Best Constants in Kahane-Khintchine Inequalities for Complex Steinhaus Functions

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1 Proc. Amer. Math. Soc. Vol. 3, No. 0, 995, (30-3) Prerit Ser. No. 4, 993, Math. Ist. Aarhus Best Costats i Kahae-Khitchie Iequalities for Comlex Steihaus Fuctios GORAN PESKIR Let f' k g k be a sequece of ideedet radom variables uiformly distributed o [0; [, ad let k k deote the Orlicz orm iduced by the fuctio (x) = ex(jxj ) 0. The: i' k = jzk j for all z ;... ; z C ad all. The costat is show to be the best ossible. The method of roof relies uo a combiatorial argumet, Taylor exasio, ad the cetral limit theorem. The result is additioally stregtheed by showig that the uderlyig fuctios are Schur-cocave. The roof of this fact uses a result o multiomial distributio of Riott, ad Schur s roositio o sum of covex fuctios. The estimates obtaied throughout are show to be the best ossible. The result exteds ad geeralizes to rovide similar iequalities ad estimates for other Orlicz orms.. Itroductio Let f" i g i be a Beroulli sequece of radom variables defied o the robability sace (; F ; P ), ad let k k deote the Orlicz orm iduced by the fuctio (x) = ex(jxj )0. Thus, wheever is a radom variable defied o (; F ; P ), we have: 8 9 kk = if C > 0 j E (jj=c) with covetio (.) if ; = +. The: a i " i i= r 8 3 = ja i j i= 8=3 is the best ossible (see [3]). for all a ;... ; a R ad all. The costat The iequality (.) is called Kahae-Khitchie s iequality i (exoetial) Orlicz sace, ad is kow to have a umber of alicatios. The reset aer is motivated by the followig observatios. Cosider radom variables " i i (.). Recall that P f" i = 6g = =. Thus " i may be iterreted as uiformly distributed o the uit shere S = f0; +g i R. Now let f k g k be a sequece of ideedet radom variables uiformly distributed o the uit shere S i R. The the roblem aears worthy of cosideratio: Does the aalogue of (.) remai valid, ad what is the best ossible costat i this case? It is the urose of the aer to reset solutio for this roblem. AMS 980 subject classificatios. Primary 4A44, 4A50, 46E30, 60E5. Secodary 60G50. Key words ad hrases: Kahae-Khitchie iequality, comlex Steihaus sequece, Beroulli sequece, Schur-covex (-cocave), the gauge orm, Orlicz orm, Gaussia distributio, multiomial distributio. gora@imf.au.dk

2 I this cotext we fid it coveiet to relace R with the set of comlex umbers C. The k = e i' k for k, where ' k s are ideedet ad uiformly distributed o [0; [. The aalogue of (.) may be stated as follows: (.) k i' C = jz k j where z ;... ; z C ad. I order to rove (.) recall that (.) geeralizes to the form: (.3) r i 8 i= 3 k i k = i= where f i g i is ay sequece of ideedet symmetric a.s. bouded radom variables (see [3]). Notice that i'k jz k je i'k, thus there is o restrictio i (.) to assume that z k R + for all k. Combiig this fact with (.3), we obtai by triagle iequality: k i' for all z ;... ; z C = z k cos ' k + i z k si ' k r 8 3 = jz k j ad all. Thus (.) is valid with C = 8=3 = 3:6... However, it is clear that this costat is far from beig the best ossible i (.). Our mai aim i this aer is to reset a method of roof which establishes (.) with the best ossible costat C. We fid it useful here to clarify the mai oits of the aroach. For this cosider (.) with give ad fixed a;... ; a R ad. Deote S = P P i= a i"i, ad throughout assume that i= ja ij =. Note that (.) follows as soo as we obtai: js (.4) E j ex C with C = 8=3. Thus the roblem reduces to estimate the left side of (.4) i a otimal way. It turs out that the best estimate is as follows: js j (.5) jzj E ex C E ex C = C C 0 which is valid for all C > ad where Z N(0; ) is stadard Gaussia variable. Idetifyig C= C 0 =, oe gets C = 8=3 ad comletes the roof of (.). Probably the best uderstadig of (.5) may be obtaied through the cocet of Schur-covexity i the theory of majorizatio, which we fid istructive here to exlai i more detail. It should be oted that i this rocess we also clarify the reaso for which (.5) serves the best ossible costat i (.). First ote that exadig the itegrads i (.5) ito Taylor series, it suffices to show: (.6) EjSj k EjZj k for all k. This iequality follows by the cetral limit theorem from a itermediate fact which is by itself of theoretical iterest: (.7) The ma (x ;... ; x) 8 70! E i= xi "i k is Schur-cocave o R +. Roughly seakig, this meas that 8(x) domiates 8(y) wheever the comoets of vector x

3 are less sread out or more early equal tha the comoets of vector y. More recisely, if z 3... z3 deote the comoets of vector z i decreasig order, the (.7) meas: k (.7 ) x 3 i k yi 3 (k = ;... ; 0) & xi = yi ) 8(x) 8(y) i= i= i= i= wheever x; y R +. I articular, we obtai: (.8) E i= ai"i k E i= "i k + E + i= "i k for all k. Fially, assig to the limit i (.8), we get (.6) by the cetral limit theorem. Moreover, i this way we also see that iequality (.6) evetually becomes a equality ( with the choice ai = = for i = ;... ; ). The same fact carries over to (.5), ad exlais how (.5) serves the best ossible costat i (.). We remid that (.6) is due to Khitchie [8], while (.7) is due artially to Efro [], ad fully to Eato []. It should be oited out that two iterestig geeralizatios of these results may be ecoutered, where g(x) = jxj k for k is relaced by a wider class of fuctios (see []). For more iformatio about the theory of majorizatio ad its alicatios just idicated, we refer the reader to []. I articular, we fid it istructive to remid o Schur s coditio which characterizes Schur s covexity (cocavity) i terms of first artial derivatives (see [].57). We tur back to the origial roblem of fidig the otimal method for the roof of (.). To the best of our kowledge the theory of majorizatio described P above does ot suort our eeds exlicitly for the otimal (.). I articular, uttig S = k i' for give z ;... ; z P C such that jz kj =, othig seems i geeral to be kow about the aalogue of (.6) ad (.7) i this case, as well as about their geeralizatios to a wider class of fuctios as metioed above. It should be oted that for we have:! E k i' = E z k cos ' k + z k si ' k To coclude, we may otice that actually two searate roblems have aeared. The first oe is to rove (.) i a otimal way which will rovide the best ossible costat C. The secod ad more geeral oe is to rove the aalogue of (.6) ad (.7) for a wider class of fuctios as idicated above which will iclude g(x) = jxj for. I this aer we fid comlete solutio for the first roblem, ad artial solutio for the secod roblem which covers our eeds for the first roblem. The aroach makes o attemt to obtai a more geeral solutio for the secod roblem. It is left as worthy of cosideratio.. Prelimiary facts I this sectio we itroduce otatio ad collect facts eeded for the mai results i the ext sectio. Throughout we deote (x) = ex(jxj )0 ad work with the Orlicz orms: kk = if f C > 0 j E (jj=c) g kk T = if f C > 0 j E (jj=c) C g kk7 = E (jj) 3

4 where is a radom variable defied o the robability sace (; F ; P ). Recall that k k is called the gauge orm. We remark that the quatity kk T emerged i the study [5]. Its iterest relies uo the fact that for more geeral fuctios, the ma kk eed ot be a Fréchet orm, but kk T is so (see [5].7,8). The quatity kk 7 is of a itermediate value for both kk ad kk T. For more iformatio about the Orlicz orm just itroduced we refer to [5] ad [3]. We tur to the cocet of Schur-covexity i the theory of majorizatio. Let z 3... z3 deote the comoets of vector z R i decreasig order. Give x; y R, we say that x is majorized by y ad write x y, if the coditios are fulfilled: x i k = yi & xi 3 k yi 3 i= i= i= i= for all (.) (.) (.3) k = ;... ; 0. For istace, we have: ;... ; 0... ; ; ;... ; 0 ;0 0 ; ; 0;... ; 0 (; 0;... ; 0) (a ;... ; a ) (; 0;... ; 0) wheever a i P 0 ad 0 0P i= i x (;... ; ) (x;... ; x ) wheever x i 0. i= a i = Let D R be a set, ad let 8 : D! R be a fuctio. The 8 is said to be Schur-covex o D, if 8(x) 8(y) wheever x; y D ad x y. The ma 8 is said to be Schur-cocave, if 08 is Schur-covex. It is easily verified that if 8 is Schur-covex, ad D is symmetric, the 8 is symmetric as well. Moreover, if 8 is symmetric ad covex, the 8 is Schur-covex. The followig well-kow result combied with (.)-(.3) may rovide a lot of iterestig iequalities (for roof see [].64). Proositio. (Schur; Hardy-Litlewood-Pólya) If I R is a iterval, ad g : I! R is covex, the the fuctio: is Schur-covex o I. 8(x) = g(x i ) i= The ext fudametal theorem characterizes Schur-covexity (cocavity) i terms of first artial derivatives (for roof see [].57). Theorem. (Schur-Ostrowski) Let I R be a iterval, ad let 8 : I! R be cotiuously differetiable. The 8 is Schur-covex o I, if ad oly if the followig two coditios are satisfied: (.4) 8 is symmetric o I (.5) (x i 0x j (x) 0 for all x I ad all i 6= j. Moreover, wheever (.4) is satisfied, (.5) may be relaced by the coditio: (.5 ) (x 0 (x)0 (x) 0 for all 4

5 The same characterizatio remais valid for Schur-cocave fuctios 8 with iequalities (.5) ad (.5 ) beig reversed. The followig result o multiomial distributio of Riott is show to be useful. It might be roved i a straightforward way by verifyig Schur s coditio (.5 ) i Theorem. (see [5]). (.6) Let = ( ;... ; ) be a radom vector from the P multiomial distributio with arameters z = (z ;... ; z ) [0; ] ad, where i= z i =. I other words: P f = ;... ; = g =!!...! z... z P for ;... ; Z with + i= i =. If 9 is Schur-covex (-cocave), the the fuctio z 7! E z 9() is Schur-covex (-cocave). We coclude with a few facts o the comlex Steihaus sequece of radom variables. Let f' k g k be a sequece of ideedet radom variables uiformly distributed o [0; [. The fe i' k g k is a sequece of radom variables uiformly distributed o the uit shere S i C. We deote k = e i' k for k, ad the sequece f k g k is called a comlex Steihaus sequece. Note that E cos ' = E si ' = E cos ' i si ' j = 0 for i 6= j, ad E cos ' = E si ' = =. Thus by the two-dimesioal cetral limit theorem we have: (.7) e i'k as!, where Z Z N (0; =) 0! Z + iz (.8) f (Z ;Z ) (x; y) = ex(0x 0y ) are ideedet Gaussia variables with joit desity: for x + iy C. It should be recalled that C is toologically the same as R. Thus week covergece i (.7) coicides for both C ad R. From (.8) we get: (.9) E ex jz + iz j =C Z Z = 0 = 0x 0 ex + y =C Z 0 ex 0 0 C = E ex 0Z + Z for all C >. Idetifyig C =C 0 =, we obtai C = (.0) kz + iz k =. Similarly, from (.8) we get: (.) EjZ + iz j = E(Z + Z Z ) Z = 0 0 Z (x Z = 0 0 =C ex(0x 0y ) dx dy x dx = C 0 C = C 0 + y ) ex(0x 0 y ) dx dy r + ex(0r ) dr = Z 0. Thus we have: r + ex(0r ) dr =! 5

6 for all. Fially, to distiguish from the real case i (.) where E(" i ) = for i, it is iterestig to observe that E( k ) = 0, although Ej k j = for k. 3. Kahae-Khitchie iequalities for comlex Steihaus variables I this sectio we reset the mai results of the aer. We begi with the followig basic fact. Lemma 3. Let fe i'k g k be a comlex Steihaus sequece. The the iequality is satisfied: (3.) E k i'! jz k j for all z;... ; z C, ad all itegers,. The costat! is the best ossible. Proof. Sice 'k jz k je ' k for k, it is o restrictio to assume that the give umbers z ;... ; z belog to R + for. I order to clarify the combiatorial argumet i the geeral case below, we first verify (3.) for = =. For this, ote that we have: (3.) Ee i('j0' k+'l0'm) = ; if 0 ; otherwise (j; l) f(k; m); (m; k)g for all E k i' = E k i' 4 = E j= j; k; l; m. From this fact we obtai: = j= l= m= z j k) i('j0' l= m= zj zkzlzmee i('j0'k+'l0'm) = z 4 + 4z z + z4 z l z m e i(' l0' m) = z 4 = + z z + z4 = (z + z ) ad the roof of (3.) i this case is comlete. The geeral case follows from the aalogue of (3.) by the same combiatorial atter: (3.3) E zke k i' = (3.4) C ;...; = i0; +...+= C ;...; z beig valid for all, where i 0 ad P E k i'! =! i0; +...+= jz k j... z =!!...! i= i =. Combiig (3.3) ad (3.4) we get:!!...! z... z for all. Thus the roof of (3.) is comlete. For the last statemet, otice that by the cetral limit theorem (.7) with (.) we obtai: 6

7 E lim! e k i' = EjZ + izj =! for all. Thus the last statemet follows by uttig z =... = z = = i (3.), ad assig to the limit whe!. These facts comlete the roof. (3.5) Theorem 3. Let fe i' k g k for all z ;... ; z be a comlex Steihaus sequece. The the iequality is satisfied: C0P jz kj = sese described i the roof below). i'k7 C C 0 0 C, all, ad all C >. The estimate is the best ossible (i the Proof. It is o restrictio to assume that the give umbers z ;... ; z belog to R + for P, as well as that jz kj =. Deote S = P z ke i' k, the by (3.) ad Taylor exasio we get: (3.6) E ex js j C = =0 EjS j! C =0 C = C 0 C for all C >. Hece (3.5) follows straightforward by defiitio of the Orlicz orm k k 7. For the last statemet otice that uttig z =... = z = = i (3.6), we obtai by the cetral limit theorem (.7) with (.9): j (3.7) C lim C = C 0! E ex js for all C >. Hece we see that with this choice iequality (3.5) evetually becomes a equality. This fact comletes the roof. (3.8) Theorem 3.3 Let fe i'k g k be a comlex Steihaus sequece. The the iequality is satisfied: k i' for all z;... ; z C, ad all. The costat = jz k j is the best ossible. Proof. It is o restrictio to assume that the give umbers z ;... ; z belog to R + for P jz kj =. Idetifyig C =C 0 = i (3.6), we obtai C =., as well as that Thus (3.8) is satisfied, ad the roof of the first art is comlete. For the last statemet take z =... = z = = i (3.8), the by (3.7) we easily fid: lim! This fact comletes the roof. e i'k =. 7

8 (3.9) Corollary 3.4 Let fe i' k g k be a comlex Steihaus sequece. The the iequality is satisfied: k i' + ( 0 ) = jz k j for all z;... ; z C, all, ad all > 0. Proof. It follows from (3.8) by Jese s iequality for, ad the fact that x 7! x = is subadditive o R + for 0 < <. (3.0) Theorem 3.5 Let fe i' k g k be a comlex Steihaus sequece. The the iequality is satisfied: 0P + 5 jz kj = zke i'k T for all z ;... ; z C, ad all. The costat (+ 5)= is the best ossible. Proof. It is o restrictio to assume that the give umbers z ;... ; z belog to R + for P, as well as that jz kj =. Idetifyig C =C 0 = C i (3.6), we obtai C = (+ 5)=. Thus (3.0) is satisfied, ad the first art of the roof is comlete. For the last statemet take z =... = z = = i (3.0), the by (3.7) we easily fid: This fact comletes the roof. lim! e i'kt = + 5 The recedig results may be additioally stregtheed by the ext two facts which are also of iterest i themselves (recall (.)-(.3)). Theorem 3.6 Let fe i' k g k is Schur-cocave o R + be a comlex Steihaus sequece. The the fuctio: (jz j;... ; jz j) 70! E jzk j e i' k, for all, ad all.. Proof. Let the fuctio be deoted by 8. It is o restrictio to assume that 8 the set D of all (jz j;... ; jz j) R + such that P jz kj =. Put: is defied o 9(;... ; ) =!!...! for all ;... ; Z + with P i= i =. The by (3.3) ad (3.4) we fid: 8(jz j;... ; jz j) = E jzj 9() 8

9 where = (;... ; ) is a radom vector P from the multiomial distributio with arameters jzj = (jzj;... ; jzj) ad, where jz kj =. Thus by Riott s result (.6), the roof will be comleted as soo as we show that 9 is Schur-cocave. This is evidetly true if ad oly if log 9 is Schur-cocave. Notice that: log 9(;... ; ) = log! 0 log 0(k +) P for all ;... ; Z + with i= i =. Thus the roof will follow as soo as we show that: is Schur-covex o R + is kow to be covex o Corollary 3.7 Let fe i' k g k is Schur-cocave o R + (;... ; ) 7! log 0(k +). However, this follows by Schur s roositio., sice x 7! log 0(x) ]0; [. The roof is comlete. be a comlex Steihaus sequece. The the fuctio: (jz j;... ; jz j) 70! E ex, for all. jzk j e k i' Proof. It follows from Theorem 3.6 by Taylor exasio. Ackowledgmet. The author thaks I. Pielis for oitig out [] ad []. REFERENCES [] EATON, M. L. (970). A ote o symmetric Beroulli radom variables. A. Math. Statist. 4 (3-6). [] EFRON, B. (969). Studet s t-test uder symmetry coditios. J. Amer. Statist. Assoc. 64 (78-30). [3] GLUSKIN, E. D. PIETSCH, A. ad J. PUHL. (980). A geeralizatio of Khitchie s iequality ad its alicatio i the theory of oerator ideals. Studia Math. 67 (49-55). [4] HAAGERUP, U. (978-98). The best costats i the Khitchie iequality. Oerator algebras, ideals, ad their alicatios i theoretical hysics, Proc. It. Cof. Leizig (69-79). Studia Math. 70 (3-83). [5] HOFFMANN-JØRGENSEN, J. (99). Fuctio orms. Math. Ist. Aarhus, Prerit Series No. 40 (9 ). [6] JOHNSON, W. B. SCHECHTMAN, G. ad ZINN, J. (985). Best costats i momet iequalities for liear combiatios of ideedet ad exchageable radom variables. A. Probab. 3 (34-53). [7] KAHANE, J. P. ( ). Some Radom Series of Fuctios. D. C. Heath & Co. (first 9

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