Moment estimates for chaoses generated by symmetric random variables with logarithmically convex tails

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1 Moment estmtes for choses generted by symmetrc rndom vrbles wth logrthmclly convex tls Konrd Kolesko Rf l Lt l Abstrct We derve two-sded estmtes for rndom multlner forms (rndom choses) generted by ndeendent symmetrc rndom vrbles wth logrthmclly concve tls. Estmtes re exct u to multlctve constnts deendng only on the order of chos. Keywords: Polynoml choses; Tl nd moment estmtes; Logrthmclly convex tls. AMS MSC 2010: 60E15 1 Introducton nd Mn Results. In ths er we study homogeneous choses of order d,.e. rndom vrbles of the form S = 1,..., d =1 1,..., d X 1 X d, (1) where X 1,..., X n re ndeendent rndom vrbles nd ( 1,..., d ) s multndexed symmetrc rry of rel numbers such tht 1,..., d = 0 whenever k = l for some k l. Choses of order d = 1 re just sums of ndeendent r.v s. There re numerous clsscl results rovdng bounds for moments nd tls of S n ths cse (such s Khntchne, Rosenthl, Bernsten, Hoeffdng, Prokhorov, Bennett nequltes, to nme few). However the stuton s more delcte f one looks for two-sded estmtes. In [7] two-sded bounds for L -norms S = (E S ) were found n qute generl stuton. Nmely, for ny 2 nd men zero r.v s X determnstc functon f ( 1,..., n ) ws constructed such tht 1 C f ( 1,..., n ) X Cf ( 1,..., n ), The er ws rered whle the uthor held ost-doctorl oston t Wrsw Center of Mthemtcs nd Comuter Scence. Reserch suorted by the NCN grnt DEC-2012/05/B/ST1/00692 Reserch suorted by the NCN grnt DEC-2012/05/B/ST1/

2 where C s unversl constnt. Strctly relted queston concernng two-sded bounds for tls of S ws treted n [5]. The cse d 2 s much less understood. In [8] two-sded estmtes for moments of Gussn choses were found. In [1] moment bounds were estblshed n the cse when d 3 nd X re symmetrc wth logrthmclly concve tls nd for choses of rbtrry order generted by symmetrc exonentl r.v s. The mn urose of ths note s to study the cse of symmetrc vrbles wth logrthmclly convex tls,.e. r.v s such tht functons t ln P( X t) re convex on [0, ). The clss of vrbles wth logconvex tls ncludes n rtculr vrbles wth exonentl nd hevy-tled Webull dstrbutons. Observe tht symmetrc exonentl rndom vrbles re n both clsses (logrthmclly concve nd logrthmclly convex tls) nd methods descrbed n ths note llow to vod mny techncl clcultons resented n [1] for ths rtculr cse (cf. Theorem 1 nd Remrk 3 wth Theorem 3.4 n [1]). One of the mn tools n the study of rndom choses s the decoulng technque (cf. the monogrh [2] for ts vrous lctons). It sttes tht the symtotc behvour of the homogeneous chos S s the sme s of ts decouled counterrt S defned by the formul S := 1,..., d X 1 1 X d d, 1,..., d =1 where (X j ) 1 n, j = 1,..., d re ndeendent coes of the sequence (X ) n. In rtculr the result of Kweń [6] (see lso [3] for ts more generl verson) sys tht for ny symmetrc multndexed mtrx ( 1,..., d ) such tht 1,..., d = 0 whenever k = l for some k l nd ny 1, 1 C(d) S S C(d) S, (2) where C(d) s ostve constnt, whch deends only on d. Before we stte mn results we need to ntroduce some notton. By C (res. C(d)) we denote ostve unversl constnts (ostve constnts deendng only on the rmeter d). In ll cses vlues of constnts my dffer t ech occurrence. To smlfy the notton we wrte A B (A B) f 1 C A B CA ( 1 C(d) A B C(d)A res.). The th moment of rndom vrble X s X = E X. For {1,..., n} d nd I [d] := {1,..., d} we wrte := ( k ) k I. For I [d] by P(I) we denote the fmly of ll rttons of I nto rwse dsjont subsets. If J = {I 1,..., I k } P(I) nd ( ) s multndexed mtrx we set ( ) J = ( ) I J := su Now we re redy to stte our mn theorem. k l=1 x Il : l x 2 l 1, 1 l k. 2

3 Theorem 1. Let (X j ) n,j d be ndeendent symmetrc r.v s wth logrthmclly convex tls such tht E X j 2 = 1 for ll, j. For ny multndexed mtrx ( ) nd ny 2 we hve X 1 1 X d d J /2 ( ) I c J I [d] J P(I c ) J /2 I ( ) I c J Remrk 1. Theorem 1 (nd (2)) my be used to obtn two-sded estmtes for moments of nonhomogeneous choses of the form S = d j=0 1,..., j =1 j 1,..., j X 1 X j, where 0 s number nd (j 1,..., j ) 1,..., j, j = 1,..., d re multndexed mtrces such tht j 1,..., j = 0 f k = l for some 1 k < l j. Indeed, n ths cse we hve (cf. [6, Lemm 2] or [1, Prooston 1.2]) 1 d C(d) j d 1,..., j X 1 X j S j 1,..., j X 1 X j. j=0 1,..., j =1 j=0 1,..., j =1 Remrk 2. Theorem 1 yelds lso tl bounds for choses bsed on symmetrc r.v s wth logrthmclly convex tls. Obvously we hve P( S e S ) e. Moreover, f S 2 λ S for ll 2 then by the Pley-Zygmund nequlty t s not hrd to show tht P( S S /C(λ)) e for (λ). Observe lso tht f X 2 µ X for ll then, by Theorem 1, S 2 C(d)µ d S. Remrk 3. Suose tht functons X j grow t most exonentlly,.e there exst constnts α, β such tht Then for 2, X j αe β for ll n, j d nd 2. J /2 I,α,β ( ) I c J J /2 mx ( ) I c J.. 3

4 To see ths, observe frst tht for ny γ 0 nd 3, x l x 2/ x 1 2/ l 2 2 l = e γ x 2/ e 2γ l 2 e 2γ 2 x e γ x l x 1 2/ l l e γ x l 2 + e 6γ x l. Therefore x l C(γ) ( e γ ) x l 2 + x l for 2. (3) Fx I [d] nd J P(I c ). We hve J /2 e d/2 e C(d) nd ( ) I c J 2 Hence (3), led wth γ = dβ + 1, yelds J /2 I ( ) I c J X j C(d, α, β) α 2 I e 2β I c ( ) {[d]} + J /2 mx 2 = α2 I e 2β I ( ) 2 {[d]}. ( ) I c J X j. Exmle 1. Observe tht ( j ) {1,2} s the Hlbert-Schmdt nd ( j ) {1},{2} the oertor norm of mtrx ( j ). Thus (2) nd Theorem 1 yeld the followng two-sded estmte for choses of order two (( j ) s symmetrc mtrx wth zero dgonl),,j X X j,j X 1 X 2 j,j=1,j=1 (,j ) o + 1/2 (,j ) HS + 1/2 X j +,j X X j,j. 2 j /2 Exmle 2. If X hve symmetrc exonentl dstrbuton wth the densty 1 2 e x then 4

5 X = Γ( + 1) nd we obtn by Theorem 1 nd Remrk 3 X 1 1 X d d I + J /2 I ( ) I c J I + J /2 mx ( ) I c J. Exmle 3. If X hve symmetrc Webull dstrbuton wth scle rmeter 1 nd she rmeter r (0, 1],.e. P( X t) = ex( t r ) for t 0 then X = Γ(/r + 1) nd X 1 1 X d d ( ) I Γ r + 1 J /2 I However by Strlng s formul Γ(/r + 1) r 1/r, hence X 1 1 X d d,r,r where the lst estmte follows from Remrk 3. I /r+ J /2 I ( ) I c J ( ) I c J I /r+ J /2 mx ( ) I c J, Theorem 1 my be used to derve uer moment nd tl bounds for choses bsed on vrbles whose moments re domnted by moments of vrbles wth logconvex tls. Here s smle result n such drecton. Corollry 2. Let r (0, 1], A < nd suose tht (X j ) n,j d re ndeendent centred r.v s such tht X j A 1/r for ll, j nd 2. For ny multndexed mtrx ( ) nd ny 2 we hve X 1 1 X d d C(r, d)a d Moreover for t > 0, ( ) P X 1 1 X d d t ( 2 ex mn mn ( I /r+ J /2 mx ( ) I c J. (4) t C (r, d)a d mx I ( ) I c J ) 2r ) 2 I +r J. (5). 5

6 2 Proofs The followng result ws estblshed by Htczenko, Montgomery-Smth nd Oleszkewcz [4] (see [7, Exmle 3.3] for smler roof). Theorem 3. Let X be ndeendent symmetrc r.v s wth logrthmclly convex tls. Then for ny 2, ( ) ( ) 1/2 X E X + EX 2. Snce g for 2 nd stndrd norml N (0, 1) r.v. g we mmedtely get the followng corollry. Corollry 4. Let X be ndeendent symmetrc r.v s wth logrthmclly convex tls nd vrnce one nd let g be..d. stndrd norml N (0, 1) r.v s. Then for ny sclrs, ( ) X E X + g. Remrk 4. Both Theorem 3 nd Corollry 4 my be vewed s some versons of the Rosenthl nequlty. Note however tht, contrry to Rosenthl s bound, rovded estmtes re shr u to unversl constnts tht do not deend on. Prooston 5. Let (X j ) n,j d be s n Theorem 1 nd (g j ) n,j d be..d. stndrd norml N (0, 1) r.v s. For ny multndexed mtrx ( ) nd ny 2 we hve X 1 1 X d d I [d] c c g j j. (6) Proof. We roceed by nducton wth resect to d. For d = 1 the estmte follows by Corollry 4. To show the nducton ste ssume tht d > 1 nd the bound holds for d 1. By Corollry 4 led condtonlly we get X 1 1 X d d X 1 1 X d 1 d 1 g d d + d X 1 1 X d 1 d 1 [d 1] X d d. (7) 6

7 The condtonl lcton of the nducton ssumton yelds X 1 1 X d 1 d 1 g d d d I [d 1] c c g j j (8) nd for ny d, X 1 1 X d 1 d 1 [d 1] Estmtes (7) (9) mly (6). I [d 1] g j j [d 1]\I j [d 1]\I. (9) Theorem 1 mmedtely follows by Prooston 5 nd the followng two-sded bound for moments of Gussn choses [8]. Theorem 6. For ny d nd 2 we hve g 1 1 g d d J P([d]) J /2 ( ) J. Proof of Corollry 2. Obvously t s enough to show (4) for = 2l, l = 1, 2,.... Let Y j, 1, j = 1,..., d be..d. symmetrc Webull r.v s wth scle rmeter 1 nd she rmeter r nd let (ε ) be..d. symmetrc ±1 r.v s, ndeendent of the sequence (X j ). We hve (see Exmle 3) X j = ε X j A 1/r C(r)A Y j for ll 2. Thus, for ny ostve nteger l nd ny sclrs (b ), b X j 2 b ε X j 2C(r)A 2l 2l b Y j, 2l where the frst nequlty follows by the stndrd symmetrzton rgument (cf. [2, Lemm 1.2.6]). Esy nducton shows tht X 1 1 X d d (2C(r)A) 2l d Y 1 1 Y d d 2l nd (4) (for = 2l) follows by Exmle 3. The tl bound (5) follows by (4) nd Chebyshev s nequlty. 7

8 References [1] R. Admczk nd R. Lt l, Tl nd moment estmtes for choses generted by symmetrc rndom vrbles wth logrthmclly concve tls, Ann. Inst. Henr Poncré Probb. Stt. 48 (2012), [2] V. H. de l Peñ nd E. Gné, Decoulng: From Deendence to Indeendence, Srnger, New York, [3] V. H. de l Peñ nd S. J. Montgomery-Smth, Decoulng nequltes for the tl robbltes of multvrte U-sttstcs, Ann. Probb. 23 (1995) [4] P. Htczenko, S. J. Montgomery-Smth nd K. Oleszkewcz, Moment nequltes for sums of certn ndeendent symmetrc rndom vrbles, Stud Mth. 123 (1997), [5] P. Htczenko nd S. J. Montgomery-Smth, Mesurng the mgntude of sums of ndeendent rndom vrbles, Ann. Probb. 29 (2001), [6] S. Kweń, Decoulng nequltes for olynoml chos, Ann. Probb. 15 (1987), [7] R. Lt l, Estmton of moments of sums of ndeendent rel rndom vrbles, Ann. Probb. 25 (1997), [8] R. Lt l, Estmtes of moments nd tls of Gussn choses, Ann. Probb. 34 (2006), Konrd Kolesko Instytut Mtemtyczny Unwersytet Wroc lwsk Pl. Grunwldzk 2/ Wroc lw, Polnd kolesko@mth.un.wroc.l Rf l Lt l Insttute of Mthemtcs Unversty of Wrsw Bnch Wrszw, Polnd rltl@mmuw.edu.l 8