf k f k f (b) 3 f 3 f (b) 3 = f k (b) Lemma It takes lace the followig iequality: (77) max f (b) 3 f 3 = + 4d 3 Proof We shall first show that for ay stoig time : (78) f 3 4d (b) f Ideed we have: ad so: (79) f (b) = b Id + IdjF f (b) f = I d +IdjF 4

25 Sice (d jf ) = the (I d jf ) + ( I d jf ) = Hece from (79): (7) ~g = (I k d k jf k) = ( I k d k jf k) ad it follows: f (b) I d + ~g d I jd j d 3 + d I jd jd 3 F d 3 + d3 = 4d3 which roves (78) i the case Let ow be a redictable stoig time ad let the martigale f = (f ; F ) be uiformly itegrable The agai (d jf ) = ad the chai of iequalities (7) remais valid with chagig to But if is a arbitrary stoig time the from (7) it follows that it is sufficiet oly to check that ~g d 3 With this i goal ote that ~g = (~g ) is a redictable sequece ad cosequetly the times of its jums are redictable So either ~g = or as already roved ~g d 3 if f is a uiformly itegrable martigale The case of arbitrary martigale reduces to the revious oe by hel of the usual rocedure of localizatio cosistig of this that oe ca fid such a sequece of stoig times ( k ) k that k " ad the stoed martigales f (k) = (f ^k ; F ) are already uiformly itegrable (see for examle [89] 73) The established roerty (78) ermits us to claim that the martigale f (b) = (f (b); F ) is locally square-itegrable ie there exists such a sequece of stoig times ( k ) k k " that the stoed sequece f^ k (b) forms a square-itegrable martigale ie such a martigale that: f su ^ k (b) < ( For examle for localizig stoig times it is sufficiet to take the followig oes: k = if 8 : jf (b)j _ 4d3 k 9 ) The fact that f (b) ad f are local square-itegrable martigales ermits us to state that for ay stoig time : (7) If we deote: f (7) = (b) f (b) 3 f 3 f (b) the from (7) we have i articular that: (73) Y Y = f 3 max f m m 5

26 for ay stoig time This roerty is widely called the roerty of L-domiatio ( of rocess by rocess Y ) or the roerty of domiatio of Leglart; see [69] 35 or [89] 68 Accordig to Theorem 4 o 68 i [89] the roerty of L-domiatio ermits us to assert that if Y D where D = (D ) is a icreasig adated rocess the for ay > ; > ad stoig time it takes lace the followig iequality: (74) P max Y +D ^ Let us aly this to the case (7) the from (74) we fid that: (75) P f (b) = P max f 3 +D ^ max f (b) + P Y +D f 3 + P +D where oe may take D k = 4d 3 k sice by the force of (78): Deote: f 3 k = f k Dk = f 3 + D The (75) takes the followig form: ad so: P max Z (76) max f (b) f (b) ^ + P ^ d + Z P d Note that: = = Z + d + + = 3 Z Z d + P d q f 3 q f +D 3 q f + D 3 + 4d3 which together with (76) roves the desired iequality (77) The lemma is roved Lemma 3 It takes lace the followig iequality: (77) f 3 = 5 max jf j 6

27 Proof From (7) we have: f max 3 f ie it takes lace the roerty of L-domiatio (73) with: The from (74) we get: m f = f 3 Y = max (78) P f 3 = P max m max f m + D m m f 3m! ^ + P where oe may take D = 4d 3 sice the max m (f m ) D which imlies the validity of the iequality (74) Let: The from (78): = max jf m j + m 4d3 max f m + D m f 4d 3 = ad so: P f 3= f 3= ^ + P f g Z ^ d + Z Z = d + d + = + + = 3 Hece together with (7) we fid that: f 3 = 3 max m jf m j + d3 The lemma is roved 3 3 max m jf mj + 3 max m jf mj + 4 max m jf m j + 4 max m jf mj + 4 From Lemma Lemma 3 ad (76) we fid that: (79) max m f m (b) 3 3 f = + 4d 3 max m jf mj max m jf mj = 5 max jf mj m 7

28 3 5 max jf mj + 8 max jf mj m m Together with estimate (7) from (79) we fid: max m fm (b) 85 max jf mj m = 77 max m jf mj which roves the iequality (79) with costat G 3 = 85 (The questio about the value for the best costat G 3 still remais oe) 4 Let us formulate the above metioed result of Burkholder ( see for examle [8] [9] []) for the case of discrete time Let > ad let f = (f ) ad g = (g ) be two martigales such that P -as for all : g f (see (74) ad (76)) The for all : where: G = 3 g G f 3 = max (; q) = + =q = Moreover the costat G = 3 is the best ossible 8 O the best costats i the Khitchie s iequalities I accordace with () ad (5) for ay ad ay sequece of umbers a = (a; a; ) : (8) m m a k " k m ja k j where N (; ) ad cosequetly: Put i (8): m = (m)! m m! (8) a = = a = The we fid that: (83) By the cetral limit theorem: m " k m 8

29 (84) d " k! d (! deotes covergece i distributio ) From the iequalities of Khitchie (4) it follows that for ay > : ad cosequetly the family: su m " k " k m < is uiformly itegrable Hece from (84) it follows that: lim! " k m= m From this it follows that i the iequality (8) beig cosidered for all ad all sequeces a = (a; a; ) the costat Bm = m = (m)!! ) is the best ossible ( i the sese exlaied i the ed of art of Sectio 7 ) Rewrite the iequality (8) to the form: (85) where a k; " k m m a k; = a k P ja kj P ad cosequetly a k; = It is clear that by symmetry of distributios of radom variables " k it is sufficiet to cosider oly the case of o-egative values a k excludig i this case whe a = = a = The cosideratio give above shows that the choice of values a k = = k has some extreme roerties Ideed as it will be show below for ay (a; ; a ) : ad moreover the umbers: a k; " k m m " k m " k are mootoe-icreasig to m The exlaatio of this heomeo ad its aalogues i other iterestig for us cases might be obtaied if we tur to the otio of covexity i the sese of Schur itroduced by Schur 9

30 i the aer [3] ublished as it turs out i the same year whe the aer of Khitchie [74] with the iequalities of Khitchie has bee aeared Let us itroduce cocets ad facts related to the otio of Schur-covexity ad the theory of majorizatio The motivatio for itroducig the corresodig otios relates with a atural desire to have a right defiitio of the roerty that the comoets of a vector x are less sread out tha are the comoets of a vector y A recise defiitio of this vague itetio oe may obtai by itroducig the cocet (Hardy Littlewood Pólya 99) which states that x is majorized by y ( i the otatio: x y ) Let z = (z ; ; z) be a vector i R ad let z 3 ; ; z3 be the comoets of vector z i decreasig (more recisely o-icreasig) order: z 3 z3 z3 For the two vectors x ; y R we say that x is majorized by y ad we write x y if: k i= for all k = ; ; ; ad: (86) (87) (88) For examle: i= x 3 i xi = k i= i= y 3 i yi ; ; ; ; ; ; ; ; ; (; ; ; ) P ; ; ; (a ; ; a) (; ; ; ) for ai with P i= x i ; ; P i= x i (x; ; x ) for x i i= a i = ; Let D R be a set ad let 8 = 8(x) be a fuctio defied o D with values i R We will say that the fuctio 8 = 8(x) osses the roerty of covexity i the sese of Schur ( Schur-covex ) o D if: 8(x) 8(y) for x ; y D such that x y A fuctio 8 = 8(x) x D is said to be Schur-cocave if the fuctio 8 = 8(x) is Schur-covex Note if the fuctio 8 = 8(x) is symmetric ad covex the it is Schur-covex (this i essece was cotaied i the aer of Schur [3] 93) For our aims the followig result (together with (86)-(88)) leads to may iterestig corollaries ( Schur [3] 93; Hardy Littlewood Pólya [6] 99): Let I R be a iterval ad suose that the fuctio g = g(x) defied o I with values i R is covex The the fuctio: is Schur-covex o I 8(x) = i= g(x i ) 3

31 The followig fudametal theorem (Schur [3] 93; Ostrowski [3] 95) ermits us to verify the roerty of covexity i the sese of Schur i terms of a roerty of the first artial derivatives Let I R be a iterval ad let a real valued fuctio 8 = 8(x) x I be cotiuously differetiable The 8 = 8(x) is Schur-covex o I if ad oly if the followig two coditios are satisfied: (i) 8 is symmetric o I (ii) (x i x j i j for all x I ad all i 6= j Uder the validity of coditio (i) the coditio (ii) is equivalet to the coditio: (ii) (x @x (x) for all x I A aalogous characterizatio takes lace for cocavity i the sese of Schur as well with chagig the iequalities i (ii) ad (ii) to the reversed oes Note that coditio (ii) (or (ii) ) is widely called the Schur coditio Remark The aer of Schur [3] 93 cotais a first sufficietly comrehesive study of fuctios 8 = 8(x) for which x y ) 8(x) 8(y) I this aer oe fids the Schur coditio ( o R + which later was exteded by Ostrowski [3] 95 to R ) Fuctios satisfyig that roerty Schur called covex as oosite to the fuctios which were covex i the sese of Jese Now the last oes are called covex ad covex fuctios by Schur accordig to Ostrowski [3] 95 are called Schur-covex Remark A imortat aalytic roerty i the theory of majorizatio was established by Hardy Littlewood ad Pólya i 99 This roerty states that x y if ad oly if x = P y for some double stochastic matrix P = [ ij ] ( ij ; Pi ij = Pj ij = ; 8i; j ) I fact this roerty was used by Schur as defiitio Let us give two iterestig examles of fuctios havig the roerty of covexity i the sese of Schur ad the roerty of cocavity i the sese of Schur P xamle Let = ( ; ; ) ; i ; i= i = The etroy of the vector = ( ; ; ) is by defiitio the quatity: H() = i= i log i This fuctio is (strictly) Schur-cocave I articular: xamle Put: H(; ; ; ) H() H( ; ; ) x = i= xi 3

32 Cosider the stadard deviatio of the vector x ; ; x : = xi x i= The this fuctio is Schur-covex 3 Now we are i ositio to be ready to exlai why the choice of values for a ; ; a made i (9) ( a = = a = = ) osses some extreme roerties Cosider the fuctio: (89) ' m (x ; ; x ) = m xk " k m It turs out that this fuctio cosidered o R + is Schur-cocave which is directly verified by usig the Schur-coditio (ii) Hece by usig (87) we fid that: (8) ' m ( ; ; ) ' m(x ; ; x ) x i P i= x i = This recisely meas that the choice made i (8) osses the extreme roerty such that with this choice the left-had side i (85) takes its maximal value Returig back to (86) we fid that the Schur-cocavity of the fuctio ' m (x ; ; x ) leads to a iterestig roerty that: m " k m " k Thus cosequetly additioally to the statemet of the cetral limit theorem (84) we fid that the momets: m " k are mootoe-icreasig to m 4 Now deote: (8) ' (x ; ; x ) = xk " k As a matter of fact the fuctio ' (x ; ; x ) is Schur-cocave o R + ot oly for = m ; m but also for all 3 This was roved by a straightforward cosideratio (without referrig to or metioig covexity i the sese of Schur ) i the work of P: Wittle [53] 96 (I this work it was roved the validity of this roerty for < < 3 as well which is like writte i [53] wrog sice oe ca give a examle of failure of the iequality P i= xi " i P (= ) i= " P i for those values of ad i= x i = x i ) The argumetatio based uo the cocet of covexity i the sese of Schur is cotaied i the articles of M: ato [39] 97 ad Komorowski [79] 988 3

33 I this way the tur to cocavity i the sese of Schur of the fuctio ' (x; ; x ) ermits us to make some secial coclusios cocerig the values for the best costats B () i the iequality: (8) a k " B k () for every ad fg [ [3; [ : (83) B () = max( (84) " k = ja k j = a k " k : = k= ja k j = ) = C k jkj = + = " k!! Note that the case ]; [ is trivial (with B () = ) sice k P a k" k k k P a k" k k = ( P ja kj ) = ad takig a = ; a = = a = we have k P a k" k k = Cocerig the case ]; 3[ as far as we kow about the values for the best costats B () are ot kow Cosequetly for ay : (85) B () = 8< : Let us tur to the estimate from below: A () = = P ja k j = ; <? ; < < 3 k= C k = jkj ; 3 < : a k " k The case is simle: A () = sice ( P ja kj ) = = k P a k" k k k P a k" k k ad for a = ; a = = a = the iequality turs ito a equality I the case < < the value for A () as far as we kow about is ot kow Cosequetly for ay :? ; < < (86) A () = ; < : 5 Let us retur to the roblem about the best (uiversal) costats A ad B i the iequalities of Khitchie: (87) A ja k j = a k " B k = ja k j beig cosidered ot just for oe cocrete ad fixed but for all About the costats B the followig statemet holds true: 33

34 (88) B = 8 < : ; < + = ; < ; where ((( + )=)= ) = = kk N (; ) As already oted above i the case = m m : =m B m = kk m = (m)!! The questio about the values for the (best) costats B (exosed i (88) has a log history The case of otimality of B for = m has bee established by S: B: Stechki [39] 96 (by a direct calculatio without ivokig the cetral limit theorem as reseted above) From the article of Wittle [53] it ca be cocluded that for 3 the best costat B i (87) is less or equal to ((( + )=)= ) = The otimality of the costats B for 3 follows from the cosideratios metioed above about cocavity i the sese of Schur of the fuctio ' (x ; ; x ) It follows from this (as well as for = m ) that: B = = lim! B () = lim! = = +! =! = " k Notice that the case 3 was cosidered (by aother method) i the aer of Youg [56] 976 Fially the case ]; 3[ does ot fit uder the aalysis based uo Schur-cocavity ad was ivestigated i the aer of Haageru [57] 98 i which the value of the costat B has bee obtaied by direct (rather comlicated) calculatios It is iterestig to observe that with! : r B e About the costats A the followig is kow: (89) A = 8 >< >: where is the root of the equatio: + = == ; < + = ; ; ; < ; < < ( = :8474 ) Note that the case is trivial agai The case = was ivestigated by Szarek i the aer [4] 978 (I Hall [58] 975 the history of this roblem is described; esecially that 34

35 A equals to = was cojectured by Littlewood) The case ]; [ [ ]; [ has bee ivestigated by Haageru [57] i 98 His method (like i the case of costat B for ]; 3[) is of a comutatioal character ad it is ot see ca oe exlai his results by basig them uo some geeral cocet of tye of the covexity i the sese of Schur 9 The iequalities of Khitchie i the (exoetial) Orlicz saces I the first art of this sectio we demostrate how the kowledge about the best costats i the (classical) iequalities of Khitchie eables oe to derive the aalogue of such iequalities (with best costats) i the case of some saces of Orlicz Let us recall a few of eeded facts A fuctio = (x) defied o R + ad with values i R + is called a Youg fuctio if it is covex icreasig () = ad () = For such a Youg fuctio defie the Orlicz sace L (P ) as a vector sace of all radom variables = (!) defied o (; F ; P ) ad satisfyig for some C > the roerty: jj (9) < I the sace L (P ) oe itroduces the Orlicz orm: C (9) kk = if 8 C > : jj=c 9 with resect to which the sace is becomig a Baach sace If (x) = x ; < the L (P ) coicides with the sace L (P ) Aother iterestig examle is rovided by the exoetial fuctio: (x) = ex x < The cosideratio give below cocerig the case = will relate the questio o how to comute the orm: (93) ja k j = a k " k P P Deote S = a k" k ; A = ja kj I accordace with (9) for fidig the orm we are iterested i it is ecessary to cosider the quatity: js j js = ex j C A C A ad to fid such a miimal C for which this quatity is less or equal to By force of Taylor exasio ad kowledge about the values of the best costats B = m ; m i the iequalities of Khitchie we fid that: for 35