Hindwi Publishing Corportion Journl of Inequlities nd Applictions Volume 2009, Article ID 271265, 14 pges doi:10.1155/2009/271265 Reserch Article Moment Inequlities nd Complete Moment Convergence Soo Hk Sung Deprtment of Applied Mthemtics, Pi Chi Uversity, Tejon 302-735, South Kore Correspondence should be ddressed to Soo Hk Sung, sungsh@pcu.c.kr Received 22 August 2009; Accepted 26 September 2009 Recommended by Andrei Volodin Let { i, 1 i n} nd {Z i, 1 i n} be sequences of rndom vribles. For ny ɛ > 0 nd >0, bounds for E n i Z i ɛ nd E mx k i Z i ɛ re obtined. From these results, we estblish generl methods for obting the complete moment convergence. The results of Chow 1988, Zhu 2007, nd Wu nd Zhu 2009 re generlized nd extended from independent or dependent rndom vribles to rndom vribles stisfying some mild conditions. Some pplictions to dependent rndom vribles re discussed. Copyright q 2009 Soo Hk Sung. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. 1. Introduction Let { n,n 1} be sequence of rndom vribles defined on fixed probbility spce Ω, F,P. The most interesting inequlities to probbility theory re probbly Mrcinkiewicz- Zygmund nd Rosenthl inequlities. For sequence { i, 1 i n} of i.i.d. rndom vribles with E 1 q < for some q>1, Mrcinkiewicz nd Zygmund 1 nd Rosenthl 2 1 <q 2ndq>2, resp. proved tht there exist positive constnts A q nd B q depending only on q such tht E i E i q A q E i q for 1 <q 2, 1.1 E i E i q q/2 B q E i q E i 2 for q>2. 1.2
2 Journl of Inequlities nd Applictions The following Mrcinkiewicz-Zygmund nd Rosenthl type mximl inequlities re well known. For sequence { i, 1 i n} of i.i.d. rndom vribles with E 1 q < for some q>1, there exist positive constnts C q nd D q depending only on q such tht Emx i E i Emx i E i q q C q E i q for 1 <q 2, 1.3 q/2 D q E i q E i 2 for q>2. 1.4 Note tht 1.3 nd 1.4 imply 1.1 nd 1.2, respectively. The bove inequlities hve been obtined for dependent rndom vribles by mny uthors. Sho 3 proved tht 1.3 nd 1.4 hold for negtively ssocited rndom vribles. Asdin et l. 4 proved tht 1.1 nd 1.2 hold for negtively orthnt dependent rndom vribles. For sequence of some mixing rndom vribles, 1.4 holds. However, the constnt D q depends on both q nd the sequence of mixing rndom vribles. Sho 5 obtined 1.4 for φ-mixing identiclly distributed rndom vribles stisfying φ1/2 2 n <. Sho 6 lso obtined 1.4 for ρ-mixing identiclly distributed rndom vribles stisfying ρ2/q 2 n <. Utev nd Peligrd 7 obtined 1.4 for ρ -mixing rndom vribles. The concept of complete convergence ws introduced by Hsu nd Robbins 8. A sequence { n,n 1} of rndom vribles is sid to converge completely to the constnt θ if P n θ >ɛ < ɛ>0. 1.5 In view of the Borel-Cntelli lemm, this implies tht n θ lmost surely. Therefore the complete convergence is very importnt tool in estblishing lmost sure convergence of summtion of rndom vribles. Hsu nd Robbins 8 proved tht the sequence of rithmetic mens of i.i.d. rndom vribles converges completely to the expected vlue if the vrince of the summnds is fite. Erdös 9 proved the converse. The result of Hsu-Robbins-Erdös hs been generlized nd extended in severl directions. Bum nd Ktz 10 proved tht if { n,n 1} is sequence of i.i.d. rndom vribles with E 1 <, E 1 pt < 1 p<2, t 1 is equivlent to n P t 2 i E i >ɛn1/p < ɛ>0. 1.6 Chow 11 generlized the result of Bum nd Ktz 10 by showing the following complete moment convergence. If { n,n 1} is sequence of i.i.d. rndom vribles with E 1 pt <
Journl of Inequlities nd Applictions 3 for some 1 p<2ndt>1, then n E t 2 1/p i E i ɛn1/p < ɛ>0, 1.7 where mx{, 0}. Notetht 1.7 implies 1.6 see Remrk 2.6. Recently, Zhu 12 obtined complete convergence for ρ -mixing rndom vribles. Wu nd Zhu 13 obtined complete moment convergence results for negtively orthnt dependent rndom vribles. In this pper, we give generl methods for obting the complete moment convergence by using some moment inequlities. From these results, we generlize nd extend the results of Chow 11, Zhu 12, nd Wu nd Zhu 13 from independent or dependent rndom vribles to rndom vribles stisfying some conditions similr to 1.1 1.4. 2. Complete Moment Convergence for Rndom Vribles In this section, we give generl methods for obting the complete moment convergence by using some moment inequlities. The first two lemms re simple inequlities for rel numbers. Lemm 2.1. For ny rel numbers, b, c, the inequlity holds b c c b. 2.1 Proof. The result follows by n elementry clcultion. The following lemm is slight generliztion of Lemm 2.1. Lemm 2.2. Let { i, 1 i n} nd {b i, 1 i n} be two sequences of rel numbers. Then for ny rel number c, the inequlity holds mx i b i c mx i c mx b i. 1 i n 1 i n 1 i n 2.2 Proof. By Lemm 2.1, weobtin mx i b i c mx i b i c { mx i c b i } 1 i n 1 i n 1 i n mx 1 i n i c mx 1 i n b i mx i c mx b i. 1 i n 1 i n 2.3 The next two lemms ply essentil roles in the pper. Lemm 2.3 gives moment inequlity for the sum of rndom vribles.
4 Journl of Inequlities nd Applictions Lemm 2.3. Let { i, 1 i n} nd {Z i, 1 i n} be sequences of rndom vribles. Then for ny q>1, ɛ>0, >0, E i Z i ɛ 1 ɛ q 1 1 E q q 1 i E Z i. 2.4 Proof. By Lemm 2.1, E i Z i ɛ E i ɛ E Z i. 2.5 On the other hnd, we hve by Mrkov s inequlity tht E i ɛ P 0 i ɛ > t dt P 0 i >ɛ t dt P i >ɛ t dt P i >ɛ P i >t dt E n i q ɛ q q 1 1 ɛ q 1 1 iq E E q 1 i q. 1 t q dt 2.6 Substituting 2.6 into 2.5, we hve the result. The following lemm gives moment inequlity for the mximum prtil sum of rndom vribles. Lemm 2.4. Let { i, 1 i n} nd {Z i, 1 i n} be sequences of rndom vribles. Then for ny q>1, ɛ>0, >0, E mx i Z i ɛ 1 ɛ q 1 1 q Emx q 1 i Emx Z i. 2.7
Journl of Inequlities nd Applictions 5 Proof. By Lemm 2.2, E mx i Z i ɛ E mx i ɛ Emx Z i. 2.8 The rest of the proof is similr to tht of Lemm 2.3 nd is omitted. Now we stte nd prove one of our min results. The following theorem gives generl method for obting the complete moment convergence for sums of rndom vribles stisfying 2.9. The condition 2.9 is well known Mrcinkiewicz-Zygmund inequlity. Theorem 2.5. Let {, 1 i n, n 1} be n rry of rndom vribles with E < for 1 i n, n 1. Let{ n,n 1} nd {,n 1} be sequences of positive rel numbers. Suppose tht the following conditions hold. i For some 1 <q 2, there exists positive constnt C q depending only on q such tht E q E C q E q n 1, 2.9 where I n n I > n n I < n. ii q n n E q I n <. iii 1 n n E I > n <. Then b n E n E ɛ n < ɛ>0. 2.10 Proof. Observe tht E q E q I n q np > n E q I n q 1 n E I > n, E E n I > n n I < n E I > n. 2.11 2.12
6 Journl of Inequlities nd Applictions Then we hve by Lemm 2.3, 2.9, 2.11,nd 2.12 tht b n E n E ɛ n E 1 ɛ q 1 E n q n 1 C q ɛ q 1 1 C q ɛ q 1 E q E q n q n { 1 C q ɛ q 1 } 2 E q 2 n E q I n n E E I > n. 2.13 The bove two series converge by ii nd iii. Hence the result is proved. Remrk 2.6. If 2.10 holds, then P n E >ɛ n < for ll ɛ>0, since E E ɛ n 0 ɛn 0 P E ɛ n >t dt P E >ɛ n t dt ɛ n P E > 2ɛ n. 2.14 Hence complete moment convergence is more generl thn complete convergence. When q>2, we hve the following theorem. Condition 2.15 is well-known Rosenthl inequlity. Theorem 2.7. Let {, 1 i n, n 1} be n rry of rndom vribles with E < for 1 i n, n 1. Let{ n,n 1} nd {,n 1} be sequences of positive rel numbers. Suppose tht the following conditions hold.
Journl of Inequlities nd Applictions 7 i For some q>2, there exists positive constnt C q depending only on q such tht E q E C q E q E q/2 2 n 1, 2.15 where I n n I > n n I < n. ii q n n E q I n <. iii 1 n n E I > n <. iv n E r / r n q/2 < for some 0 <r 2. Then 2.10 holds. Proof. The proof is sme s tht of Theorem 2.5 except tht q n E q/2 2 n E 2 q/2 n n E b n r q/2 since n 1 2.16 n E b n r q/2 <. Corollry 2.8. Let { n,n 1} be sequence of positive rel numbers. Let {, 1 i n, n 1} be n rry of rndom vribles stisfying 2.15 for some q>2. Suppose tht the following conditions hold. i n q n E q I n <. ii n 1 n E I > n <. iii n E r / r n s < for some 0 <r 2 nd 0 <s q/2. Then 1 E n E ɛ n < ɛ>0, 2.17
8 Journl of Inequlities nd Applictions nd hence, P E >ɛ n < ɛ>0. 2.18 Proof. By Remrk 2.6, 2.17 implies 2.18. To prove 2.17,wepplyTheorem 2.7 with 1. Since 0 <s q/2, E r q/2 r E r s q/ 2s n r <. 2.19 n Hence the result follows by Theorem 2.7. The following theorem gives generl method for obting the complete moment convergence for mximum prtil sums of rndom vribles stisfying condition 2.20. Theorem 2.9. Let {, 1 i n, n 1} be n rry of rndom vribles with E < for 1 i n, n 1. Let{ n,n 1} nd {,n 1} be sequences of positive rel numbers. Suppose tht the following conditions hold. i For some 1 <q 2, there exists positive constnt C q depending only on q such tht Emx q E C q E q n 1, 2.20 where I n n I > n n I < n. ii q n n E q I n <. iii 1 n n E I > n <. Then E mx n E ɛ n < ɛ>0. 2.21
Journl of Inequlities nd Applictions 9 Proof. The proof is similr to tht of Theorem 2.5. We hve by Lemm 2.4, 2.20, ii,nd iii tht E mx n E ɛ n Emx 1 ɛ q 1 Emx n Hence the result is proved. q n 1 C q ɛ q 1 1 C q ɛ q 1 E q E q n q n { 1 C q ɛ q 1 } 2 E q 2 n E q I n n E E I > n <. 2.22 Remrk 2.10. If 2.21 holds, then P mx k E >ɛ n < for ll ɛ>0, since, s in Remrk 2.6, E mx E ɛ n ɛ n P mx E > 2ɛ n. 2.23 When q>2, we hve the following theorem. Theorem 2.11. Let {, 1 i n, n 1} be n rry of rndom vribles with E < for 1 i n, n 1. Let{ n,n 1} nd {,n 1} be sequences of positive rel numbers. Suppose tht the following conditions hold. i For some q>2, there exists positive constnt C q depending only on q such tht Emx q E C q E q E q/2 2 for n 1, 2.24 where I n n I > n n I < n. ii q n n E q I n <. iii 1 n n E I > n <. iv n E r / r n q/2 < for some 0 <r 2. Then 2.21 holds.
10 Journl of Inequlities nd Applictions Proof. The proof is similr to tht of Theorem 2.9 nd is omitted. Corollry 2.12. Let { n,n 1} be sequence of positive rel numbers. Let {, 1 i n, n 1} be n rry of rndom vribles stisfying 2.24 for some q>2. Suppose tht the following conditions hold. i n q n E q I n <. ii n 1 n E I > n <. iii n E r / r n s < for some 0 <r 2 nd 0 <s q/2. Then 1 E mx n E ɛ n < ɛ>0, 2.25 nd hence, P mx E >ɛ n < ɛ>0. 2.26 Proof. By Remrk 2.10, 2.25 implies 2.26. As in the proof of Corollry 2.8, E r q/2 r E r s q/ 2s n r <. 2.27 n Hence the result follows by Theorem 2.11 with 1. 3. Corollries In this section, we estblish some complete moment convergence results by using the results obtined in the previous section. Throughout this section, let {Ψ n t, n 1} be sequence of positive even functions stisfying Ψ n t t, Ψ n t t p s t 3.1 for some p>1. To obtin complete moment convergence results, the following lemms re needed. Lemm 3.1. Let be rndom vrible nd {Ψ n t, n 1} sequence of positive even functions stisfying 3.1 for some p>1. Then for ll >0 nd n 1, the followings hold. i If q p,thene q I / q EΨ n /Ψ n. ii E I > / EΨ n /Ψ n.
Journl of Inequlities nd Applictions 11 Proof. First note by Ψ n t / t tht Ψ n t is n incresing function. If q p, then Ψ n t / t p implies Ψ n t / t q,ndso Ψ n Ψ n Ψ n I Ψ n q I q. 3.2 Hence i holds. Since Ψ n t / t, Ψ n Ψ n Ψ n I > Ψ n I >. 3.3 So ii holds. Lemm 3.2. Let {, 1 i n, n 1} be n rry of rndom vribles with E < for 1 i n, n 1. Let{ n,n 1} nd {,n 1} be sequences of positive rel numbers. Assume tht {Ψ n t, n 1} is sequence of positive even functions stisfying 3.1 for some p>1 nd EΨ i Ψ i n <. 3.4 Then the followings hold. i If q p,then q n n E q I n <. ii 1 n n E I > n <. Proof. The result follows from Lemm 3.1. By using Lemm 3.2, we cn obtin Corollries 3.3, 3.4, 3.5, 3.6 from Theorem 2.5, Corollry 2.8, Theorem 2.9, Corollry 2.12, respectively. Corollry 3.3. Let { n,n 1} nd {, n 1} be sequences of positive rel numbers {Ψ n t, n 1} sequence of positive even functions stisfying 3.1 for some 1 <p 2. Assume tht {, 1 i n, n 1} is n rry of rndom vribles stisfying 2.9 for q p nd 3.4. Then 2.10 holds. Corollry 3.4. Let { n,n 1} be sequence of positive rel numbers {Ψ n t, n 1} sequence of positive even functions stisfying 3.1 for some p>2. Assume tht {, 1 i n, n 1} is n rry of rndom vribles stisfying 2.15 for some q mx{p, 2s} s isthesmesin 3.6, EΨ i Ψ i n <, 3.5 E r s r < for some 0 <r 2, s > 0. 3.6 n Then 2.17 holds nd hence, 2.18 holds.
12 Journl of Inequlities nd Applictions Corollry 3.5. Let { n,n 1} nd {, n 1} be sequences of positive rel numbers {Ψ n t, n 1} sequence of positive even functions stisfying 3.1 for some 1 <p 2. Assume tht {, 1 i n, n 1} is n rry of rndom vribles stisfying 2.20 for q p nd 3.4.Then 2.21 holds. Corollry 3.6. Let { n,n 1} be sequence of positive rel numbers {Ψ n t, n 1} sequence of positive even functions stisfying 3.1 for some p>2. Assume tht {, 1 i n, n 1} is n rry of rndom vribles stisfying 2.24 for some q mx{p, 2s} s isthesmesin 3.6, 3.5, nd 3.6. Then 2.25 holds nd hence, 2.26 holds. Remrk 3.7. Mrcinkiewicz-Zygmund nd Rosenthl type inequlities hold for dependent rndom vribles s well s independent rndom vribles. 1 For n rry {, 1 i n, n 1} of rowwise negtively ssocited rndom vribles, condition 2.20 holds if 1 <q 2, nd 2.24 holds if q>2bysho s 3 results. Note tht {, 1 i n, n 1} is still n rry of rowwise negtively ssocited rndom vribles. Hence Corollries 3.3 3.6 hold for rrys of rowwise negtively ssocited rndom vribles. 2 For n rry {, 1 i n, n 1} of rowwise negtively orthnt dependent rndom vribles, condition 2.9 holds if 1 <q 2, nd 2.15 holds if q>2bytheresultsof Asdin et l. 4. Hence Corollries 3.3 nd 3.4 hold for rrys of rowwise negtively orthnt dependent rndom vribles. These results lso were proved by Wu nd Zhu 13. Hence Corollries 3.3 nd 3.4 extend the results of Wu nd Zhu 13 from n rry of negtively orthnt dependent rndom vribles to n rry of rndom vribles stisfying 2.9 nd 2.15. 3 For n rry {, 1 i n, n 1} of rowwise ρ -mixing rndom vribles, condition 2.24 does not necessrily hold if q > 2. As mentioned in Section 1, Utevnd Peligrd 7 proved 1.4 for ρ -mixing rndom vribles. However, the constnt D q depends on both q nd the sequence of ρ -mixing rndom vribles. Hence condition 2.24 holds for n rry of rowwise ρ -mixing rndom vribles under the dditionl condition tht D q depending on the sequence of rndom vribles in ech row re bounded. So Corollry 3.6 holds for rrys of rowwise ρ -mixing rndom vribles stisfying this dditionl condition. Zhu 12 obtined only 2.26 in Corollry 3.6 when the rry is rowwise ρ -mixing rndom vribles stisfying the dditionl condition. This dditionl condition should be dded in Zhu 12. Hence Corollry 3.6 generlizes nd extends Zhu s 12 result from ρ -mixing rndom vribles to more generl rndom vribles. Finlly, we pply the complete moment convergence results obtined in the previous section to sequence of identiclly distributed rndom vribles. Corollry 3.8. Let { n, n 1} be sequence of identiclly distributed rndom vribles with E 1 pt < for some 1 p < 2 nd t>1. Assume tht for ny q > 2, there exists positive constnt C q depending only on q such tht E q E C q E q E q/2 2, 3.7 where ii i n 1/p n 1/p I i >n 1/p n 1/p I i < n 1/p.The.7 holds.
Journl of Inequlities nd Applictions 13 Proof. Let i for 1 i n, n 1. We pply Theorem 2.7 with n n 1/p nd n t 2. Tke r nd q>2 such tht p<r min{2,pt}, q/p t>0, nd r/p 1 q/2 t 1 > 0. Then it is esy to see tht n t 2 q/p E i q I i n 1/p <, n t 2 1/p E i I i >n 1/p <, 3.8 n t 2 E i r q/2 <. n r/p Hence the result follows from Theorem 2.7. Corollry 3.9. Let { n, n 1} be sequence of identiclly distributed rndom vribles with E 1 pt < for some 1 p < 2 nd t>1. Assume tht for ny q > 2, there exists positive constnt C q depending only on q such tht Emx q E C q E q E q/2 2, 3.9 where ii i n 1/p n 1/p I i >n 1/p n 1/p I i < n 1/p.Then n t 2 1/p E mx i E i ɛn1/p < ɛ>0. 3.10 Proof. As in the proof of Corollry 3.8, 3.8 re stisfied. So the result follows from Theorem 2.11. Remrk 3.10. If { n,n 1} is sequence of i.i.d. rndom vribles, then conditions 3.7 nd 3.9 re stisfied when q > 2. Hence Corollries 3.8 nd 3.9 generlize nd extend the result of Chow 11. There re mny sequences of dependent rndom vribles stisfying 3.7 for ll q>2. Exmples include sequences of negtively orthnt dependent rndom vribles, negtively ssocited rndom vribles, ρ -mixing rndom vribles, φ- mixing identiclly distributed rndom vribles stisfying φ1/2 2 n <, ndρ-mixing identiclly distributed rndom vribles stisfying ρ2/q 2 n <. The bove sequences of dependent rndom vribles except negtively orthnt dependent rndom vribles lso stisfy 3.9 when q>2. Hence Corollries 3.8 nd 3.9 hold for mny dependent rndom vribles s well s independent rndom vribles.
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