On weighted averages of double sequences

Transcription

1 nnales Matheaticae et Inforaticae 39 0) pp. 7 8 Proceedings of the Conference on Stochastic Models and their pplications Faculty of Inforatics, University of Derecen, Derecen, Hungary, ugust 4, 0 On weighted averages of doule sequences István Fazekas a, Tior Tóács a University of Derecen, Faculty of Inforatics Derecen, Hungary e-ail: fazekasi@inf.unide.hu Eszterházy Károly College, Institute of Matheatics and Coputer Science Eger, Hungary e-ail: toacs@ektf.hu Dedicated to Mátyás rató on his eightieth irthday. Introduction The well known Kologorov strong law of large nuers states the following. If X, X,... are independent identically distriuted i.i.d.) rando variales with finite expectation and E X 0, then the average X + + X n )/n converges to 0 alost surely a.s.). However, if we consider a doule sequence, then we need another condition. ctually, if X ij ) is a doule sequence of i.i.d. rando variales with E X 0, then E X log + X < iplies that i n j X ij ) /n) converges to 0 a.s., as n, tend to infinity see Sythe [6]). For a doule nuerical sequence x ij there are different notions of convergences. One can consider a strong version of convergence when x ij converges as one of the indices i, j goes to infinity this type of convergence was used in Fazekas []). nother version when x ij converges as oth indices i, j tend to infinity. However, in the second case convergence does not iply oundedness. To avoid unpleasant situations one can assue that the sequence is ounded. In this paper we shall study the so called ounded convergence of doule sequences. We shall prove two criteria for the ounded convergence of weighted averages of doule sequences. Both criteria are ased on susequences. The susequence is constructed y a well-known ethod: we proceed along a non-negative, increasing, unounded sequence and pick up a eer which is aout the doule Supported y the Hungarian Scientific Research Fund under Grant No. OTK T0798/009. 7

2 7 I. Fazekas, T. Tóács of the previous selected eer of the sequence. This ethod was applied e.g. in Fazekas Klesov []). However, this ethod is not convenient for an aritrary doule sequence of weights. Therefore we apply weights of product type it was considered e.g. in Noszály Tóács [5]). Our theores can e considered as generalizations of soe results in Fekete Georgieva Móricz [3], where haronic averages of doule sequences were considered. They otained the following theore. if and only if +n ax <k n <l n ln ln n n i j k x ij ij i + j n + L, as, n.) l x ij L ij 0, as, n..) Here eans the ounded convergence. Our Theore.4 is a generalization of this result for general weights. Our results can also e considered as extensions of certain theores of Móricz and Stadtüller [4] where ordinary that is not doule) sequences were studied. In our proofs we apply ideas of [4].. Main results Let x kl : k, l,,... ) e a sequence of real nuers, and let k : k,,... ), c l : l,,... ) e sequences of weights, that is, sequences of non-negative nuers for which B : k, as,.) C n : k n c l, as n..) l Let a kl : k c l, : n k l a kl and S n : n k l a klx kl. The weighted averages Z n of the sequence x kl ) with respect to the weights a kl ) are defined y Z n : S n for n, large enough so that > 0. We define a sequence 0 0, < < 3 <... of integers with the following property B i+ < B i B i+, i,,....3)

3 On weighted averages of doule sequences 73 Siilarly, let n 0 0, n < n < n 3 <... e a sequence of integers such that C nj+ < C nj C nj+, j,,....4) In this paper we shall also use the following notation n st : n ks+ lt+ a kl, n st S : n ks+ lt+ a kl x kl. ctually n st is an increent on a rectangle in other word two-diensional difference) of the sequence. We note that i+nj+ i+n j+ is called the oving average of the sequence x kl ) with respect to the weights a kl ). Definition.. Let y kl : k, l,,... ) e a sequence of real nuers, and let y e a real nuer. It is said that ounded convergence y kl S y, as k, l, is satisfied if i) the sequence y kl : k, l,,... ) is ounded; and ii) for every ε > 0 there exist positive integers k 0, l 0, such that y kl y < ε for k k 0, l l 0..5) Reark.. Relation.5) does not iply that y kl ) is ounded. For exaple if y l l for l and y kl y for k, l, then.5) holds ut y kl ) is unounded. Theore.3. Suppose that conditions.) and.) are satisfied. soe constant L, we have Then for Z in j L, as i, j.6) if and only if i+nj+ in j i+n j+ S L, as i, j,.7) where the sequences i ) and n j ) are defined in.3) and.4). Theore.4. ssue that B / + δ and C /c + δ for eing large enough where δ > 0. ssue that conditions.),.) are satisfied. Then for soe constant L, we have Z n L, as, n.8)

4 74 I. Fazekas, T. Tóács if and only if i+nj+ ax i< i+ n j<n n j+ n k i+ ln j+ where the sequences i ) and n j ) are defined in.3) and.4). 0, as i, j,.9) The following two corollaries characterize the strong law of large nuers for weighted averages of a sequence of rando variales with two-diensional indices. These corollaries are consequences of Theore.3 and.4. Corollary.5. Let X kl : k, l,,... ) e a sequence of rando variales. If conditions.) and.) are satisfied, then for soe constant L, we have in j i n j a kl X kl k l L, as i, j a.s. if and only if i+nj+ i+ n j+ k i+ ln j+ a kl X kl L, as i, j a.s., where the sequences i ) and n j ) are defined in.3) and.4). Corollary.6. Let X kl : k, l,,... ) e a sequence of rando variales. ssue that B / + δ and C /c + δ for eing large enough where δ > 0. ssue that conditions.) and.) are satisfied. Then for soe constant L, we have n a kl X kl L, as, n a.s. if and only if i+nj+ in j k l ax i< i+ n j<n n j+ n k i+ ln j+ a kl X kl L) 0, as i, j where the sequences i ) and n j ) are defined in.3) and.4). a.s., Reark.7. In the aove two corollaries L can e an a.s. finite rando variale, as well. Reark.8. The results of this section can e generalized for sequences with d- diensional indices.

5 On weighted averages of doule sequences Proofs of Theores.3 and.4 Proof of Theore.3. Let ε e a fixed positive real nuer. First we prove the necessity. ssue that.6) is satisfied, that is, there exist integers i 0, j 0 such that Zin j L < ε for all i i0, j j 0, furtherore Z in j ) is a ounded sequence. So, if i i 0, j j 0, then we have i+nj+ i+n j+ S L i+nj+ i+nj+ in j S L i+nj+ Si+n i+nj+ j+ L i+n j+ ) S in j+ L in j+ ) S i+n j L i+n j ) + S in j L in j ) i+n j+ Zi+n j+ L + Z in j+ L + Z i+n j L + Z in j L ) i+nj+ < 4ε i+n j+ i+nj+ 4ε B i+ C nj+ 6ε. 3.) B i+ B i C nj+ C nj Now, turn to the oundedness. Siilarly as aove i+nj+ S B i+ C nj+ Zi+n B i+ B i C nj+ C j+ + Z in j+ nj i+n j+ + Z i+n j + Z in j ) const., 3.) ecause Z in j ) is ounded. Inequalities 3.) and 3.) iply.7). Now, we turn to sufficiency. ssue that.7) is satisfied, that is, there exist integers i 0, j 0 such that S L < ε for all i i 0, j j 0, 3.3) furtherore i+nj+ i+n j+ i+ n j+ i n j i+n j+ then i+ > i0 and n j+ > n j0, so Z i+n j+ L S i+n j+ L i+n j+ ) i+n j+ i0 n j0 + i+n j+ k l ) S is a ounded sequence. If i i 0 and j j 0, i+ i+n j+ n j+ k i0 + ln j0 + i+ k n j+ l

6 76 I. Fazekas, T. Tóács + i0 n j+ k ln j0 + + i+ k i0 + l n j0 3.4) for all i i 0, j j 0. Consider the first ter in 3.4). Since i+ 0, as i, j, then n j+ there exist integers i i 0 and j j 0, such that i0 n j0 a kl x kl L) < ε for all i i, j j. 3.5) i+n j+ k l Now, turn to the secont ter in 3.4). If i k, then B k+ B k B i+ Siilarly, if j l, then ki 0 B k+ B k B k+ Hence we get fro 3.3) i+ n j+ i+n j+ i j i+n j+ ki 0 lj 0 i j B B k+ k B i+ ki 0 lj 0 i ) i k j < ε k i0 + ln j0 + lj 0 C nl+ C nl C nj+ k+ s k + tn l + B k+ B k+ B k+ B k+3. ) j l. n l+ a st x st L) C nl+ C n l C nj+ B i B i+ ) i k. L) k+n l+ k n l k+n l+ k n l S ) j l < 4ε for all i i 0, j j ) For the third ter in 3.4) we have i0 n j+ i+n j+ k ln j0 + i 0 j k+ n l+ i+n a st x st L) j+ k0 lj 0 s k + tn l + i 0 j B B k+ k C C L) nl+ n l B i+ C nj+ k+n l+ k0 lj k n l k+n l+ k n l S 0

7 On weighted averages of doule sequences 77 i 0 B k+ B k ) B i+ k0 const. B i0 B i+ i 0 k0 j lj 0 B k+ B k B i0 const. B i0 B i+ 0, as i. ) j l const. const. B i0 B i+ i 0 k0 ) i0 k Hence, there exists i i such that i0 n j+ i+n j+ < ε for all i i, j j ) k ln j0 + Siilarly, for the fourth ter in 3.4) we otain that there exists j j such that i+ n j0 i+n j+ < ε for all i i 0, j j. 3.8) By 3.4) 3.8), we have k i0 + l Z i+n j+ L < 7ε for all i i, j j. 3.9) Finally, turn to the proof of oundedness. Z in j i n j a kl x kl i j in j in k l j k0 l0 i j B k+ B k C C nl+ n l B i k0 l0 i const. k0 B k+ B k B i j l0 k+n l+ k n l C nj k+n l+ C nl+ C nl C nj S k n l k+n l+ k n l 4 const. This inequality and 3.9) iply.6). Thus the theore is proved. Proof of Theore.4. Let ε e a fixed positive real nuer. First we prove the necessity. ssue that.8) is satisfied, that is, there exist integers M 0, N 0 such that Z n L < ε for all M 0, n N 0, 3.0) furtherore Z n ) is a ounded sequence. Since we have n k i+ ln j+ Z n L) inz in L) S

8 78 I. Fazekas, T. Tóács j Z nj L) + in j Z in j L), if > i and n > n j, hence the ratio on the left-hand side in.9) is less than or equal to i+n j+ i+nj+ ax Z n L + ax Z in L i< i+ n j<n n j+ n j<n n j+ + ax Z nj L + Z in j L. 3.) i< i+ There exist integers i 0, j 0 such that if i i 0 and j j 0, than i M 0 and n j N 0. So 3.0) and 3.) iply, that the ratio on the left-hand side in.9) is less than i+n j+ i+nj+ in j 4ε 6ε for all i i 0, j j 0. 3.) On the other hand, since Z n ) is a ounded sequence, so y 3.), the ratio on the left-hand side in.9) is less than or equal to i+n j+ 4 const. 6 const. for all i, j. i+nj+ This fact and 3.) iply.9). Now we turn to sufficiency. ssue that.9) is satisfied. The ratio on the left-hand side in.9) is greater than or equal to i+nj+ i+ n j+ k i+ ln j+ i+nj+ i+n j+ S L, so.7) is satisfied. Now, applying Theore.3, we get that.6) is true. In the following parts of the proof, for fixed integers, n let i, j e integers, such that We have Z n L + i n j k l i < i+ and n j < n n j+. k l n + n j k i+ l + n k i+ ln j+ i n k ln j+. 3.3)

9 On weighted averages of doule sequences 79 Consider the asolute values of all ters of this su. For the first ter, fro.6) we get that i n j a kl x kl L) k l Z in j L Z in j L 0, as, n. 3.4) n We shall use the following relations for the coefficients. i+nj+ B i+ B i )C nj+ C nj ) B i+ B C n + ) ) i+ Cnj+ B i+ 4 B i+ B i+ C nj+ + ) i+ + c ) n j+ B i+ C nj+ To see the aove relation, we ention that B + B + + c n j+ C nj+ C nj+ B i+ C nj+ const. 3.5) B + δ, ecause of the assuptions of the theore. Therefore /B ) is a ounded sequence. Siilarly c n /C n ) is a ounded sequence, too. Consider the second ter in 3.3). Fro 3.5) and.9) we get that n k i+ ln j+ i+n j+ as, n. i+nj+ ax i<t i+ n j<s n j+ t s k i+ ln j+ 0, 3.6) Now turn to the third and fourth ters on the left hand side of 3.3). notation s n t Φ it : ax i+nt in t i<s i+ k i+ ln t + With we get that k i+ l n j j t n t k i+ ln t +

10 80 I. Fazekas, T. Tóács But j t i+nt in t Φ it B i+ B i B i+ j t C nt C nt Φ it. 3.7) C nj+ B i+ B i B i+ < i+ + B i B i+ < + B i+ B i i+ B i+ < + i+, B i+ which is ounded as we have already seen. Furtherore, for t,,..., j, C nt C nt C nj+ C n t C nt C nt C nt C nt+ Cn j C nt+ C nt+ C nj C nj C nj+ ) j t+. Hence 3.7) iplies that k i+ l n j j const. t ) j t Φ it. 3.8) By.9), Φ it 0. This and 3.8) iply that the expression on the left-hand side in 3.8) is ounded. Moreover, there exist i 0, j 0 such that Φ it < ε and at the sae tie /) t < ε for all i i 0, t j 0. Fro these facts and applying that the sequence Φ it is ounded, we get j t ) j t Φ it < const. j 0 t ) j/ j0 t ) j t Φ it + So it follows fro 3.8) that n j k i+ l j tj 0+ ) j t Φ it ) j/ t + ε < const.ε for all i i 0, j j 0. 0, as, n. 3.9) By siilar arguents, for the fourth ter in 3.3), we have i n 0, as, n. 3.0) k ln j+ Finally 3.3), 3.4), 3.6), 3.9) and 3.0) iply.8). Thus the theore is proved.

11 On weighted averages of doule sequences 8 References [] Fazekas, I., Marcinkiewicz strong law of large nuers for B-valued rando variales with ultidiensional indices, Statistics and proaility, Proc. 3rd Pannonian Syp., Visegrád/Hungary, 98, Reidel, Dordrecht, 53 6, 984. [] Fazekas, I., Klesov, O. I., general approach to the strong law of large nuers, Theory of Proaility pplications, 45/3, , 000. [3] Fekete, Á., Georgieva, I., Móricz, F., Characterizations of the convergence of haronic averages of doule nuerical sequences, Matheatical Inequalities & pplications, Volue 4, no. 3, , 0. [4] Móricz, F., Stadtüller, U., Characterization of the convergence of weighted averages of sequences and functions, Manuscript, 0. [5] Noszály, Cs., Tóács, T., general approach to strong laws of large nuers for fields of rando variales, nnales Univ. Sci. Budapest, 43, 6 78, 000. [6] Sythe, R. T., Strong laws of large nuers for r-diensional arrays of rando variales, nn. Proaility, no., 64 70, 973.