A Note on Sums of Independent Random Variables

Transcription

1 Cotemorary Mathematics Volume 00 XXXX A Note o Sums of Ideedet Radom Variables Pawe l Hitczeko ad Stehe Motgomery-Smith Abstract I this ote a two sided boud o the tail robability of sums of ideedet ad either symmetric or oegative radom variables is obtaied We utilize a recet result by Lata la o bouds o momets of such sums We also give a ew roof of Lata la s result for oegative radom variables ad imrove oe of the costats i his iequality 1 Itroductio Recetly Lata la (1997) obtaied the followig remarkable result: for a sequece of radom variables (X ) ad 1 < defie the followig Orlicz orm (11) (X k ) = if{λ > 0 : E 1 + X /λ e } Lata la roved that e 1 (12) 2e 2 (X k) ( E X k ) 1/ e (Xk ) rovided (X ) are either symmetric or ositive ad i the first case 2 ad i the secod case 1 The mai ovelty here is the fact that cotrary to the classical iequalities the costats here are ideedet of Certai articular cases of Lata la s result had bee kow earlier (see eg Hitczeko (1993) Gluski ad Kwaień (1995) or Hitczeko Motgomery-Smith ad Oleszkiewicz (1997)) but they ca be easily deduced from Lata la s iequality Of course the ultimate goal is to obtai bouds o the tail robabilities for sums of radom variables Lata la s result romted us to ivestigate that roblem This rogram has bee comleted; our methods which are based o estimates for the decreasig rearragemet of a radom variable work i a rather geeral settig As a result we were able to obtai extesios of Lata la s result i various directios The details of that aroach will be reseted elsewhere The goal of this ote is quite differet; we will reset a very simle argumet that allows 1991 Mathematics Subject Classificatio Primary 60G50 60E15; Secodary 46E30 Key words ad hrases sums of ideedet radom variables tail distributios The first author was artially suorted by NSF grat DMS The secod author was artially suorted by NSF grat DMS ad by the Uiversity of Missouri Research Board 1 c XXXX America Mathematical Society

2 2 PAWE L HITCZENKO AND STEPHEN MONTGOMERY-SMITH oe to deduce tail bouds from Lata la s result As a matter of fact this aroach formally does ot really deed o Lata la s result but it requires a kowledge of his bouds o momets i order to be emloyed successfully We will also reset a short roof (based o decoulig techiques) of Lata la s result for o-egative radom variables Our roof gives a slightly better costat o the left-had side of (12) Our otatio is stadard; for a sequece (z k ) we let z = max z k The letters 1 k c ad C deote absolute costats whose values may chage from oe use to the ext We will write S = k=1 X k ad S = k=1 X k ad S = (E S ) 1/ 2 Tail estimates via momet estimates I this sectio we will to obtai two-sided estimates for tails of sums of ideedet radom variables For the sake of brevity we will cocetrate o symmetric radom variables although it will be clear that our argumets work for oegative radom variables as well I certai secial cases tail iequalities have bee obtaied from momet iequalities (see Gluski ad Kwaień (1995) Hitczeko ad Kwaień (1994) or Hitczeko Motgomery-Smith ad Oleszkiewicz (1997)) Also i the case of multiles of Rademacher radom variables two-sided estimates have bee obtaied by Motgomery ad Odlyzko (1988) ad Motgomery-Smith (1990) Theorem 21 There exist ositive costats c C α ad δ such that for all sequeces of ideedet symmetric radom variables (X ) ad for all t such that ( t 1 2 X i I( X i t) = 1 1/2 X i I( X i t) ) i=1 the followig holds: Let t be the least such that The we have the iequalities X i I( X i t) 2t (21) P( S > t) c { P(X > t) + ex( α t ) } ad (22) P( S > 4t) C { P(X > t) + ex( δ t ) } If t 1 2 X i I( X i t) 2 the P( S > t) c Proof For a give t let Y i = X i I( X i t) ad let s = j=1 Y j Notice that s is a cotiuous icreasig fuctio of ad that s 2 2t Hece either 2 t < ad s t = 2t or t = ad s 2t Let us start by rovig (21) It follows from Levy s iequality ad cotractio ricile (see eg Kwaień Woyczyński (1992 Proositios 112 ad 121)) that ad P(X > t) 2P( S > t) P( s > t) 2P( S > t)

3 A NOTE ON SUMS OF INDEPENDENT RANDOM VARIABLES 3 Hece (23) P( S > t) 1 4 ( ) P(X > t) + P( s > t) Now we ca see that if t = the the iequality is established I the case that t < we eed to obtai a lower estimate for the tail robability of a maximum of artial sums of uiformly bouded symmetric radom variables (Y i ) But for such radom variables the followig iequality is true (cf Hitczeko (1994)): for all q 1 we have s q C q } { s + Y q (24) C q { } s + t We also use the Paley-Zygmud iequality that states that for ay o-egative radom variable Z ad 0 < λ < 1 Sice t = 1 2 s t we have that P(Z > λez) (1 λ) 2 (EZ)2 EZ 2 P( s > t) P( s t > 2 t s t t ) (1 2 t ) 2 s 2t t s 2t 2 t It follows from (24) that the deomiator is o more tha Therefore we get the estimate C 2t { s t + t} 2t ( 3 2 C)2t s 2t t P( s t) (1 2 t ) 2 ( 3 2 C) t ex( α t ) which together with (23) gives (21) Iequality (22) is a easy cosequece of Chebyshev s iequality If t < the P( S > 4t) P(X > t) + P( s > 4t) P(X > t) + E s t P(X > t) + 2 t = P(X > t) + ex( δ t ) (4t) t If t = we use the same ideas oticig that P( s > 2t) = 0 Fially if t 1 2 s 2 we aly the cotractio ricile the Paley-Zygmud iequality ad (24) to get 2P( S > t) P( s 2 > 1 4 E s 2 ) 9 s s s C 4 ( s 2 + t) 4 which is bouded below by a uiversal costat

4 4 PAWE L HITCZENKO AND STEPHEN MONTGOMERY-SMITH Remark The above theorem allows us to aroximate tails of the sums of ideedet radom variables i terms of tails of the idividual summads This follows from the fact that i view of Lata la s result t ca be aroximated usig oly iformatio about margial distributios ad from the well kow iequality P( Xi > u) (25) 1 + P( X i > u) P( Xi > u) P(X > u) P( X i > u) which gives tails of X i terms of tails of idividual summads 3 Aother roof of Lata la s result for oegative rv s Here we ited to give aother roof of Lata la s formula cocerig S for oegative radom variables Theorem 31 Let (X ) be a sequece of ositive ideedet radom variables The for all 1 we have that (31) κ (X ) S (e 1) 1/ (X ) where (X ) is give by (11) ad κ is the ositive umber for which f(κ) = e where (2k + 1) k f(x) = x k k! k=0 Proof First ote that if (X ) 1 the sice 1+ X (1+X ) we have that S e 1 This roves the secod iequality i (31) To rove the first we use certai results cocerig decoulig These ideas aear ofte i the literature (usually i the cotext of mea-zero or symmetric radom variables see eg Kwaień ad Woyczyński (1992)) However sice we will eed cotrol of costats we cite the followig which is a secial case of de la Peña Motgomery- Smith ad Szulga (1994 Theorem 21) Lemma 32 Let (X ) be a sequece of real valued ideedet radom variables Let (X (l) ) be ideedet coies of (X ) for 1 l k Furthermore let f i1i k be elemets of a Baach sace such that f i1i k = 0 uless the i 1 i k are distict The for ay 1 we have that f i1i k X i1 X ik (2k + 1) k f i1i k X (1) i 1 X (k) i k i 1i k i 1i k Now let us fiish the roof of Theorem 31 Note that E 1 + X = (1 + X ) ad so by Mikowski s iequality we have that (1 + X ) 1 + X i1 X ik i 1< <i k k=1

5 A NOTE ON SUMS OF INDEPENDENT RANDOM VARIABLES 5 But if k 1 X i1 X ik i 1< <i k = 1 k! i 1 i k distict (2k + 1)k k! X i1 X ik i 1 i k distict X (1) i 1 X (k) i k (where (X (l) ) are ideedet coies of (X ) for 1 l < ) (2k + 1)k k! i 1 X (k) i k i 1i k X (1) Hece So if S κ the that is = (2k + 1)k S k k! (1 + X ) f( S ) (1 + X ) e (X ) 1 Remark Our costat i the secod iequality of (31) is essetially the same as Lata la s costat But i the first iequality our costat which may umerically be show to be about is slightly better tha Lata la s costat which is about Refereces [1] V de la Peña SJ Motgomery-Smith ad J Szulga Cotractio ad decoulig iequalities for multiliear forms ad U-statistics A of Probab 22 (1994) [2] ED Gluski ad S Kwaień Tail ad momet estimates for sums of ideedet radom variables with logarithmically cocave tails Studia Math 114 (1995) [3] P Hitczeko Domiatio iequality for martigale trasforms of Rademacher sequece Israel J Math 84 (1993) [4] P Hitczeko O a domiatio of sums of radom variables by sums of coditioally ideedet oes A Probab 22 (1994) [5] P Hitczeko ad S Kwaień O the Rademacher series Probability i Baach Saces Nie Sadbjerg Demark (J Hoffma-Jørgese J Kuelbs MB Marcus ed) Birkhäuser Bosto [6] P Hitczeko SJ Motgomery-Smith ad K Oleszkiewicz Momet iequalities for sums of certai ideedet symmetric radom variables Studia Math 123 (1997) [7] S Kwaień ad WA Woyczyński Radom Series ad Stochastic Itegrals Sigle ad Multile Birkhäuser Bosto 1992 [8] R Lata la Estimatio of momets of sums of ideedet radom variables A Probab 25 (1997)

6 6 PAWE L HITCZENKO AND STEPHEN MONTGOMERY-SMITH [9] HL Motgomery ad AM Odlyzko Large deviatios of sums of ideedet radom variables Acta Arithmetica 49 (1988) [10] SJ Motgomery-Smith The distributio of Rademacher sum Proc Amer Math Soc 109 (1990) Deartmet of Mathematics North Carolia State Uiversity Raleigh NC address: awel@mathcsuedu URL: htt://mathcsuedu/~awel/ Deartmet of Mathematics Uiversity of Missouri Columbia Columbia MO address: stehe@mathmissouriedu URL: htt://mathmissouriedu/~stehe