Optimal Constants in the Rosenthal Inequality for Random Variables with Zero Odd Moments.

Transcription

1 Optma Costats the Rosetha Iequaty for Radom Varabes wth Zero Odd Momets. The Harvard commuty has made ths artce opey avaabe. Pease share how ths access beefts you. Your story matters Ctato Ibragmov, Rustam ad Marat Ibragmov Optma costats the Rosetha equaty for radom varabes wth zero odd momets. Statstcs ad Probabty Letters 78(): Pubshed Verso Ctabe Terms of Use Ths artce was dowoaded from Harvard Uversty s DASH repostory, ad s made avaabe uder the terms ad codtos appcabe to Other Posted Matera, as set forth at rs.harvard.edu/ur-3:hul.istrepos:dash.curret.terms-ofuse#laa

2 OPTIMAL CONSTANTS IN THE ROSENTHAL INEQUALITY FOR RANDOM VARIABLES WITH ZERO ODD MOMENTS Marat Ibragmov Departmet of Probabty Theory, Tashet State Ecoomcs Uversty Rustam Ibragmov Departmet of Ecoomcs, Harvard Uversty Abstract. We obta estmates for the best costat the Rosetha equaty m E ξ C m Eξ E ( )max, ξ = = = for depedet radom varabes ξ,...,ξ m wth zero frst odd momets,. The estmates are sharp the extrema cases = ad =m, that s, the cases of radom varabes wth zero mea ad radom varabes wth m zero frst odd momets. Rosetha (970) proved the foowg equaty: E = t t / t ξ C( t )max E ξ, Eξ () = = The authors are gratefu to a aoymous referee for hepfu commets ad suggestos. Correspodg author. Departmet of Ecoomcs, Harvard Uversty, Lttauer Ceter, 805 Cambrdge St., Cambrdge MA 038. Phoe: (67) Fax: (67) Ema: rbragm@fas.harvard.edu

3 for a postve tegers ad a depedet radom varabes (r.v. s) ξ,...,ξ wth Eξ = 0, t Eξ <, =,...,, t >, where C( t ) s a costat depedg oy o t. A umber of papers have focused o refemets ad extesos of equaty () ad reated probems (see Prohorov, 96; Sazoov, 974; Nagaev ad Pes, 977; Pes, 980, 994; Pes ad Utev, 984; Johso, Schechtma ad Z, 985; Utev, 985; Htczeo, 990, 994; Bestseaya ad Utev, 99, Ibragmov ad Sharahmetov, 995, 997, 00a, b; Fge, Htczeo, Johso, Schechtma ad Z, 997; Ibragmov, 997; de a Peña ad Gé, 999; ad de a Peña, Ibragmov ad Sharahmetov, 003). Fge et a. (997) ad Ibragmov ad Sharahmetov (995, 997) derved the foowg expressos for the best costat C * sym( t ) equaty () for symmetrc r.v. s: a Γ( a) = x e 0 x dx t / t + Γ * = + C ( t ) sym, <t<4, π * t C sym ( t ) Eθ θ =, t 4, where ad θ, θ are depedet Posso r.v. s wth parameter 0.5. The proof of the expressos for C * sym( t ) Ibragmov ad Sharahmetov (995, 997) rees o the wor by Utev (985), who obtaed, amog other resuts, sharp upper ad ower bouds o E = t ξ, t 4, where ξ,...,ξ are depedet symmetrc r.v. s wth fte tth momet, terms of E ξ = t ad = Eξ t /. Bestseaya ad Utev (99) derved a smar upper boud o eve momets of sums depedet mea-zero r.v. s ξ,...,ξ, from whch the best costat geera Rosetha s equaty () the case t= ca be deduced. Usg a dfferet proof techque, the expresso for the best costat geera equaty () for eve momets t= of sums of

4 3 mea-zero r.v. s was depedety obtaed Ibragmov ad Sharahmetov (00a). Ibragmov ad Sharahmetov (00b) obtaed the best costat the aaogue of equaty () for oegatve r.v. s. The resuts Ibragmov ad Sharahmetov (995, 997, 00a, b) were aso preseted Ibragmov (997). de a Peña et a. derved sharp aaogues of the Burhoder Rosetha equates ad reated estmates for the expectatos of fuctos of sums of depedet oegatve r.v. s ad codtoay symmetrc martgae dffereces wth bouded codtoa momets as we as for sums of mutear forms. The preset paper deas wth estmatg the best costats the Rosetha s equaty for r.v. s wth zero frst odd momets. Namey, et C * ( t ) deote the best costats equaty () for a postve tegers ad a depedet r.v.'s s ξ,...,ξ wth Eξ = 0, s =,,...,. The the foowg theorem hods. Theorem. If t=, m N, the j r j * ( m! ) C ( m ) ( )! j j = r = =!, () where the er sum s tae over a atura m >m >...>m r ad j,...,j r satsfyg the codtos m j +...+m r j r =, j +...+j r =j, m s, =,,..., r, s =,,...,.

5 4 Remar. The vaue j r j ( m! ) ( m )! equaty () has a smpe j j = r = =! combatora sese (e.g., Sachov, 996): t equas the umber of parttos of a set cosstg of eemets to parts the umber of eemets whch s ot equa to s, s =,,...,. Remar. As foows from the resuts Pes ad Utev (984), Bestseaya ad Utev (99) ad Ibragmov ad Sharahmetov (997, 00a), bouds () are sharp for = ad =m; addto, whe =m, the rght-had sde of () equas to the best costat C * sym( ) the Rosetha s equaty for symmetrc r.v. s. It s aso terestg to ote that, the case =0, the expresso o the rght-had sde of (), wth the er sum tae over a atura m>m >...>m r ad j,...,j r satsfyg the codtos m j +...+m r j r =, j +...+j r =j, equas to the best costat the aaogue of equaty () for oegatve r.v. s (see Ibragmov ad Sharahmetov, 00b). Smar to Remar, the atter expresso equas to the tota umber of parttos of a set cosstg of eemets (the -th Be umber). Let us formuate some auxary resuts eeded for the proof of Theorem. The foowg emma foows from Coroary Utev (985) ad the formua represetg momets by sem varats. Lemma. Let s ξ,..., ξ be depedet r.v. s wth Eξ = 0, s =,,...,. Set A = Eξ, = E = /, =,,...,, B = A. The foowg equaty hods:, r j j Am, ( m! ) ξ ( )!, (5) r= 0 = j!

6 B 5 where the er sum s tae over a atura m >m >...>m r ad j,...,j r, satsfyg the codtos m j +...+m r j r =, j +...+j r =j, m s, =,,..., r, s =,,...,. Let A, B, D>0. Deote M ( m, A, B) = sup ES, ξ m, where sup s tae over postve tegers ad a depedet r.v. s s ξ,...,ξ wth Eξ = 0, s =,,..., ad fxed A, = A, BB=B; M ( m, A, B ) = sup ES, ξ m, where sup s tae over postve tegers s ad a depedet r.v. s ξ,...,ξ wth Eξ = 0, s =,,...,, for whch A, A, B B; M ( m, D) = sup ES m, where sup s tae over postve tegers ad a depedet r.v. s, ξ s ξ,...,ξ wth Eξ = 0, s =,,...,, ad fxed max( A,, B ) = D. The foowg emma s we-ow (see, e.g., Pes ad Utev, 984). Lemma. For <s< m s ( m s) ( ) A A B s,, /( ). (6) Reato (5) ad Lemma mpy the foowg Lemma 3. For A, B>0

7 6 m M j r j (!) m A m B m j ( A m j m B m j /( m ) ( ) (,, ) ( )! ) j r!, = = = =,, where the er sum s tae over a atura m >m >...>m r ad j,...,j r, satsfyg the codtos m j +...+m r j r =, j +...+j r =j, m s, =,,..., r, s =,,...,. Proof of Theorem. From Lemma 3 ad the evdet equaty / m M ( m, D) M ( m, D, D ) t foows that M ( m, D) ( )! j j = r = r = ( m!) j j! D, where the er sum s tae over a atura m >m >...>m r ad j,..., j r, satsfyg the codtos m j +...+m r j r =, j +...+j r =j, m s, =,,..., r, s =,,...,. Sce * M ( m, D) C ( ) = sup, (7) D ths mpes (). D> 0

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