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1 GLASNIK MATEMATIČKI Vol. 52(72)(207), THE DAVIS-GUT LAW FOR INDEPENDENT AND IDENTICALLY DISTRIBUTED BANACH SPACE VALUED RANDOM ELEMENTS Pigya Che, Migyag Zhag ad Adrew Rosalsky Jia Uversity, P. R. Chia ad Uiversity of Florida, USA Abstract. A aalog of the Davis-Gut law for a sequece of idepedet ad idetically distributed Baach space valued radom elemets is obtaied, which exteds the result of Li ad Rosalsky (A supplemet to the Davis-Gut law. J. Math. Aal. Appl. 330 (2007), ).. Itroductio Let {X,X, } be a sequece of idepedet ad idetically distributed radom variables. The followig theorem, which is related to the classical Hartma-Witer law of the iterated logarithm (see, Hartma ad Witer, [6]), is well kow. As usual we let logt = log e max{e,t} for t 0. (.) (.2) (.3) Theorem.. The followig three statemets are equivalet: EX = 0 ad EX 2 =, { P X k > (+ε) }{ <, if ε > 0 2loglog =, if ε < 0, loglog P { X k > (+ε) }{ <, if ε > 0 2loglog =, if ε < 0. This result is referred to as the Davis-Gut law. The implicatio (.) (.2) was formulated by Davis ([3]) with a ivalid proof which was corrected by Li et al. ([]). The implicatio (.2) (.) was obtaied 200 Mathematics Subject Classificatio. 60F5. Key words ad phrases. Davis-Gut law, law of the iterated logarithm, sequece of idepedet ad idetically distributed Baach space valued radom elemets. 35

2 352 P. CHEN, M. ZHANG AND A. ROSALSKY by Gut ([5]). The equivalece betwee (.) ad (.3) was established by Li ([9]). Necessary ad sufficiet coditios for (.3) i a Baach space settig were obtaied by Li ([9]). For movig average processes, the implicatios (.) (.2) ad (.) (.3) were obtaied by Che ad Wag ([]). Li ad Rosalsky ([0]) provided the followig supplemet to the Davis- Gut law. Whe h(t), it yields the equivalece betwee (.) ad (.2). Theorem.2. Let h( ) be a positive odecreasig fuctio o (0, ) such that (th(t)) dt =. Write Ψ(t) = t (sh(s)) ds,t. The (.) ad { (.4) h() P X k > (+ε) }{ <, if ε > 0 2logΨ() =, if ε < 0 are equivalet. Recetly, Liu et al. ([2]) exteded Theorem.2 to movig average processes which the exteds the work of Che ad Wag ([]) by establishig the implicatio (.2) (.) for movig average processes. I this paper, we will exted Theorem.2 for a sequece of idepedet ad idetically distributed Baach space valued radom elemets. 2. Prelimiaries ad Lemmas Let B be a real separable Baach space with orm ad let B deote the topological dual space of B. We let B deote the uit ball of B. Let (Ω,F,P) be a probability space. A radom elemet X takig values i B is defied as a F-measurable fuctio from (Ω,F) ito B equipped with the Borel sigma-algebra; we call it a B-valued radom elemet for short. The expected value or mea of a B-valued radom elemet X is defied to be the Bocher itegral ad is deoted by EX. Lemma 2.. Let {k, } be a sequece of positive itegers ad {X k, k k, } a array of rowwise idepedet B-valued radom elemets. Suppose that there exists δ > 0 such that X k δ a.s. for all k k,. If k X k 0 i probability, the E k X k 0 as. Proof. Let {X k, k k, } be a idepedet copy of {X k, k k, }. The by Lemma 2.2 i Che ad Wag ([2]), it suffices to show that k (2.) E (X k X k ) 0 as.

3 DAVIS-GUT LAW FOR BANACH SPACE VALUED RANDOM ELEMENTS 353 It is easy to show that k (X k X k) 0 i probability ad X k X k 2δ. Therefore by Lemma 2. i Hu et al. ([7]), (2.) holds ad the proof is completed. Lemma 2.2. Let 0 < b, ad {X,X, } a sequece of idepedet ad idetically distributed B-valued radom elemets. If b X k 0 i probability, the E b X ki( X k b ) 0 as. Proof. Let {X,X, } be a idepedet copy of {X,X, }. The (2.2) b (X k X k ) 0 i probability. By Lévy s iequality (see display (2.7) i Ledoux ad Talagrad [8, p. 47]), for every t > 0, P{ max X k X k > t} 2P{ (X k X k) > t}, k which by (2.2) esures that (2.3) P{ max k X k X k > b /2} 0 as. By Lemma 2.6 of Ledoux ad Talagrad [8, p. 5], P{ X X > b /2} = P{ X k X k > b /2} (2.4) 2P{ max X k X k > b /2} k whe is sufficietly large. By display (6.) i Ledoux ad Talagrad [8, p. 50], (2.5) P{ X > b } 2P{ X X > b /2} whe is sufficietly large. Therefore by (2.3), (2.4), ad (2.5), (2.6) P{ X > b } 0 as. Note that for ay ε > 0 { } { } P X k I( X k b ) > εb P{ X > b }+P X k > εb.

4 354 P. CHEN, M. ZHANG AND A. ROSALSKY The by (2.6) ad b X k 0 i probability, it follows that b X k I( X k b ) 0 i probability. The coclusio the follows from Lemma 2.. The followig lemma is due to Eimahl ad Li ([4]). Lemma 2.3. Let Z,...,Z be idepedet B-valued radom elemets with mea zero such that for some s > 2, E Z k s <, k. The we have for 0 < η, δ > 0, ad t > 0, { } m P max Z k (+η)e Z k +t m { t 2 } exp (2+δ)Λ 2 +C E Z k s /t s, where Λ 2 = sup{ Ef2 (Z k ) : f B } ad C is a positive costat depedig o η,δ ad s. Lemma 2.4. Let h(t) ad Ψ(t) be as i Theorem.2. Suppose that X is a B-valued radom elemet with (2.7) h() P{ X > logψ()} <. The for ay s > 2, h() (logψ()) s/2 E X s I( X logψ()) <. Proof. Set b 0 = 0 ad b = logψ(),. Note that Ψ() ad therefore b /. The b k /b k/ wheever k. Hece, h() = = (logψ()) s/2 E X s I( X logψ()) h()b s E X s I(b k < X b k ) h()b s b s k P{b k < X b k } b s kp{b k < X b k } h()b s =k

5 DAVIS-GUT LAW FOR BANACH SPACE VALUED RANDOM ELEMENTS 355 k s/2 P{b k < X b k } s/2 h() C C h() +C =k k h(k) P{b k < X b k } [ k + h(k +) k h(k) C h() +C h(k) P{ X > b k} <, where C = (s/2 ). The proof is completed. ] P{ X > b k } Lemma 2.5. Let h(),ψ() be as i Theorem.2. The for ay B-valued radom elemet X, (2.7) is equivalet to (2.8) for some M > 0. h() P{ X > M logψ()} < Proof. It suffices to prove that (2.7) implies (2.8) for all 0 < M <. Set b = logψ(),. Note that Ψ() ad therefore b /. The b 2 /2 b 2 for. Hece, ad h(2) P{ X > 2 /2 b 2 } h() P{ X > b } h(2+) P{ X > 2 /2 b 2+ } h(2) P{ X > 2 /2 b 2 } h() P{ X > b }, which esures that h() P{ X > 2 /2 b } = h() P{ X > 2 /2 b } + h(2) P{ X > 2 /2 b 2 }+ h() P{ X > 2 /2 b }+2 h(2+) P{ X > 2 /2 b 2+ } h() P{ X > logψ()} <.

6 356 P. CHEN, M. ZHANG AND A. ROSALSKY The by mathematical iductio, for ay iteger k, h() P{ X > 2 k/2 b } <. The proof is completed. 3. The Mai Result ad its Proof We ow state ad prove the mai result. Theorem 3.. Let h(t) ad Ψ(t) be as i Theorem.2. Let {X,X, } be a sequece of idepedet ad idetically distributed B-valued radom elemets. Suppose that ( logψ()) X k 0 i probability. (i) Suppose that (2.7) holds ad (3.) EX = 0, Ef 2 (X) < f B. (3.2) The { h() P X k > (+ε) }{ 2σ 2 <, if ε > 0 logψ() =, if ε < 0, where σ 2 = sup{ef 2 (X) : f B}. (ii) Coversely, suppose that { (3.3) h() P X k > M } logψ() < holds for some M > 0. The (2.7) ad (3.) hold. Proof. Set a = 2σ 2 logψ(),b = logψ(), ad X k = X k I( X k b ), Z k = X k EX k, k,. (i) Suppose that (2.7) ad (3.) hold. We first prove that { } (3.4) h() P X k > (+ε)a < ε > 0. Note that for ay ε > 0, { } { } P X k > (+ε)a P{ X > b }+P X k > (+ε)a.

7 DAVIS-GUT LAW FOR BANACH SPACE VALUED RANDOM ELEMENTS 357 Hece, by (2.7), to prove (3.4), it suffices to prove that { } (3.5) h() P X k > (+ε)a < ε > 0. b By Lemma 2.2, EX k E X k 0 as b ad E Z k 2 E X k 0 as. b b The to prove (3.5), it suffices to prove that { } (3.6) h() P Z k > 2E Z k +(+ε)a < ε > 0. By Lemma 2.3, for some s > 2 ad ay δ > 0 { } P Z k > 2E Z k +(+ε)a (3.7) { exp (+ε)2 a 2 } (2+δ)Λ 2 + C b s E Z k s, where Λ 2 = sup{ Ef2 (Z k ) : f B }. Note that for all f B, Ef 2 (Z k ) = Ef 2 (X k ) (Ef(X k )) 2 Ef 2 (X k ) Ef 2 (X), k,. Therefore Λ 2 σ 2,. Choose δ > 0 small eough so that t = 2( + ε) 2 /(2+δ) >. The (3.8) { h() exp (+ε)2 a 2 } (2+δ)Λ { h() exp (+ε)2 a 2 } (2+δ)Λ h() exp{ tlogψ()} h() (Ψ()) t <,

8 358 P. CHEN, M. ZHANG AND A. ROSALSKY sice dx/[xh(x)ψ t (x)] <. By the C r -iequality, Hölder s iequality, ad Lemma 2.4, (3.9) h() b s E Z k s h() (logψ()) s/2 E X s I( X logψ()) <. By (3.7), (3.8), ad (3.9), (3.6) holds ad hece (3.4) holds as was argued above. Now we prove that (3.0) h() P { } X k > (+ε)a = ε < 0. For ay f B, by (3.), Ef(X) = 0 ad Ef 2 (X) <. The by the implicatio (.) (.4) i Theorem.2, for all ε < 0 { } (3.) h() P f(x k ) > (+ε) 2Ef 2 (X)logΨ() =. Note that for ay f B, f(x k) X k ad so it follows from (3.) that for all f B, for all ε < 0 { (3.2) h() P X k > (+ε) } 2Ef 2 (X)logΨ() =. Hece (3.0) holds by (3.2) ad σ 2 = sup{ef 2 (X) : f B }. Combiig (3.4) ad (3.0) yields (3.2). (ii) Assume that (3.3) holds for some M > 0. The for ay f B, h() P { } f(x k ) > Mb <. The by the implicatio (2.3) (2.4) of Li ad Rosalsky ([0]), it follows that Ef(X) = 0 ad Ef 2 (X) <. Hece (3.) holds. Let {X,X, } be a idepedet copy of {X,X, }. The by the same argumet as i the proof of Lemma 2.2, { } P{ X > 4Mb } 8P (X k X k ) > 2Mb { } 6P X k > Mb,

9 DAVIS-GUT LAW FOR BANACH SPACE VALUED RANDOM ELEMENTS 359 which by (3.3) esures that h() P{ X > 4Mb } < ad so (2.7) holds by Lemma 2.5. The proof is completed. Remark 3.2. A sufficiet coditio for (2.7) is E X 2 <. Ideed, h() P{ X > logψ()} P{ X > } h() h() E X 2 <. Remark 3.3. Some examples of momet coditios which are equivalet to (2.7) for various choices of h( ) will ow be give. Case (i). Set h(t) = (loglogt) b where b 0. The logψ(t) loglogt as t ad (2.7) is equivalet to E X 2 /(loglog X ) b+ <. Case (ii). Set h(t) = (logt) r where 0 r <. The logψ(t) (r )loglogt ast ad(2.7)is equivalet toe X 2 /[(log X ) r loglog X ] <. Case (iii). Set h(t) = logt. The logψ(t) logloglogt as t ad (2.7) is equivalet to E X 2 /[(log X )logloglog X ] <. Case (iv). I Case (i), take b = 0, or i Case (ii), take r = 0. The (2.7) is equivalet to E X 2 /loglog X <. Ackowledgemets. The research of Che is supported by the Natioal Natural Sciece Foudatio of Chia (No ). Refereces [] P.Y. Che ad D.C. Wag, Covergece rates for probabilities of moderate deviatios for movig average processes, Acta Math. Si. (Egl. Ser.) 24 (2008), [2] P.Y.Che ad D.C.Wag, L r covergece for B-valued radom elemets, Acta Math. Si. (Egl. Ser.) 28 (202), [3] J.A. Davis, Covergece rates for the law of the iterated logarithm, A. Math. Statist. 39 (968), [4] U. Eimahl ad D. Li, Characterizatio of LIL behavior i Baach space, Tras. Amer. Math. Soc. 360 (2008), [5] A. Gut, Covergece rates for probabilities of moderate deviatios for sums of radom variables with multidimesioal idices, A. Probab. 8 (980), [6] P. Hartma ad A. Witer, O the law of the iterated logarithm, Amer. J. Math. 63 (94), [7] T.-C. Hu, A. Rosalsky, D. Szyal ad A.I. Volodi, O complete covergece for arrays of rowwise idepedet radom elemets i Baach spaces, Stochastic Aal. Appl. 7 (999), [8] M. Ledoux ad M. Talagrad, Probability i Baach spaces. Isoperimetry ad processes, Spriger-Verlag, Berli, 99.

10 360 P. CHEN, M. ZHANG AND A. ROSALSKY [9] D.L. Li, Covergece rates of law of iterated logarithm for B-valued radom variables, Sci. Chia Ser. A 34 (99), [0] D. Li ad A. Rosalsky, A supplemet to the Davis-Gut law, J. Math. Aal. Appl. 330 (2007), [] D.L.Li, X. C. Wag ad M.B. Rao, Some results o covergece rates for probabilities of moderate deviatios for sums of radom variables, Iterat. J. Math. Math. Sci. 5 (992), [2] X. Liu, H. Qia ad L. Cao, The Davis-Gut law for movig average processes, Statist. Probab. Lett. 04 (205), 6. P. Che Departmet of Mathematics Jia Uversity Guagzhou, P. R. Chia tchepy@ju.edu.c M. Zhag Departmet of Mathematics Jia Uiversity Guagzhou, P. R. Chia zmy02@qq.com A. Rosalsky Departmet of Statistics Uiversity of Florida Gaiesville, FL 326 USA rosalsky@stat.ufl.edu Received: