Almost Sure Convergence of the General Jamison Weighted Sum of B-Valued Random Variables

Transcription

1 Acta Mathematica Sinica, English Series Feb., 24, Vol.2, No., pp Almost Sure Convergence of the General Jamison Weighted Sum of B-Valued Random Variables Chun SU Tie Jun TONG Department of Statistics and Finance, University of Science and Technology of China, Hefei 2326, P.R.China suchun@ustc.edu.cn, tong@pstat.ucsb.edu Abstract In this paper, two new functions are introduced to depict the Jamison weighted sum of random variables instead using the common methods, their properties and relationships are systematically discussed. We also analysed the implication of the conditions in previous papers. Then we apply these consequences to B-valued random variables, and greatly improve the original results of the strong convergence of the general Jamison weighted sum. Furthermore, our discussions are useful to the corresponding questions of real-valued random variables. Keywords Almost sure convergence, B-valued random variable, General Jamison weighted sum, p-smooth Banach space, Banach space of type p MR(2) Subject Classification 6F5 Introduction There has been much research work (for example, [, 2] and [3]) about the almost sure convergence of the general Jamison weighted sum of real-valued independent random variables and negatively associated random variables, while the articles discussing the same problem in Banach space are very few. Recently, Liu Jingjun and Gan Shixin have done some significant work in this field (see [4]), but as compared with real-valued random variables, there still remains much to be desired. The purpose of this article is to make some progress in this situation. And the concepts in this article are the same as in [4]. In this paper, we let {Ω, F, P} be a complete probability space and B be a real separable Banach space with norm. The Banach space B is called type p ( p 2) if there exists a c = c p > such that p E X i c E X i p, n, i= i= where the independent B-valued random variables X,..., X n have mean zero and E X i p <, i=,..., n. Received April, 2, Revised April 9, 2, Accepted June 8, 2 Research supported by National Science Foundation of China (No.78) and special financial support of Chinese Academy of Sciences

2 82 Su C. and Tong T. J. Also in this paper, c denotes a finite positive constant which may be different at different places; {X n } X means sup n P ( X n >x) cp (X > x), where x>andx is some real-valued random variable. Let {a k,k N}and {b k,k N}besequences of real numbers, in which a k,<b k. {X n,n N}is a sequence of B -valued random variables. We will discuss the conditions satisfying a i X i = a.s. i= As compared with [4], there are several differences in our article: We remove the requirement that {a k,k N}is a sequence of positive numbers. 2 We remove the requirement that {b k / a k,k N}is strictly increasing. 3 We haven t any additional requirement about the convergence order of k=n /αp k, where α k = b k / a k,p>. First we introduce several notations: α k = b k a k for k N, and N(x) =#{k : α k x} for x>, where #A denotes the element number of set A, and we suppose N(x) <, x >. Denote x =inf{x : N(x) > }. Clearly, from N() <, we know that there are only finite elements in {k : α k }. Hence x =inf{α k } >. Now we define two other functions: x N(t) x L(x) = x t 2 dt = N(t) t 2 dt, and R N(t) p(x) = dt, x tp+ for x x and p>. The function N(x) is familiar, we can see it in many references (for example see [4]), but L(x)andR p (x) are unfamiliar, here we need to introduce their background. The following condition was used many times in [4]: max = O(n), (.) k n αp k α p k=n k where p>andα k = b k / a k. Clearly, the necessary condition of (.) is <. (.2) First, we have the following Lemma. Condition (.2) implies that where p is the same as that in (.2). α p k R p (x) <, x x, (.3) First we have > α p k α p n=2 k:n <α k n k n=2 N(n) N(n ) n p

3 Almost Sure Convergence of the General Jamison Weighted Sum 83 2 p p 2 p n=2 n=2 N(n) N(n ) (n ) p =2 p N(n) n n n=2 ( N(n) y p+ dy 2 p N() p 2 p (n ) p n p n n=2 n ) 2 p N() N(y) y p+ dy 2 p N() = p N(y) 2 p y p+ dy 2 p N() = pr p() 2 p 2 p N(). Because N() <, wehaver p () <. Noting that R p (x) is a non-increasing function, R p (x) <, for all x. Trivially, for <x <, N(y) N() dy x yp+ x p+ <, which implies R p (x) <, for all x (, ). This lemma explains the background of R p (x) very well. Moreover, we have Lemma.2 p>, Then we have and Suppose X is a non-negative real-valued random variable such that, for some EX p R p (X) <. (.4) EX r R r (X) <, r >p, (.5) EN(X) <. (.6) For x>x,wehave x r R r (x) =x r N(y) y r p N(y) dy xp x yr+ x y r+ Hence (.4) implies (.5) for any r>p.also N(y) R p (x) = dy N(x) yp+ x x dy = x p x dy y p+ = N(x) p x p, N(y) y p+ dy = xp R p (x). so (.4) implies (.6). For L(x), obviously L(x) <, x x,andifx is a non-negative real-valued random variable, we have the following lemma: Lemma.3 Suppose X is a non-negative real-valued random variable. Then the next two conditions are equivalent to each other: EXL(X) <, (.7) ( ) X EN dt <, (.8) t and each of them implies that EN(X/t) <, a.e. t. Obviously, EN ( ) X dt = t ( ( ) ) x N dp (X x) dt t

4 84 Su C. and Tong T. J. ( ( ) ) x = N dt dp (X x) t ( x ) N(y) = x y 2 dy dp (X x)(let y = x/t) = EXL(X). So we see that (.7) and (.8) are equivalent to each other, and each of them implies EN(X/t) < a.e. t. 2 More Remarks on Condition (.) In order to further our discussion and make comparison with [4], we will make more remarks on Condition (.). For this reason, we denote d n =max k n α k. Clearly d n is non-decreasing, and (.) is equivalent to = O(n), (2.) d p n α p k=n k where p>. It is easy to see that (2.) implies <, (2.2) and α p k d p n d p k=n k = O(n). (2.3) (2.2) indicates that k, {α k } is not a finite accumulative point, and α k as k. Hence N(x) is a right continuous ascending step function, and we can construct a continuous function Ñ(x) that satisfies { N(x) if x is a jumping point of N(x), Ñ(x) = Linear if x lies between two jumping points of N(x). And let L(x) = x Ñ(t) x t dt, and R 2 p (x) = Ñ(t) x t dt, for x x p+ and p>. Clearly, Ñ(x) and L(x) are strictly increasing, R p (x) is strictly decreasing, and N(x) Ñ(x),L(x) L(x),R p (x) R p (x). Now we prove several lemmas. Lemma 2. Suppose there exists some p> such that R p (x ) <. Theng p (x) =x p Rp (x), x x must be a strictly increasing function of x. By noting that Ñ(x) is continuous, we have g p(x) =px p R p (x)+x p d R p (x) = px p dx x = ) (px p Ñ(t) dt Ñ(x) > x tp+ x = (Ñ(x) Ñ(x)) =. x So g p (x) is strictly increasing when x x. Next we estimate the order of N(d n ). x Ñ(t) t ( px p Ñ(x) dt p+ xñ(x) ) dt Ñ(x) tp+ x

5 Almost Sure Convergence of the General Jamison Weighted Sum 85 Lemma 2.2 where c>. If (.) holds, then n N(d n ) cn, (2.4) Because d n =max k n α k,wehave{, 2,..., n} {k : α k d n }. So N(d n )=#{k : α k d n } n. On the other hand, for s>, denoting A n (s) ={α k : α k s, k n}, then by (2.), there must exist c > such that c n d p n d p n Hence α p k=n k B n (s) ={α k : α k s, k<n}, α p k A k n(d n) #A n (d n ). N(d n )=#A n (d n )+#B n (d n ) c n + n =(c +)n =cn, where c>. Using Lemmas 2. and 2.2, we have: Lemma 2.3 Suppose X is a non-negative real-valued random variable. If (.) holds, then (.6) is equivalent to EX p R p (X) EX p Rp (X) <. (2.5) By Lemma.2, we need to prove only that under Condition (.), the condition (.6) implies (2.5). From (2.), we know that (2.3) and (2.4) hold. We notice that d j, j must be the jumping points of N(x), and Ñ(x) is a strictly increasing function. n N, wehave Ñ(y) R p (d n )= d n y p+ dy = dn+ Ñ(y) k=n d n y p+ dy [ Ñ(d k+ ) p d p ] k=n k d p k+ = [ N(d k+ ) ] (by definition of Ñ(t)) p c k=n [ k k=n d p k d p k ] d p k+ d p k+ = cn d p n + c d p n k=n+. (2.6) Noting that g p (x) =x p Rp (x) is a strictly increasing function, we have EX p R p (X) EX p R p (X) d p R(d )+ EX p R p (X)I(d n X<d n+ ) d p R(d )+ d p R(d )+c + c n= d p R n+ p (d n+ )P (d n X<d n+ ) n= (n +)P (d n X<d n+ ) n= P (d n X<d n+ )d p n+ n=. d p k=n+ k

6 86 Su C. and Tong T. J. So by the condition (2.3), we have EX p R p (X) EX p Rp (X) d p R(d )+c np (d n X<d n+ ) m= n= = d p R(d )+c P (X d m ) d p R(d )+c P (N(X) N(d m )) d p R(d )+c m= P (N(X) m) =d p R(d )+cen(x) <. m= 3 Preinary Work In this section, we prove several valuable lemmas. As compared with [2], our method is much concise. Lemma 3. If X is a non-negative real-valued random variable such that EN(X) <, then P (X α k) <. We have P (X α k )= = dp (X x) = I(x α k )dp (X x) α k I(α k x)dp (X x) = N(x)dP (X x) = EN(X) <. Lemma 3.2 If X is a non-negative real-valued random variable such that E(X p R p (X)) <, for some <p 2, then EX p I(X α k ) <. α p k First, from Lemma.2, we know EN(X) pex p R p (X) <. Because {α k } is not necessarily monotone, we re-align α, α 2,..., α n as α n, α n,2 α n,n, if α i = α j when i<j, then assume α i precedes α j. It is clear that N(α n,k ) k, and using the inequality α p k EX p I(X t) p we have, for all n N, EX p I(X α k ) p = p α p k ( X u p P u >α k t αk ) du = p x p P (X >x)dx, x p P (X >x)dx ( ) X u p P u >α n,k du

7 Almost Sure Convergence of the General Jamison Weighted Sum 87 ( ( ) ) X p u p P N N(α n,k ) du (since N(x) is non-decreasing) u ( ( ) ) X ( ( ( ) )) X p u p P N k du p u p P N k du u u ( ) X ( ) X = p u p EN du = pe u p N du u u ( ( ) ) x = p u p N du dp (X x) u ( ) = p x p N(t) dt dp (X x) =pe(x p R x tp+ p (X)) <. The right hands of the above inequality is independent of n, so α p EX p I(X α k ) = k α p EX p I(X α k ) <. k Lemma 3.3 If X is a non-negative real-valued random variable such that EXL(X) <, then EXI(X α k ) EXL(X) <. α k Omitted. Remark 3. The method of the proof is similiar to that of Lemma 3.2. Lemma 3.4 Suppose that X is a B-valued random variable, and P ( X t) cp (X t), t >, where X is a non-negative real-valued random variable. Then q >,t>, we have E X q I( X t) ct q P (X >t)+cex q I(X t), E X q I( X >t) cex q I(X >t). Omitted. 4 Main Results In this section, we suppose that X is a non-negative real-valued random variable, and B is a Banach space. On the basis of the results in the above sections, we can improve Theorems in [4] substantially. Theorem 4. Suppose that {a k,k N}and {b k,k N}are sequences of real numbers such that a k and <b k.let{x n,n N}be a sequence of B-valued random variables with {X n } X. IfX satisfies EXR (X) <, (4.) then a k X k = a.s. (4.2)

8 88 Su C. and Tong T. J. Define First, from Lemma.2, we know that (4.) implies EN(X) <. (4.3) Y n = X n I( X n α n ), Z n = X n I( X n >α n ), where α n = / a n,n N.Notingthat{X n } X, so by (4.3) and Lemma 3., we have P ( Z n )= P (X n = Z n ) P ( X n α n ) P (X α n ) <. n= n= n= By the Borel Cantelli lemma we get P ( Z n, i.o.) =. Then a n Z n Z n < a.s. α n n= n= Hence, by the Kronecher lemma we have a k Z k a.s. n. (4.4) n= On the other hand, from Lemma 3.4 and {X n } X, weknow E Y n = E X n I( X n α n ) α n P (X >α n )+EXI(X α n ). Hence, by Lemmas 3. and 3.2 we have E Y n P (X >α n )+ EXI(X α n ) <. α n α n So By the Kronecker lemma again n= n= and using the C r -inequality, we can get So α n Y n < n= a.s. a k Y k a.s., a k Y k a k Y k. n= a k Y k a.s., n. (4.5) Then from (4.4) and (4.5), we obtain a k X k a.s., n. Remark 4. Comparing with Theorem 2. in [4], we require neither a k > norα k = b k / a k to be strictly increasing, and Conditions () and (2) in Theorem 2. [4] implies (4.), so our conditions are much weaker than those in Theorem 2. [4]. In addition, (4.) is very concise.

9 Almost Sure Convergence of the General Jamison Weighted Sum 89 Theorem 4.2 Suppose that {a k,k N}, {b k,k N}are the same as in Theorem 4.. Let B be a p-smooth Banach space for some p 2, and{x n,n N}be a sequence of B-valued integrable random variables with {X n } X. IfX satisfies E(X p R p (X)) <, (4.6) EXL(X) <, (4.7) then where F = {φ, Ω}, F k = σ{x,..., X k }, k. a k (X k E(X k F k )) = a.s., First we define {Y n }, {Z n } as in Theorem 4.. To prove the result, we need to prove only that a k (Y k E(Y k F k )) = a.s., (4.8) Noting that { m a i i= i= a k (Z k E(Z k F k )) = a.s. (4.9) a i b i (Y i E(Y i F i )), F m,m } is a martingale, we need to prove only b i U i converges a.s. in order to prove (4.8), where U i := Y i E(Y i F i ),i. And by the B-valued martingale convergence theorem [5] we need to prove only that { m a k b k U k,m } is L p -bounded. Using the property of a p-smooth Banach space, we have m E a k p m U k b k c E m α p U k p c p 2 p E k α p Y k p k m c E X k p I( X k α k ) c α p k α p k E X k p I( X k α k ) c P (X >α k )+c EX p I(X <α k ). α p k Hence, using the same method as in Theorem 4., we can get ( m sup E a k p) p U k n b k <. Now we prove (4.9). Let V k = Z k E(Z k F k ), k. Then E V k E Z k +E E(Z k F k 2E Z k, k. So by Lemma 3.3 and (4.7), ( ) a k E V k b k E V k 2 E Z k α k α k

10 9 Su C. and Tong T. J. This implies 2 c α k α k P ( X k t)dt 2 α k α k EXI(X α k ) EXL(X) <. a k V k b k < Hence, by the Kronecher lemma again, (4.9) holds. a.s. α k cp (X t)dt Remark 4.2 This is the same as Theorem 4., here we remove the requirement that a k > and α k = b k / a k is strictly increasing. Conditions () and (3) in Theorem 2.2 [4] imply (4.6), and Condition (2) in Theorem 2.2 [4] is equivalent to (4.7) by Lemma.3. Corollary 4. Suppose that {a n,n }, {,n } are the same as in Theorem 4.. LetB be a p-smooth Banach space for some p 2, and{x n, F n,n } be a martingale difference sequence with {X n } X. IfX satisfies E(X p R p (X)) <, and EXL(X) <, then a k X k = a.s. Theorem 4.3 Suppose that {a n,n }, {,n } are the same as in Theorem 4.. Let B be of type p for some p 2, and {X n,n } be a sequence of B-valued independent random variables with EX n =,n and {X n } X. If X satisfies E(X p R p (X)) <, and EXL(X) <, then a k X k = a.s. Omitted. Remark 4.3 The method of the proof is similar to that of Theorem 4.2. Corollary 4.2 Suppose that {a n,n }, {,n } are the same as in Theorem 4.. Let B be of type p for some p 2, and{x n,n } be a sequence of B-valued independent random variables with {X n } X. IfX satisfies E(X p R p (X)) <, then there exists c n B, n =, 2,..., such that a k X k c n a.s. b n Define Y n and Z n as being the same as in the proof of Theorem 4.. First it is easy to know that a k (Y k EY k ) a.s. b n On the other hand, E(X p R p (X)) < implies EN(X) < by Lemma.2. So by Lemma 3.

11 Almost Sure Convergence of the General Jamison Weighted Sum 9 we can get b n a k Z k Then by letting c n = n a key k,wehave a k X k c n The proof is finished. b n a.s. a.s. Remark 4.4 Howell, Taylor and Woyczynshi [6] (98) proved that, under the conditions a i > fori, EN(X) < and t p N(y) P (X >t) dydt <, t yp+ there exists c n B, n =, 2,..., such that a k X k c n a.s. b n Seeing that we have removed the conditions EN(X) < and a i > fori, together with a trivial fact that t p N(y) P (X >t) dydt = t p P (X >t)r t yp+ p (t)dt EX p R p (X), so we say that Corollary 4.2 improves their result. Finally, as a supplement to Theorem 4.2, we will offer a result based on EXL(X) =. (4.) Theorem 4.4 <b k and Let {a n,n N} and {,n N} be sequences of positive numbers with a k = O( ), n. (4.) Let B be of type p for some p 2, and{x n,n N} be a sequence of B-valued independent random variables with EX n =and {X n } X. IfX satisfies EXL(X) =, E(X p R p (X)) <, EX <, (4.2) then (4.2) holds. Using the same method as in Theorem 4.2, we can know that (4.4) and (4.5) hold, so it suffices to show that a k EY k a.s. (4.3) Since EX k =,k N, it suffices to show that a k EZ k a.s. (4.4) Noting that a k is non-negative, and EZ k = EX k I( X k α k ) E X k I( X k α k ) cexi(x α k ),

12 92 Su C. and Tong T. J. hence, it suffices to show that n= a k EXI(X α k ) a.s. (4.5) By (4.), there exists c such that <c <, sothat a k c, (4.6) for all n N. From Lemma.2, we know that (4.6) implies EN(X) <. So it is easy to know that α k. ε >, there exists β> such that EXI(X β) < ε, c where c is the same as in (4.6). Since α k, there exists n N such that Hence, n >n,wehave min α k >β, k>n k=n + sup EXI(X α k ) < ε. k>n c ( a k EXI(X α k ) < on the other hand, clearly we have n a k EXI(X α k ), So (4.4) holds. ) ε a k <ε; (4.7) c n. References [] Chen, X. R, Zhu, L. X., Fang, K. T.: Almost sure convergence of weighted sum. Statistica Sinica, 6(2), (996) [2] Wang, D. C., Su, C., Leng, J. S.: Almost sure convergence of general Jamison s sum. Submitted to Journal of Applied Math. [3] Su, C., Wang, Y. B.: Strong Convergence of Identified N.A. Random Variable. Chinese Journal of Applied Probab. and Stat., 4(2), 3 4 (998) [4] Liu, J. J., Gan, S. X.: Strong Convergence of Weighted Sum of Random Variables. Acta Mathematica Sinica, 4(4), (998) [5] Hu, D. H., Gan S. X.: Modern Martingale Theory, Wuhan University Press, Wuhan, 994 [6] Howell, J., Taylor, R. L., Woyczynski, W. A.: Stability of linear forms in independent random variables in Banach spaces, Lecture Notes in Math., 86, , Springer-Verlag, New York, 98 [7] Wang, Y. B., Liu, X. G., Liang, Q.: Strong Stability of General Jamison Weighted Sum of N.A. random variables. Chinese Science Bulletin, 42(22), (997) [8] Liang H. Y., Su, C.: Strong Laws for Weighted Sum of Random Elements. Statistica Sinica, (3), 9 (2) [9] Adler, A., Rosalsky, A., Taylor, R. L.: A Weak law for normed weighted sum of random elements in Rademacher type P Banach spaces. J. Multivariate Anal., 37, (99)