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Complete q th moment convergence for arrays of random variables

Complete q th moment convergence for arrays of random variables Department of Applied Let {X ,1 ≤ i ≤ n, n ≥ 1} be an array of random variables with EX =0 and E|X | < ∞ ni ni ni Mathematics, PaiChaiUniversity, Taejon, 302-735, South Korea for some q ≥ 1. For any sequences {a , n ≥ 1} and {b , n ≥ 1} of positive real numbers, n n sets of sufficient conditions are given for complete qth moment convergence of the –q q ∞ k form b a E(max | X | – a ) < ∞, ∀ >0, where x = max{x,0}. n n 1≤k≤n ni n + + n=1 i=1 From these results, we can easily obtain some known results on complete qth moment convergence. Keywords: complete convergence; complete moment convergence; L -convergence; dependent random variables 1 Introduction The concept of complete convergence was introduced by Hsu and Robbins []. A sequence {X , n ≥ } of random variables is said to converge completely to the constant θ if P |X – θ | >  < ∞ for all  >. n= Hsu and Robbins [] proved that the sequence of arithmetic means of i.i.d. random vari- ables converges completely to the expected value if the variance of the summands is finite. Erdös []provedthe converse. The result of Hsu, Robbins, and Erdös has been generalized and extended in several directions. Baum and Katz []provedthatif {X , n ≥ } is a sequence of i.i.d. random pt variables with EX =, E|X | < ∞ ( ≤ p <, t ≥ ) is equivalent to   ∞ n t– /p n P X > n < ∞ for all  > . (.) n= i= Chow [] generalized the result of Baum and Katz [] by showing the following complete moment convergence. If {X , n ≥ } is a sequence of i.i.d. random variables with EX = n  pt and E(|X | + |X | log( + |X |)) < ∞ for some  < p <, t ≥ , and pt ≥ , then    ∞ k t––/p /p n E max X – n < ∞ for all  >, (.) ≤k≤n n= i= where x = max{x,}.Notethat(.)implies (.). Li and Spătaru [] gave a refinement of the result of Baum and Katz [] as follows. Let {X , n ≥ } be a sequence of i.i.d. random © 2013 Sung; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Sung Journal of Inequalities and Applications 2013, 2013:24 Page 2 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 variables with EX =, and let  < p <, t ≥ , q >, and pt ≥ . Then E|X | < ∞ if q > pt, pt (.) E|X | log( + |X |)< ∞ if q = pt,   pt E|X | < ∞ if q < pt, if and only if ∞ n t– /q /p n P X > x n dx < ∞ for all  >. n= i= Recently, Chen and Wang [] proved that for any q >, any sequences {a , n ≥ } and {b , n ≥ } of positive real numbers and any sequence {Z , n ≥ } of random variables, n n /q b P |Z | > x a dx < ∞ for all  > n n n n= and –q b a E |Z | – a < ∞ for all  >, n n n n= are equivalent. Therefore, if {X , n ≥ } is a sequence of i.i.d. random variables with EX = n  and  < p <, t ≥ , q >, and pt ≥ , then the moment condition (.)isequivalentto ∞ n t––q/p /p n E  X  – n < ∞ for all  >. (.) n= i= + When q =, the complete qth moment convergence (.) is reduced to complete moment convergence. The complete qth moment convergence for dependent random variables was estab- lished by many authors. Chen and Wang [] showed that (.)and (.)are equivalent for ϕ-mixing random variables. Zhou and Lin [] established complete qth moment con- vergence theorems for moving average processes of ϕ-mixing random variables. Wu et al. [] obtained complete qth moment convergence results for arrays of rowwise ρ -mixing random variables. The purpose of this paper is to provide sets of sufficient conditions for complete qth moment convergence of the form ∞ k –q b a E max  X  – a < ∞ for all  >, (.) n ni n ≤k≤n n= i= where q ≥ , {a , n ≥ } and {b , n ≥ } are sequences of positive real numbers, and n n {X , ≤ i ≤ n, n ≥ } is an array of random variables satisfying Marcinkiewicz-Zygmund ni and Rosenthal type inequalities. When q = , similar results were established by Sung []. From our results, we can easily obtain the results of Chen and Wang []and Wu et al. []. Sung Journal of Inequalities and Applications 2013, 2013:24 Page 3 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 2 Main results In this section, we give sets of sufficient conditions for complete qth moment convergence (.). The following theorem gives sufficient conditions under the assumption that the array {X , ≤ i ≤ n, n ≥ } satisfies a Marcinkiewicz-Zygmund type inequality. ni Theorem . Let  ≤ q < and let {X , ≤ i ≤ n, n ≥ } be an array of random variables ni with EX = and E|X | < ∞ for  ≤ i ≤ nand n ≥ . Let {a , n ≥ } and {b , n ≥ } be ni ni n n sequences of positive real numbers. Suppose that the following conditions hold: (i) for some s ( ≤ q < s ≤ ), there exists a positive function α (x) such that E max X (x)– EX (x) ni ni ≤k≤n i= ≤ α (n) E X (x) for all n ≥  and x >, (.) ni i= /q /q /q /q /q where X (x)= X I(|X |≤ x )+ x I(X > x )– x I(X <–x ), ni ni ni ni ni ∞ n –s s (ii) b a α (n) E|X | I(|X |≤ a )< ∞, n s ni ni n n= i= ∞ –q n (iii) b a ( + α (n)) E|X | I(|X | > a )< ∞, n n s ni ni n n= i= (iv) E|X |I(|X | > a )/a → . ni ni n n i= Then (.) holds. Proof It is obvious that ∞ k –q b a E max X – a n ni n ≤k≤n n= i= ∞ k –q /q = b a P max X > a + x dx n ni n ≤k≤n n= i= ∞ k k a   ∞ –q /q ≤ b a P max  X  > a dx + P max  X  > x dx n ni n ni ≤k≤n ≤k≤n  a n= i= i= := I + I .   We first show that I < ∞.For  ≤ i ≤ n and n ≥ , define X = X I |X |≤ a + a I(X > a )– a I(X <–a ), X = X – X . ni ni n n ni n n ni n ni ni ni ni Then we have by EX = , Markov’s inequality, and (i) that ni P max X > a ni n ≤k≤n i= = P max X – EX + X – EX > a ni ni ni ni ≤k≤n i= k k ≤ P max X – EX > a / + P max X – EX > a / n n ni ni ni ni ≤k≤n  ≤k≤n i= i= Sung Journal of Inequalities and Applications 2013, 2013:24 Page 4 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 k k s –s –s   – – ≤   a E max  X – EX  + a E max  X – EX n ni ni n ni ni ≤k≤n  ≤k≤n i= i= n n s –s –s  – – ≤   a α (n) E X + a E X n ni n ni i= i= s –s –s s s ≤   a α (n) E|X | I |X |≤ a + a P |X | > a s ni ni n ni n n n i= – – + a E|X |I |X | > a ni ni n i= n n s –s –s s s –s –q q ≤   a α (n) E|X | I |X |≤ a +  a α (n) E|X | I |X | > a s ni ni n s ni ni n n n i= i= – –q q + a E|X | I |X | > a . ni ni n i= It follows that ∞ k I = b P max  X  > a  n ni n ≤k≤n n= i= ∞ n s –s –s s ≤   b a α (n) E|X | I |X |≤ a n s ni ni n n= i= ∞ n –q s –s – q + b a   α (n)+ E|X | I |X | > a . n s ni ni n n= i= Hence I < ∞ by (ii) and (iii). We next show that I < ∞. By the definition of X (x), we have that ni /q P max  X  > x ni ≤k≤n i= n k /q  /q ≤ P |X | > x + P max  X (x) > x . ni ni ≤k≤n i= i= We also have by EX = and (iv) that ni –/q sup max x EX (x) ni q ≤k≤n x≥a n i= –/q = sup max x E X – X (x) ni ni q ≤k≤n x≥a n i= –/q /q ≤ sup x E|X |I |X | > x ni ni x≥a i= – ≤ a E|X |I |X | > a → . ni ni n i= Sung Journal of Inequalities and Applications 2013, 2013:24 Page 5 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 Hence to prove that I < ∞, it suffices to show that ∞ n –q /q I := b a P |X | > x dx < ∞,  n ni n= i= ∞ k –q   /q I := b a P max X (x)– EX (x) > x / dx < ∞.  n n ni ni ≤k≤n n= i= /q /q If x > a ,then P(|X | > x )= P(|X |I(|X | > a )> x )and so n ni ni ni n ∞ ∞ /q /q P |X | > x dx = P |X |I |X | > a > x dx ni ni ni n q q a a n n /q q ≤ P |X |I |X | > a > x dx = E|X | I |X | > a , ni ni n ni ni n which implies that ∞ n –q q I ≤ b a E|X | I |X | > a .  n ni ni n n= i= Hence I < ∞ by (iii). Finally, we show that I < ∞. We get by Markov’s inequality and (i) that ∞ k s –q –s/q I ≤  b a x E max X (x)– EX (x) dx  n n ni ni ≤k≤n n= i= ∞ n s –q –s/q ≤  b a α (n) x E X (x) dx n s n ni n= i= ∞ n s –q –s/q s /q s/q /q = b a α (n) x E|X | I |X |≤ x + x P |X | > x dx n s ni ni ni n= i= ∞ n s –q s –s/q = b a α (n) E|X | I |X |≤ a x dx n s ni ni n n= i= ∞ n s –q –s/q s /q + b a α (n) x E|X | I a < |X |≤ x dx n s ni n ni n= i= ∞ n s –q /q + b a α (n) P |X | > x dx := I + I + I . n s ni    n= i= Using a simple integral and Fubini’s theorem, we obtain that ∞ n s –s s I = b a α (n) E|X | I |X |≤ a ,  n s ni ni n s – q n= i= ∞ n s –q –s/q s /q I = b a α (n) x E|X | I a < |X |≤ x dx  n s ni n ni n= i= ∞ n s –q q = b a α (n) E|X | I |X | > a . n s ni ni n s – q n= i= Sung Journal of Inequalities and Applications 2013, 2013:24 Page 6 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 Similarly to I , ∞ n s –q q I ≤  b a α (n) E|X | I |X | > a .  n s ni ni n n= i= Hence I < ∞ by (ii) and (iii). The next theorem gives sufficient conditions for complete qth moment convergence (.) under the assumption that the array {X , ≤ i ≤ n, n ≥ } satisfies a Rosenthal type ni inequality. Theorem . Let q ≥  and let {X , ≤ i ≤ n, n ≥ } be an array of random variables ni with EX = and E|X | < ∞ for  ≤ i ≤ nand n ≥ . Let {a , n ≥ } and {b , n ≥ } be ni ni n n sequences of positive real numbers. Suppose that the following conditions hold: (i) for some s > max{, q/r} (r is the same as in (v)), there exist positive functions β (x) and γ (x) such that E max  X (x)– EX (x) ni ni ≤k≤n i= s/ n n s  ≤ β (n) E X (x) + γ (n) E X (x) s s ni ni i= i= for all n ≥  and x >, (.) /q /q /q /q /q where X (x)= X I(|X |≤ x )+ x I(X > x )– x I(X <–x ), ni ni ni ni ni ∞ n –s s (ii) b a β (n) E|X | I(|X |≤ a )< ∞, n s ni ni n n= n i= ∞ –q n (iii) b a ( + β (n)) E|X | I(|X | > a )< ∞, n n s ni ni n n= i= (iv) E|X |I(|X | > a )/a → , ni ni n n i= ∞ n –r r s/ (v) b γ (n)( a E|X | ) < ∞ for some < r ≤ . n s ni n= i= n Then (.) holds. Proof Theproof is similartothatofTheorem .. As in the proof of Theorem ., ∞ k –q b a E max X – a n ni n ≤k≤n n= i= ∞ k k a ∞ –q /q ≤ b a P max X > a dx + P max X > x dx n ni n ni ≤k≤n  ≤k≤n  a n= i= i= := J + J .   Similarly to I in the proof of Theorem .,wehavebythe c -inequality that  r P max  X  > a ni n ≤k≤n i= s –s –s s s ≤   a β (n) E|X | I |X |≤ a + a P |X | > a s ni ni n ni n n n i= Sung Journal of Inequalities and Applications 2013, 2013:24 Page 7 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 s/ s –s –s   +  a γ (n) E|X | I |X |≤ a + a P |X | > a s ni ni n ni n n n i= – – + a E|X |I |X | > a ni ni n i= s –s –s s ≤   a β (n) E|X | I |X |≤ a s ni ni n i= s –s – –q q +   β (n)+ a E|X | I |X | > a s ni ni n i= s/ s/– –s –r r +  γ (n) a E|X | I |X |≤ a s ni ni n i= s/ s/– –s –r r +  γ (n) a E|X | I |X | > a . s ni ni n i= Hence J < ∞ by (ii), (iii), and (v). As in the proof of Theorem .,toprove that J < ∞, it suffices to show that ∞ n –q /q J := b a P |X | > x dx < ∞,  n ni n= i= ∞ k –q   /q J := b a P max X (x)– EX (x) > x / dx < ∞.  n n ni ni ≤k≤n n= i= The proof of J < ∞ is same as that of I in the proof of Theorem ..   For J , we have by Markov’s inequality and (i) that ∞ k s –q –s/q J ≤  b a x E max X (x)– EX (x) dx  n n ni ni ≤k≤n n= i= s/ ∞ n n s  s –q –s/q ≤  b a x β (n) E X (x) + γ (n) E X (x) dx n s s n ni ni n= i= i= := J + J .   Similarly to I in the proof of Theorem .,weget that ∞ n s –s s J ≤  b a β (n) E|X | I |X |≤ a  n s ni ni n s – q n= i= ∞ n s –q q + + b a β (n) E|X | I |X | > a . n s ni ni n s – q n= i= Hence J < ∞ by (ii) and (iii).  Sung Journal of Inequalities and Applications 2013, 2013:24 Page 8 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 Finally, we show that J < ∞.Bythe c -inequality,  r s/ ∞ n s –q –s/q  /q /q /q J = b a γ (n) x E|X | I |X |≤ x + x P |X | > x dx  n s ni ni ni n= i= s/ ∞ n s/– –q –s/q  /q ≤  b a γ (n) x E|X | I |X |≤ x dx n s ni ni n= i= s/ ∞ n s/– –q /q + b a γ (n) P |X | > x dx n s ni n= i= s/ ∞ n s/– –q –s/q r (–r)/q ≤  b a γ (n) x E|X | x dx n s ni n= i= s/ ∞ n s/– –q –r/q r + b a γ (n) x E|X | dx n s ni n= i= s/ ∞ n q s/ –r r = b γ (n) a E|X | . n s ni rs –q n= i= Hence J < ∞ by (v). Remark . Marcinkiewicz-Zygmund and Rosenthal type inequalities hold for depen- dent random variables as well as independent random variables. () Let {X , ≤ i ≤ n, n ≥ } be an array of rowwise negatively associated random vari- ni s –s ables. Then, for  < s ≤ , (.)holds for α (n)=  =. For s >, (.)holds for s s s β (n)= (s/ log s) and γ (n)=(s/ log s) (see Shao []). Note that α (n)and β (n) s s s s s   s s  s are multiplied by the factor  since E|X (x)– EX (x)| ≤  E|X (x)| . ni ni ni () Let {X , ≤ i ≤ n, n ≥ } be an array of rowwise negatively orthant dependent ran- ni dom variables. By Corollary . of Asadian et al. []and Theoremof Móricz [], (.) s s s holds for α (n)= C (log n) ,and (.)holds for β (n)= C (log n) and γ (n)= C (log n) , s  s  s  where C and C are constants depending only on s.   () Let {X , n ≥ } be a sequence of identically distributed ϕ-mixing random variables. Set X = X for  ≤ i ≤ n and n ≥ . By Shao’s []result, (.) holds for a constant function ni i / n β (x) and a slowly varying function γ (x). In particular, if ϕ ( )< ∞,then(.) s s n= holds for some constant functions β (x)and γ (x). s s () Let {X , n ≥ } be a sequence of identically distributed ρ-mixing random variables. Set X = X for  ≤ i ≤ n and n ≥ . By Shao’s []result, (.) holds for some slowly ni i /s n varying functions β (x)and γ (x). In particular, if ρ ( )< ∞,then(.)holds for s s n= some constant functions β (x)and γ (x). s s () Let {X , n ≥ } be a sequence of ρ -mixing random variables. Set X = X for  ≤ i ≤ n n ni i and n ≥ . By the result of Utev and Peligrad [], (.) holds for some constant functions β (x)and γ (x). s s 3 Corollaries In this section, we establish some complete qth moment convergence results by using the results obtained in the previous section. Sung Journal of Inequalities and Applications 2013, 2013:24 Page 9 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 Corollary . (Chen and Wang []) Let {X , n ≥ } be a sequence of identically distributed ϕ-mixing random variables with EX =, and let t ≥ ,  < p <, q ≥ , and pt ≥ . Assume that (.) holds. Furthermore, suppose that / n ϕ  < ∞ n= if t = and max{q, pt} <. Then ∞ k t––q/p /p n E max X – n < ∞ for all  >. ≤k≤n n= i= /p t– Proof Let a = n and b = n for n ≥ , and let X = X for  ≤ i ≤ n and n ≥ . Then, for n n ni i s ≥ , (.) holds for a constant function β (x) and a slowly varying function γ (x)(see Re- s s / n mark .()). Under the additional condition that ϕ ( )< ∞,(.) holds for some n= constant functions β (x)and γ (x). In particular, for s =, (.) holds for a constant func- s s tion α (x) under this additional condition. By a standard method, we have that t––s/p s /p pt n E|X | I |X |≤ n ≤ CE|X | if pt < s,    n= CE|X | if q > pt, ⎪  t––q/p q /p pt n E|X | I |X | > n ≤ CE|X | log( + |X |)if q = pt,     n= pt CE|X | if q < pt, –/p /p –t pt /p n E|X |I |X | > n ≤ n E|X | I |X | > n if pt ≥ ,     where C is a positive constant which is not necessarily the same one in each appear- ance. Hence, the conditions (i)-(iv) of Theorem . hold if we take s > max{pt,,q/r}. / n Under the additional conditions that max{q, pt} < and ϕ ( )< ∞, all condi- n= tions of Theorem . hold if we take s = . Therefore, the result follows from Theo- rems . and . if we only show that the condition (v) of Theorem . holds when t > or max{q, pt}≥ . To do this, we take r = if max{q, pt}≥ and r = max{q, pt} if max{q, pt} <. If t > or max{q, pt}≥ , then r > p and so we can choose s > large enough such that t –+(– r/p)s/ < . Then s/ ∞ n ∞ s/ –r r r t–+(–r/p)s/ b γ (n) a E|X | = E|X | γ (n)n < ∞. n s ni  s n= i= n= Hence the condition (v) of Theorem . holds. Let { (x), n ≥ } be a sequence of positive even functions satisfying (|x|) (|x|) n n ↑ and ↓ as |x|↑ (.) q s |x| |x| for some  ≤ q < s. Sung Journal of Inequalities and Applications 2013, 2013:24 Page 10 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 Corollary . Let { (x), n ≥ } be a sequence of positive even functions satisfying (.) for some  ≤ q < s ≤ . Let {X , ≤ i ≤ n, n ≥ } be an array of random variables satisfying ni EX = for  ≤ i ≤ nand n ≥ , and (.) for some constant function α (x). Let {a , n ≥ } ni s n and {b , n ≥ } be sequences of positive real numbers. Suppose that the following conditions hold: ∞ n (i) b E (|X |)/ (a )< ∞, n i ni i n n= i= (ii) E (|X |)/ (a ) → . i ni i n i= Then (.) holds. q s Proof First note by (|x|)/|x| ↑ that (|x|) is an increasing function. Since (|x|)/|x| ↓, i i i |X | I(|X |≤ a ) (|X |I(|X |≤ a )) (|X |) ni ni n i ni ni n i ni ≤ ≤ . a (a ) (a ) i n i n Since q ≥ and (|x|)/|x| ↑, |X |I(|X | > a ) |X | I(|X | > a ) (|X |I(|X | > a )) (|X |) ni ni n ni ni n i ni ni n i ni ≤ ≤ ≤ . a (a ) (a ) n n i n i n It follows that all conditions of Theorem . are satisfied and so the result follows from Theorem .. Corollary . Let { (x), n ≥ } be a sequence of positive even functions satisfying (.) for some q ≥  and s > max{, q}. Let {X , ≤ i ≤ n, n ≥ } be an array of random variables ni satisfying EX = for  ≤ i ≤ nand n ≥ , and (.) for some constant functions β (x) and ni s γ (x). Let {a , n ≥ } and {b , n ≥ } be sequences of positive real numbers. Suppose that the s n n following conditions hold: ∞ n (i) b E (|X |)/ (a )< ∞, n i ni i n n= i= (ii) E (|X |)/ (a ) → , i ni i n i= ∞ n –  s/ (iii) b ( a E|X | ) < ∞. n ni n= i= n Then (.) holds. Proof The proof is similar to that of Corollary .. By the proof of Corollary . and the condition (iii), all conditions of Theorem . are satisfied and so the result follows from Theorem .. Remark . When b = for n ≥ , the condition (i) of Corollaries . and . is reduced ∞ n to the condition E (|X |)/ (a )< ∞, and so the condition (ii) of Corollar- i ni i n n= i= ies . and . follows from this reduced condition. For a sequence of ρ -mixing random variables, (.) holds for some constant function α (x)if s =, and (.) holds for some constant functions β (x)and γ (x)if s > (see Remark .()). Wu et al. [] proved Corol- s s laries . and . when b = for n ≥ , and {X } is an array of rowwise ρ -mixing random n ni variables. Competing interests The author declares that he has no competing interests. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0013131). Sung Journal of Inequalities and Applications 2013, 2013:24 Page 11 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 Received: 9 October 2012 Accepted: 3 January 2013 Published: 17 January 2013 References 1. Hsu, PL, Robbins, H: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33, 25-31 (1947) 2. Erdös, P: On a theorem of Hsu and Robbins. Ann. Math. Stat. 20, 286-291 (1949) 3. Baum, LE, Katz, M: Convergence rates in the law of large numbers. Trans. Am. Math. Soc. 120, 108-123 (1965) 4. Chow, YS: On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin. 16, 177-201 (1988) 5. Li, D, Spataru, ˘ A: Refinement of convergence rates for tail probabilities. J. Theor. Probab. 18, 933-947 (2005) 6. Chen, PY, Wang, DC: Convergence rates for probabilities of moderate deviations for moving average processes. Acta Math. Sin. 24, 611-622 (2008) 7. Chen, PY, Wang, DC: Complete moment convergence for sequences of identically distributed ϕ-mixing random variables. Acta Math. Sin. 26, 679-690 (2010) 8. Zhou, XC, Lin, JG: Complete q-moment convergence of moving average processes under ϕ-mixing assumption. J. Math. Res. Expo. 31, 687-697 (2011) 9. Wu, Y, Wang, C, Volodin, A: Limiting behavior for arrays of rowwise ρ -mixing random variables. Lith. Math. J. 52, 214-221 (2012) 10. Sung, SH: Moment inequalities and complete moment convergence. J. Inequal. Appl. 2009, Article ID 271265 (2009) 11. Shao, QM: A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theor. Probab. 13, 343-356 (2000) 12. Asadian, N, Fakoor, V, Bozorgnia, A: Rosenthal’s type inequalities for negatively orthant dependent random variables. J. Iran. Stat. Soc. 5, 69-75 (2006) 13. Móricz, F: Moment inequalities and the strong laws of large numbers. Z. Wahrscheinlichkeitstheor. Verw. Geb. 35, 299-314 (1976) 14. Shao, QM: A moment inequality and its applications. Acta Math. Sin. Chin. Ser. 31, 736-747 (1988) 15. Shao, QM: Maximal inequalities for partial sums of ρ-mixing sequences. Ann. Probab. 23, 948-965 (1995) 16. Utev, S, Peligrad, M: Maximal inequalities and an invariance principle for a class of weakly dependent random variables. J. Theor. Probab. 16, 101-115 (2003) doi:10.1186/1029-242X-2013-24 Cite this article as: Sung: Complete qth moment convergence for arrays of random variables. Journal of Inequalities and Applications 2013 2013:24. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Inequalities and Applications Springer Journals

Complete q th moment convergence for arrays of random variables

Sung, Soo
Journal of Inequalities and Applications , Volume 2013 (1) – Jan 17, 2013
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Copyright © 2013 by Sung; licensee Springer
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Mathematics; Analysis; Applications of Mathematics; Mathematics, general
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10.1186/1029-242X-2013-24
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Abstract

Department of Applied Let {X ,1 ≤ i ≤ n, n ≥ 1} be an array of random variables with EX =0 and E|X | < ∞ ni ni ni Mathematics, PaiChaiUniversity, Taejon, 302-735, South Korea for some q ≥ 1. For any sequences {a , n ≥ 1} and {b , n ≥ 1} of positive real numbers, n n sets of sufficient conditions are given for complete qth moment convergence of the –q q ∞ k form b a E(max | X | – a ) < ∞, ∀ >0, where x = max{x,0}. n n 1≤k≤n ni n + + n=1 i=1 From these results, we can easily obtain some known results on complete qth moment convergence. Keywords: complete convergence; complete moment convergence; L -convergence; dependent random variables 1 Introduction The concept of complete convergence was introduced by Hsu and Robbins []. A sequence {X , n ≥ } of random variables is said to converge completely to the constant θ if P |X – θ | >  < ∞ for all  >. n= Hsu and Robbins [] proved that the sequence of arithmetic means of i.i.d. random vari- ables converges completely to the expected value if the variance of the summands is finite. Erdös []provedthe converse. The result of Hsu, Robbins, and Erdös has been generalized and extended in several directions. Baum and Katz []provedthatif {X , n ≥ } is a sequence of i.i.d. random pt variables with EX =, E|X | < ∞ ( ≤ p <, t ≥ ) is equivalent to   ∞ n t– /p n P X > n < ∞ for all  > . (.) n= i= Chow [] generalized the result of Baum and Katz [] by showing the following complete moment convergence. If {X , n ≥ } is a sequence of i.i.d. random variables with EX = n  pt and E(|X | + |X | log( + |X |)) < ∞ for some  < p <, t ≥ , and pt ≥ , then    ∞ k t––/p /p n E max X – n < ∞ for all  >, (.) ≤k≤n n= i= where x = max{x,}.Notethat(.)implies (.). Li and Spătaru [] gave a refinement of the result of Baum and Katz [] as follows. Let {X , n ≥ } be a sequence of i.i.d. random © 2013 Sung; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu- tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Sung Journal of Inequalities and Applications 2013, 2013:24 Page 2 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 variables with EX =, and let  < p <, t ≥ , q >, and pt ≥ . Then E|X | < ∞ if q > pt, pt (.) E|X | log( + |X |)< ∞ if q = pt,   pt E|X | < ∞ if q < pt, if and only if ∞ n t– /q /p n P X > x n dx < ∞ for all  >. n= i= Recently, Chen and Wang [] proved that for any q >, any sequences {a , n ≥ } and {b , n ≥ } of positive real numbers and any sequence {Z , n ≥ } of random variables, n n /q b P |Z | > x a dx < ∞ for all  > n n n n= and –q b a E |Z | – a < ∞ for all  >, n n n n= are equivalent. Therefore, if {X , n ≥ } is a sequence of i.i.d. random variables with EX = n  and  < p <, t ≥ , q >, and pt ≥ , then the moment condition (.)isequivalentto ∞ n t––q/p /p n E  X  – n < ∞ for all  >. (.) n= i= + When q =, the complete qth moment convergence (.) is reduced to complete moment convergence. The complete qth moment convergence for dependent random variables was estab- lished by many authors. Chen and Wang [] showed that (.)and (.)are equivalent for ϕ-mixing random variables. Zhou and Lin [] established complete qth moment con- vergence theorems for moving average processes of ϕ-mixing random variables. Wu et al. [] obtained complete qth moment convergence results for arrays of rowwise ρ -mixing random variables. The purpose of this paper is to provide sets of sufficient conditions for complete qth moment convergence of the form ∞ k –q b a E max  X  – a < ∞ for all  >, (.) n ni n ≤k≤n n= i= where q ≥ , {a , n ≥ } and {b , n ≥ } are sequences of positive real numbers, and n n {X , ≤ i ≤ n, n ≥ } is an array of random variables satisfying Marcinkiewicz-Zygmund ni and Rosenthal type inequalities. When q = , similar results were established by Sung []. From our results, we can easily obtain the results of Chen and Wang []and Wu et al. []. Sung Journal of Inequalities and Applications 2013, 2013:24 Page 3 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 2 Main results In this section, we give sets of sufficient conditions for complete qth moment convergence (.). The following theorem gives sufficient conditions under the assumption that the array {X , ≤ i ≤ n, n ≥ } satisfies a Marcinkiewicz-Zygmund type inequality. ni Theorem . Let  ≤ q < and let {X , ≤ i ≤ n, n ≥ } be an array of random variables ni with EX = and E|X | < ∞ for  ≤ i ≤ nand n ≥ . Let {a , n ≥ } and {b , n ≥ } be ni ni n n sequences of positive real numbers. Suppose that the following conditions hold: (i) for some s ( ≤ q < s ≤ ), there exists a positive function α (x) such that E max X (x)– EX (x) ni ni ≤k≤n i= ≤ α (n) E X (x) for all n ≥  and x >, (.) ni i= /q /q /q /q /q where X (x)= X I(|X |≤ x )+ x I(X > x )– x I(X <–x ), ni ni ni ni ni ∞ n –s s (ii) b a α (n) E|X | I(|X |≤ a )< ∞, n s ni ni n n= i= ∞ –q n (iii) b a ( + α (n)) E|X | I(|X | > a )< ∞, n n s ni ni n n= i= (iv) E|X |I(|X | > a )/a → . ni ni n n i= Then (.) holds. Proof It is obvious that ∞ k –q b a E max X – a n ni n ≤k≤n n= i= ∞ k –q /q = b a P max X > a + x dx n ni n ≤k≤n n= i= ∞ k k a   ∞ –q /q ≤ b a P max  X  > a dx + P max  X  > x dx n ni n ni ≤k≤n ≤k≤n  a n= i= i= := I + I .   We first show that I < ∞.For  ≤ i ≤ n and n ≥ , define X = X I |X |≤ a + a I(X > a )– a I(X <–a ), X = X – X . ni ni n n ni n n ni n ni ni ni ni Then we have by EX = , Markov’s inequality, and (i) that ni P max X > a ni n ≤k≤n i= = P max X – EX + X – EX > a ni ni ni ni ≤k≤n i= k k ≤ P max X – EX > a / + P max X – EX > a / n n ni ni ni ni ≤k≤n  ≤k≤n i= i= Sung Journal of Inequalities and Applications 2013, 2013:24 Page 4 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 k k s –s –s   – – ≤   a E max  X – EX  + a E max  X – EX n ni ni n ni ni ≤k≤n  ≤k≤n i= i= n n s –s –s  – – ≤   a α (n) E X + a E X n ni n ni i= i= s –s –s s s ≤   a α (n) E|X | I |X |≤ a + a P |X | > a s ni ni n ni n n n i= – – + a E|X |I |X | > a ni ni n i= n n s –s –s s s –s –q q ≤   a α (n) E|X | I |X |≤ a +  a α (n) E|X | I |X | > a s ni ni n s ni ni n n n i= i= – –q q + a E|X | I |X | > a . ni ni n i= It follows that ∞ k I = b P max  X  > a  n ni n ≤k≤n n= i= ∞ n s –s –s s ≤   b a α (n) E|X | I |X |≤ a n s ni ni n n= i= ∞ n –q s –s – q + b a   α (n)+ E|X | I |X | > a . n s ni ni n n= i= Hence I < ∞ by (ii) and (iii). We next show that I < ∞. By the definition of X (x), we have that ni /q P max  X  > x ni ≤k≤n i= n k /q  /q ≤ P |X | > x + P max  X (x) > x . ni ni ≤k≤n i= i= We also have by EX = and (iv) that ni –/q sup max x EX (x) ni q ≤k≤n x≥a n i= –/q = sup max x E X – X (x) ni ni q ≤k≤n x≥a n i= –/q /q ≤ sup x E|X |I |X | > x ni ni x≥a i= – ≤ a E|X |I |X | > a → . ni ni n i= Sung Journal of Inequalities and Applications 2013, 2013:24 Page 5 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 Hence to prove that I < ∞, it suffices to show that ∞ n –q /q I := b a P |X | > x dx < ∞,  n ni n= i= ∞ k –q   /q I := b a P max X (x)– EX (x) > x / dx < ∞.  n n ni ni ≤k≤n n= i= /q /q If x > a ,then P(|X | > x )= P(|X |I(|X | > a )> x )and so n ni ni ni n ∞ ∞ /q /q P |X | > x dx = P |X |I |X | > a > x dx ni ni ni n q q a a n n /q q ≤ P |X |I |X | > a > x dx = E|X | I |X | > a , ni ni n ni ni n which implies that ∞ n –q q I ≤ b a E|X | I |X | > a .  n ni ni n n= i= Hence I < ∞ by (iii). Finally, we show that I < ∞. We get by Markov’s inequality and (i) that ∞ k s –q –s/q I ≤  b a x E max X (x)– EX (x) dx  n n ni ni ≤k≤n n= i= ∞ n s –q –s/q ≤  b a α (n) x E X (x) dx n s n ni n= i= ∞ n s –q –s/q s /q s/q /q = b a α (n) x E|X | I |X |≤ x + x P |X | > x dx n s ni ni ni n= i= ∞ n s –q s –s/q = b a α (n) E|X | I |X |≤ a x dx n s ni ni n n= i= ∞ n s –q –s/q s /q + b a α (n) x E|X | I a < |X |≤ x dx n s ni n ni n= i= ∞ n s –q /q + b a α (n) P |X | > x dx := I + I + I . n s ni    n= i= Using a simple integral and Fubini’s theorem, we obtain that ∞ n s –s s I = b a α (n) E|X | I |X |≤ a ,  n s ni ni n s – q n= i= ∞ n s –q –s/q s /q I = b a α (n) x E|X | I a < |X |≤ x dx  n s ni n ni n= i= ∞ n s –q q = b a α (n) E|X | I |X | > a . n s ni ni n s – q n= i= Sung Journal of Inequalities and Applications 2013, 2013:24 Page 6 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 Similarly to I , ∞ n s –q q I ≤  b a α (n) E|X | I |X | > a .  n s ni ni n n= i= Hence I < ∞ by (ii) and (iii). The next theorem gives sufficient conditions for complete qth moment convergence (.) under the assumption that the array {X , ≤ i ≤ n, n ≥ } satisfies a Rosenthal type ni inequality. Theorem . Let q ≥  and let {X , ≤ i ≤ n, n ≥ } be an array of random variables ni with EX = and E|X | < ∞ for  ≤ i ≤ nand n ≥ . Let {a , n ≥ } and {b , n ≥ } be ni ni n n sequences of positive real numbers. Suppose that the following conditions hold: (i) for some s > max{, q/r} (r is the same as in (v)), there exist positive functions β (x) and γ (x) such that E max  X (x)– EX (x) ni ni ≤k≤n i= s/ n n s  ≤ β (n) E X (x) + γ (n) E X (x) s s ni ni i= i= for all n ≥  and x >, (.) /q /q /q /q /q where X (x)= X I(|X |≤ x )+ x I(X > x )– x I(X <–x ), ni ni ni ni ni ∞ n –s s (ii) b a β (n) E|X | I(|X |≤ a )< ∞, n s ni ni n n= n i= ∞ –q n (iii) b a ( + β (n)) E|X | I(|X | > a )< ∞, n n s ni ni n n= i= (iv) E|X |I(|X | > a )/a → , ni ni n n i= ∞ n –r r s/ (v) b γ (n)( a E|X | ) < ∞ for some < r ≤ . n s ni n= i= n Then (.) holds. Proof Theproof is similartothatofTheorem .. As in the proof of Theorem ., ∞ k –q b a E max X – a n ni n ≤k≤n n= i= ∞ k k a ∞ –q /q ≤ b a P max X > a dx + P max X > x dx n ni n ni ≤k≤n  ≤k≤n  a n= i= i= := J + J .   Similarly to I in the proof of Theorem .,wehavebythe c -inequality that  r P max  X  > a ni n ≤k≤n i= s –s –s s s ≤   a β (n) E|X | I |X |≤ a + a P |X | > a s ni ni n ni n n n i= Sung Journal of Inequalities and Applications 2013, 2013:24 Page 7 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 s/ s –s –s   +  a γ (n) E|X | I |X |≤ a + a P |X | > a s ni ni n ni n n n i= – – + a E|X |I |X | > a ni ni n i= s –s –s s ≤   a β (n) E|X | I |X |≤ a s ni ni n i= s –s – –q q +   β (n)+ a E|X | I |X | > a s ni ni n i= s/ s/– –s –r r +  γ (n) a E|X | I |X |≤ a s ni ni n i= s/ s/– –s –r r +  γ (n) a E|X | I |X | > a . s ni ni n i= Hence J < ∞ by (ii), (iii), and (v). As in the proof of Theorem .,toprove that J < ∞, it suffices to show that ∞ n –q /q J := b a P |X | > x dx < ∞,  n ni n= i= ∞ k –q   /q J := b a P max X (x)– EX (x) > x / dx < ∞.  n n ni ni ≤k≤n n= i= The proof of J < ∞ is same as that of I in the proof of Theorem ..   For J , we have by Markov’s inequality and (i) that ∞ k s –q –s/q J ≤  b a x E max X (x)– EX (x) dx  n n ni ni ≤k≤n n= i= s/ ∞ n n s  s –q –s/q ≤  b a x β (n) E X (x) + γ (n) E X (x) dx n s s n ni ni n= i= i= := J + J .   Similarly to I in the proof of Theorem .,weget that ∞ n s –s s J ≤  b a β (n) E|X | I |X |≤ a  n s ni ni n s – q n= i= ∞ n s –q q + + b a β (n) E|X | I |X | > a . n s ni ni n s – q n= i= Hence J < ∞ by (ii) and (iii).  Sung Journal of Inequalities and Applications 2013, 2013:24 Page 8 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 Finally, we show that J < ∞.Bythe c -inequality,  r s/ ∞ n s –q –s/q  /q /q /q J = b a γ (n) x E|X | I |X |≤ x + x P |X | > x dx  n s ni ni ni n= i= s/ ∞ n s/– –q –s/q  /q ≤  b a γ (n) x E|X | I |X |≤ x dx n s ni ni n= i= s/ ∞ n s/– –q /q + b a γ (n) P |X | > x dx n s ni n= i= s/ ∞ n s/– –q –s/q r (–r)/q ≤  b a γ (n) x E|X | x dx n s ni n= i= s/ ∞ n s/– –q –r/q r + b a γ (n) x E|X | dx n s ni n= i= s/ ∞ n q s/ –r r = b γ (n) a E|X | . n s ni rs –q n= i= Hence J < ∞ by (v). Remark . Marcinkiewicz-Zygmund and Rosenthal type inequalities hold for depen- dent random variables as well as independent random variables. () Let {X , ≤ i ≤ n, n ≥ } be an array of rowwise negatively associated random vari- ni s –s ables. Then, for  < s ≤ , (.)holds for α (n)=  =. For s >, (.)holds for s s s β (n)= (s/ log s) and γ (n)=(s/ log s) (see Shao []). Note that α (n)and β (n) s s s s s   s s  s are multiplied by the factor  since E|X (x)– EX (x)| ≤  E|X (x)| . ni ni ni () Let {X , ≤ i ≤ n, n ≥ } be an array of rowwise negatively orthant dependent ran- ni dom variables. By Corollary . of Asadian et al. []and Theoremof Móricz [], (.) s s s holds for α (n)= C (log n) ,and (.)holds for β (n)= C (log n) and γ (n)= C (log n) , s  s  s  where C and C are constants depending only on s.   () Let {X , n ≥ } be a sequence of identically distributed ϕ-mixing random variables. Set X = X for  ≤ i ≤ n and n ≥ . By Shao’s []result, (.) holds for a constant function ni i / n β (x) and a slowly varying function γ (x). In particular, if ϕ ( )< ∞,then(.) s s n= holds for some constant functions β (x)and γ (x). s s () Let {X , n ≥ } be a sequence of identically distributed ρ-mixing random variables. Set X = X for  ≤ i ≤ n and n ≥ . By Shao’s []result, (.) holds for some slowly ni i /s n varying functions β (x)and γ (x). In particular, if ρ ( )< ∞,then(.)holds for s s n= some constant functions β (x)and γ (x). s s () Let {X , n ≥ } be a sequence of ρ -mixing random variables. Set X = X for  ≤ i ≤ n n ni i and n ≥ . By the result of Utev and Peligrad [], (.) holds for some constant functions β (x)and γ (x). s s 3 Corollaries In this section, we establish some complete qth moment convergence results by using the results obtained in the previous section. Sung Journal of Inequalities and Applications 2013, 2013:24 Page 9 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 Corollary . (Chen and Wang []) Let {X , n ≥ } be a sequence of identically distributed ϕ-mixing random variables with EX =, and let t ≥ ,  < p <, q ≥ , and pt ≥ . Assume that (.) holds. Furthermore, suppose that / n ϕ  < ∞ n= if t = and max{q, pt} <. Then ∞ k t––q/p /p n E max X – n < ∞ for all  >. ≤k≤n n= i= /p t– Proof Let a = n and b = n for n ≥ , and let X = X for  ≤ i ≤ n and n ≥ . Then, for n n ni i s ≥ , (.) holds for a constant function β (x) and a slowly varying function γ (x)(see Re- s s / n mark .()). Under the additional condition that ϕ ( )< ∞,(.) holds for some n= constant functions β (x)and γ (x). In particular, for s =, (.) holds for a constant func- s s tion α (x) under this additional condition. By a standard method, we have that t––s/p s /p pt n E|X | I |X |≤ n ≤ CE|X | if pt < s,    n= CE|X | if q > pt, ⎪  t––q/p q /p pt n E|X | I |X | > n ≤ CE|X | log( + |X |)if q = pt,     n= pt CE|X | if q < pt, –/p /p –t pt /p n E|X |I |X | > n ≤ n E|X | I |X | > n if pt ≥ ,     where C is a positive constant which is not necessarily the same one in each appear- ance. Hence, the conditions (i)-(iv) of Theorem . hold if we take s > max{pt,,q/r}. / n Under the additional conditions that max{q, pt} < and ϕ ( )< ∞, all condi- n= tions of Theorem . hold if we take s = . Therefore, the result follows from Theo- rems . and . if we only show that the condition (v) of Theorem . holds when t > or max{q, pt}≥ . To do this, we take r = if max{q, pt}≥ and r = max{q, pt} if max{q, pt} <. If t > or max{q, pt}≥ , then r > p and so we can choose s > large enough such that t –+(– r/p)s/ < . Then s/ ∞ n ∞ s/ –r r r t–+(–r/p)s/ b γ (n) a E|X | = E|X | γ (n)n < ∞. n s ni  s n= i= n= Hence the condition (v) of Theorem . holds. Let { (x), n ≥ } be a sequence of positive even functions satisfying (|x|) (|x|) n n ↑ and ↓ as |x|↑ (.) q s |x| |x| for some  ≤ q < s. Sung Journal of Inequalities and Applications 2013, 2013:24 Page 10 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 Corollary . Let { (x), n ≥ } be a sequence of positive even functions satisfying (.) for some  ≤ q < s ≤ . Let {X , ≤ i ≤ n, n ≥ } be an array of random variables satisfying ni EX = for  ≤ i ≤ nand n ≥ , and (.) for some constant function α (x). Let {a , n ≥ } ni s n and {b , n ≥ } be sequences of positive real numbers. Suppose that the following conditions hold: ∞ n (i) b E (|X |)/ (a )< ∞, n i ni i n n= i= (ii) E (|X |)/ (a ) → . i ni i n i= Then (.) holds. q s Proof First note by (|x|)/|x| ↑ that (|x|) is an increasing function. Since (|x|)/|x| ↓, i i i |X | I(|X |≤ a ) (|X |I(|X |≤ a )) (|X |) ni ni n i ni ni n i ni ≤ ≤ . a (a ) (a ) i n i n Since q ≥ and (|x|)/|x| ↑, |X |I(|X | > a ) |X | I(|X | > a ) (|X |I(|X | > a )) (|X |) ni ni n ni ni n i ni ni n i ni ≤ ≤ ≤ . a (a ) (a ) n n i n i n It follows that all conditions of Theorem . are satisfied and so the result follows from Theorem .. Corollary . Let { (x), n ≥ } be a sequence of positive even functions satisfying (.) for some q ≥  and s > max{, q}. Let {X , ≤ i ≤ n, n ≥ } be an array of random variables ni satisfying EX = for  ≤ i ≤ nand n ≥ , and (.) for some constant functions β (x) and ni s γ (x). Let {a , n ≥ } and {b , n ≥ } be sequences of positive real numbers. Suppose that the s n n following conditions hold: ∞ n (i) b E (|X |)/ (a )< ∞, n i ni i n n= i= (ii) E (|X |)/ (a ) → , i ni i n i= ∞ n –  s/ (iii) b ( a E|X | ) < ∞. n ni n= i= n Then (.) holds. Proof The proof is similar to that of Corollary .. By the proof of Corollary . and the condition (iii), all conditions of Theorem . are satisfied and so the result follows from Theorem .. Remark . When b = for n ≥ , the condition (i) of Corollaries . and . is reduced ∞ n to the condition E (|X |)/ (a )< ∞, and so the condition (ii) of Corollar- i ni i n n= i= ies . and . follows from this reduced condition. For a sequence of ρ -mixing random variables, (.) holds for some constant function α (x)if s =, and (.) holds for some constant functions β (x)and γ (x)if s > (see Remark .()). Wu et al. [] proved Corol- s s laries . and . when b = for n ≥ , and {X } is an array of rowwise ρ -mixing random n ni variables. Competing interests The author declares that he has no competing interests. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0013131). Sung Journal of Inequalities and Applications 2013, 2013:24 Page 11 of 11 http://www.journalofinequalitiesandapplications.com/content/2013/1/24 Received: 9 October 2012 Accepted: 3 January 2013 Published: 17 January 2013 References 1. Hsu, PL, Robbins, H: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33, 25-31 (1947) 2. Erdös, P: On a theorem of Hsu and Robbins. Ann. Math. Stat. 20, 286-291 (1949) 3. Baum, LE, Katz, M: Convergence rates in the law of large numbers. Trans. Am. Math. Soc. 120, 108-123 (1965) 4. Chow, YS: On the rate of moment convergence of sample sums and extremes. Bull. Inst. Math. Acad. Sin. 16, 177-201 (1988) 5. Li, D, Spataru, ˘ A: Refinement of convergence rates for tail probabilities. J. Theor. Probab. 18, 933-947 (2005) 6. Chen, PY, Wang, DC: Convergence rates for probabilities of moderate deviations for moving average processes. Acta Math. Sin. 24, 611-622 (2008) 7. 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