An almost sure invariance principle for trimmed sums of random vectors

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1 Proc. Idia Acad. Sci. Math. Sci. Vol. 20, No. 5, November 200, pp Idia Academy of Scieces A almost sure ivariace priciple for trimmed sums of radom vectors KE-ANG FU School of Statistics ad Mathematics, Zhejiag Gogshag Uiversity, Hagzhou 3008, Chia statzju@tom.com MS received 3 August 2009 Abstract. Let {X ; } be a sequece of idepedet ad idetically distributed radom vectors i R p with Euclidea orm, ad let X r = X m if X m is the r-th imum of { X k ; k }. Defie S = k X k ad r S = S X + +Xr. I this paper a geeralized strog ivariace priciple for the trimmed sums r S is derived. Keywords. Trimmed sums; strog ivariace priciple; law of the iterated logarithm.. Itroductio ad mai results Let {X, X ; } be a sequece of idepedet ad idetically distributed i.i.d. radom vectors i R p, ad set S = X k,. As usual, write Lt = logt e ad LLt = LLt, t 0. For iteger r 0 ad r, let X r imum of { X k ; k } 0 if r>, ad let r S = S X = X m if X m is the r-th + +X r if r be the trimmed sums, where meas the Euclidea orm o R p. Obviously, 0 S is just S. Also we deote by L p,q the space of all radom vectors ξ such that J p,q := 0 t p P ξ >t q/p t dt<. It is readily see that L p,q L p,q2 if q q 2 ad lim t t p P ξ >t q/p = 0if ξ L p,q. For the partial sums S, if the radom vector X has mea zero ad a fiite covariace matrix, may classical limit theorems icludig cetral limit theorem, Dosker s weak ivariace priciple ad Strasse s strog ivariace priciple have bee ivestigated. Recetly, Eimahl [3] established a ew strog ivariace priciple for S, where E X 2 might be ifiite. This puts a totally ew couteace upo limit theorems i R p. For trimmed sums r S, the laws of the iterated logarithm ad strog approximatio theorems were also derived by Zhag [7, 8] i real ad Baach space settig, respectively. However, the fiiteess of the secod momet was required. I this paper, we aim to establish a geeral strog ivariace priciple for the trimmed sums of idepedet radom vectors, ad further a geeral law of the iterated logarithm LIL, where the variace may be ifiite. This exteds the results of Zhag [7] ad Eimahl [2]. 6

2 62 Ke-Ag Fu To help formulate our results, we eed some extra otatios. Let the fuctio cx or the sequece c = c be a sequece of real positive umbers satisfyig the followig coditios: α /3, c / α,. ε>0 m ε > 0: c /c m /m τ, qm qm ε,.2 where τ = + r κ for some 0 <κ<ad C is a positive costat. Obviously,.2 coicides with 2.3 of Eimahl [3] as r = 0. For the choice of c, oe ca refer to Eimahl ad Li [4] for more details. Let X be a radom vector, ad Fx = P X x. Set Bx = ivcx := sup{t >0: ct<x}, ϕx = Bx/Fx ad φx = ivϕx. Recall the partial sum process S := S [t] + t [t]s [t]+, where [x] meas the iteger part of x, ad similarly, 0, if t = 0; r r S m, for t = m/; S t = Lagrage iterpolatio over 0 t,m / t m/, where m. The we have the followig strog approximatio theorem. Theorem.. Let {X, X ; } be a sequece of i.i.d. mea zero radom vectors i R p, ad r 0. Assume that ad B X L,r+,.3 c 3 E X 3 I X c <..4 The without chagig its distributio, oe ca redefie the sequece {X ; } o a richer probability space together with a d-dimesioal stadard Browia motio {Wt; t 0} such that as, r S W =oc a.s., where =sup 0 t, the sequece c satisfies coditios. ad.2, W = Wt, 0 t, ad 2 = EXi X j I X c p i,j=. Now write A := sup{ A ν : ν }, for ay p, p-matrix A. We say that A 2 is equal to the largest eigevalue of the symmetric matrix A t A, due to the fact that the largest eigevalue C of a positive semidefiite, symmetric p, p-matrix C satisfies C = sup{ ν, Cν : ν }, where, is the stadard scalar product o R p. Deote Ht := sup{e ν, X 2 I X t: ν }.5

3 Trimmed sums of radom vectors 63 for ay t>0, ad from Eimahl [3] it follows 2 = Hc. Set { α 0 = sup α 0: exp α2 c 2 } =, 2H c ad the we have a corollary as a applicatio of Theorem.. COROLLARY.2 Let {X, X ; } be a sequece of i.i.d. mea zero radom vectors i R p, ad r 0. Assume that coditios.3 ad.4 hold. The we have with probability oe, lim sup r S c = α 0, where the sequece c satisfies coditios. with α = /2 ad.2. Remark.. It is obvious that Theorem. ad Corollary.2 do ot require the variace of X is fiite. By takig r = 0, it readily implies Theorem 2. ad Corollary 2.4 of Eimahl [3]. The proof of Corollary.2 is similar to the proof of Corollary 2.4 i Eimahl [3], ad hece we omit it. 2. Proof of Theorem. I this sectio, we will preset the proof of Theorem.. I the sequel, let {X, X ; } be a sequece of i.i.d. mea zero radom vectors i R p uless metioed otherwise, ad let C deote a positive costat which may take differet values i its differet appearace. We use the otatio a b if ad oly if a /b, as. The proofs of Theorem. will be accomplished through the followig lemmas. Lemma 2.. B X L,r+ if ad oly if 0 t r F r+ εc t dt < ε>0. Ad if B X L,r+, the for k>2r + 2 ad ay δ>0, 0 t k F k φδtdt < ad for Q large eough say Q>6r + 6, Proof. See Zhag [7, 8]. 0 t φt/ct Q dt<. Remark 2.. From Lemma 2. we ca coclude that B X L,r+ implies BxFx 0,xFx 0, φx/cx 0, ψx/bx ad xfφx 0 as x. I fact, from Zhag [7, 8], oe ca get a aalog of Lemma 2. for the multidimesioal idices case.

4 64 Ke-Ag Fu Lemma 2.2. If B X L,r+, the for ay t>0, we have that as, E X t I φ X c = oc t /. 2. Additioally, if c /c m /m τ, where τ = + r κ for some 0 <κ< ad C is a positive costat, the we have that as, E X I X φ = oc /, 2.2 E X I X φk = oc. 2.3 Proof. Equatio 2.2 follows from the same lies i Liu ad Li [5] for radom variables. As for eq. 2., ote that from Remark 2., we have c t E X t I φ X c Fφ = o. For eq. 2.3, recall that c satisfies coditios. ad.2. Due to eq. 2.2, we have as, c E X I X φk = o c k /kc = o k α / = o. Lemma 2.3. Let X i, i be idepedet mea zero radom vectors o R p such that E X i <, i. Let x be fixed. If the uderlyig probability space is rich eough, oe ca costruct idepedet ormal 0,I-distributed radom vectors Y i, i such that P X i A i Y i x E X i 3 x 3, k where A i is the positive semidefiite ad symmetric matrix satisfyig A 2 i = CovX i, i ad C is a positive costat depedig o p oly. Proof. See Eimahl [3], which is based o the work of Sakhaeko [6]. We ow start to prove Theorem.. Proof of Theorem.. Let θ> ad θ i deote [θ i ]. Set S, = k X k I X k εc, ε > 0, S 2, = k X k I X k φ, ad S 3, = k X k I X k φk.

5 Trimmed sums of radom vectors 65 Observe that for ay ε>0, it follows from Lemma 2. that P m θ r S m S,m rεc θ P X r+ θ εc θ θ Fεc θ r+ Fεc r+ <, P S m θ,m S 2,m ε2r + 3c θ P { X k φθ ; k θ } 2r + 3 θ Fφθ 2r+3 Fφ 2r+3 <. Also from the Rosethal iequality ad Lemmas 2. ad 2.2, it follows that for Q large eough P S m θ 2,m S 3,m 2εc θ P X m I φm < X m φθ 2εc θ m θ P { X m I φm < X m φθ m θ E X m I φm < X m φθ } εc θ + C + C m θ E X m 2 I φm < X m φθ c 2 θ Q/2 m θ E X m Q I φm < X m φθ c Q θ φθ Q/2 m θ E X m I φm < X m φθ c θ c θ Q/2 φθ Q m θ E X m I φm < X m φθ c θ c θ φθ Q/2 φ Q/2 <. c θ c

6 66 Ke-Ag Fu Hece, coupled with the Borel Catelli lemma ad a stadard argumet, it leads to that with probability oe ad m r S m S,m =oc, S,m S 2,m =oc, m S 2,m S 3,m =oc, m as. Thus i order to complete the proof, we oly eed to demostrate that with probability oe S 3, ES 3, W =oc. Recall that S 3, is the partial sum of idepedet radom variables Z k := X k I X k φk, k, ad the for ay ε>0, P m m S 3,m ES 3,m A k Y k εc c 3 E Z k 3, 2.4 where {Y ; } is a sequece of i.i.d. stadard ormal radom vectors ad A 2 k = CovX k I X k φk. By applyig 2.4, the right had side of 2.4 is ot more tha C c 3 E X k 3 I X k φ = C c 3 E X 3 I X φ c 3 E X 3 I X c <. Thus a applicatio of Borel Catelli lemma ad a stadard argumet etails m S 3,m ES 3,m A k Y k = oc a.s. 2.5 m Note that for ay ν R p, ad ν, 2 A2 ν =E X, ν I φ X c 2 2 A2 = sup E X, ν I φ X c 2 E X 2 I φ X c. ν

7 Trimmed sums of radom vectors 67 The by applyig the iequality exp x x Q for x>0 ad Q large eough with Lemma 2., we have P k i A i Y i εc exp ε 2 c 2 2p i A i 2 exp ε 2 c 2 4pE X 2 I φ X c FφQ <, where we have used the fact that A 2 2 A2 from Theorem X.. i Bhatia []. This immediately yields P k θ i A i Y i εc θ + <, ad further via the Borel Catelli lemma, leads to i A i Y i = oc a.s. 2.6 k Now recall that exp /x + y exp /2x + exp /2y for x,y > 0, ad take a costat D satisfyig Dε 2 /p > 0α 2 0 ad D = [/D]. The by Lemma 2. with the defiitio of α 0, ad observig that i 2 E X 2 Iδ c X c for some δ > 0, we ca similarly get P k i Y i εc exp ε 2 c 2 2p i 2 exp ε 2 c 2 4p D i 2 + C exp ε 2 c 2 4p i=d + i 2

8 68 Ke-Ag Fu + C exp Dε2 c 2 4p 2 exp ε 2 c 2 4pE X 2 Iδ c X c exp Dε2 c 2 4p 2 + C Fδ c r+ <, which further guaratees that with probability oe i Y i = oc. 2.7 k Also from the same lies of Eimahl [3], we ca coclude that Y i Wk = O log a.s., k where {Wt; t 0} is a stadard d-dimesioal Browia motio, ad this, of course, leads to Y i Wk = O log = oc a.s. 2.8 k Combiig relatios , the desired result follows immediately. Ackowledgemet This project is supported by the Natioal Natural Sciece Foudatio of Chia Nos & , the Itroductio Talet Foudatio of Zhejiag Gogshag Uiversity 020XJ20096 ad the Research Grat of Zhejiag Gogshag Uiversity X0-26. Refereces [] Bhatia R, Matrix Aalysis New York: Spriger 997 [2] Eimahl U, A geeralizatio of Strasse s fuctioal LIL, J. Theor. Probab [3] Eimahl U, A ew strog ivariace priciple for sums of idepedet radom vectors, Zap. Nauch Sem S-Peterburg Otdel Mat Ist Steklov POMI [4] Eimahl U ad Li D L, Some results o two-sided LIL behavior, A. Probab [5] Liu W D ad Li Z Y, Some LIL type results o the partial sums ad trimmed sums with multidimesioal idices, Elect. Comm. Probab [6] Sakhaeko A I, A ew way to obtai estimates i the ivariace priciple, i: High dimesioal probability II, Progress i Probability Birkhäuser-Bosto 2000 vol. 47, pp [7] Zhag L X, Strog approximatio theorems for sums of radom variables whe extreme terms are excluded, Acta Math. Siica, Eglish Series [8] Zhag L X, LIL ad approximatio of rectagular sums of B-valued radom variables whe extreme terms are excluded, Acta Math. Siica, Eglish Series