In this paper we prove weighted martingale-ergodic and weighted ergodic-martingale theorems. Furthermore, analogous dominant and maximal inequalities for weighted martingale ergodic sequences and weighted ergodic martingale averages are also obtained

Ganiev, Inomjon

Othman, Raihan

English

The International Islamic University Malaysia Repository

Middle-East Journal of Scientific Research 13 (Mathematical Applications in Engineering): 70-76, 2013 ISSN 1990-9233 © IDOSI Publications, 2013 DOI: 10.5829/idosi.mejsr.2013.13.mae.99924  Corresponding Author: Ganiev  I.G.,  Faculty  of  engineering, International Islamic University Malaysia, 57528, P.O. 10, Kuala-Lumpur, Malaysia 70 Weight Martingale-Ergodic and Ergodic-Martingale Theorems  Ganiev I.G. and Othman R.  Faculty of Engineering, International Islamic University Malaysia, 57528, P.O. 10, Kuala-Lumpur, Malaysia  Abstract: In this paper we prove weighted martingale-ergodic and weighted ergodic-martingale theorems. Furthermore, analogous dominant and maximal inequalities for weighted martingale ergodic sequences and weighted ergodic martingale averages are also obtained.   Key words: Ergodic averages  martingale  ergodic-martingale  martingale-ergodic averages  INTRODUCTION   General theories unifying ergodic averages and martingales were reported by Kachurovskii [1-3]. Four different variants for theories unifying ergodic averages and martingales have been reported in [4-7]. Besides, one parameter weighted ergodic theorem and multiparameter weighted ergodic theorems have been investigated by Baxter J.H. Olsen [8] and R.L. Jones, J.H. Olsen [9], respectively. In [10], M. Lin and M. Weber considered weighted ergodic theorems and strong laws of large numbers. General ergodic theory is reported in [11].   In this paper we prove weighted martingale-ergodic and weighted ergodic-martingale theorems. Furthermore, analogous dominant and maximal inequalities for weighted martingale ergodic sequences and weighted ergodic martingale averages are also obtained.   Preliminaries: Let         be a space with a finite measure,          be a space of complex measurable functions on   ,            ∫                , with the norm       (∫        )   if      ,                       if         Let          be a monotone sequence of -subalgebras of  ,       (or                be the expectation operator,          be the Dunford-Schwartz operator. Put             ∑                                                   ∑                                        In [1] it is proved the following   Theorem 1.1 1) Let             . Then                   in   , in this case               Middle-East J. Sci. Res., 13 (Mathematical Applications in Engineering): 70-76, 2013 71 2) Let             . Then                  →     in   .  3) Let     ,                   . Then                 →     in   .  Put           ,            ∑           ,                   .  Theorem 1.2: [2] 1) Let               Then                   in    as    . 2) Let             . Then                  →     in   .  3) Let                        Then                 →     in   .   Lemma 1: [2, 12] Let                 and       in   , as    . Then                   in    as    .  Lemma 2: [2, 12] Let      →    as     and                Then             →           in      Lemma 3: [11] Let      →    as     and                Then            →    in    as     where               .   Definition 1.3: The sequence numbers      is called Besicovich sequences such that given    , there is a trigonometric polynomial       such that             ∑                       A sequence      is called bounded Besicovich if          . In [9] it was proved the following   Theorem 1.4: Let   be denoted the Dunford-Schwartz operator. Then there exists           ∑                  a.e. for every            and all bounded Besicovich sequences       In this case, if        , then           ∑                 (o) -converges in    and if      , then         (o)-converges in   .  In this paper we prove theorems analogous to 1.1 and 1.2 in the case of weighted averages.   WEIGHTED MARTINGALE-ERGODIC THEOREMS   Let          be a monotone sequence of -subalgebras of  ,       (or             be a Besicovich bounded sequence,          be the Dunford-Schwartz operator. We put,             ∑                                                 Theorem 2.1 1) Let     ,        . Then                 →     in   .  2) If      and                 , then                 →     in   .  Middle-East J. Sci. Res., 13 (Mathematical Applications in Engineering): 70-76, 2013 72   Proof: 1) Since                         ∑                            ∑                                ∑                 where               by the Akcoglu’s theorem we have                        According to Theorem 1.2 [9],           →     Therefore Lemma 2 implies                 →              in    as n  .  Let                    Then                       for all n. By Theorem 2 [13]                   Hence,                         . Therefore                 →     in   .   2) By Theorem 1.4 [9] we have            a.e. As                   by Lemma 2 we obtain                    a.e. Since the convergence a.e. is the (o)-convergence in   , we obtain                 →     in   .  WEIGHTED ERGODIC-MARTINGALE THEOREM   Let, as in Section 2,          be a monotone sequence of -subalgebras of  ,       (or             be a Besicovich bounded sequence,          is the Dunford-Schwartz operator. We put                          ∑                                        Lemma 3.1: Let      →                  and                 . Then            →    in   .   Proof: Let                    Obviously,                                                                                                  By theorems 1.2 and 1.4 [9],           →    as     in   . We will prove that                     →   as     in   . It is clear,                                     ∑                               ∑                                       Middle-East J. Sci. Res., 13 (Mathematical Applications in Engineering): 70-76, 2013 73  Since          →   and                 , we obtained from Lemma 3, that                  →   as n  . Therefore                    →   and            →    in   .   Theorem 3.2 1) Let         . Then                 →     in   as       2) Let       and                   . Then                 →     in   .  Proof: 1) According to Theorem 2 [13],                   . Therefore by Lemma 3.1, we have                  →     in    .    Let                    Then             and                            . As by the Akcoglu’s theorem                     we have                           and                            Hence,                →        2) By the Theorem 3 [13], we have             →          in     As                  , by Lemma 3.1 we obtain                  →     in    .   WEIGHTED DOMINANT AND MAXIMAL INEQUALITIES  Theorem 4.1: Let         and     . Then for      the following inequality holds:                                       where               .  Proof: Let                    Then                                                                                                           and therefore                                                By the dominant inequality for submartingale [12] we have                                    Since the conditional expectation operator is contracting in    we obtain, that                      Therefore                                                  Middle-East J. Sci. Res., 13 (Mathematical Applications in Engineering): 70-76, 2013 74  It follows from Akcoglu’s theorem that                               Thus, for      the inequality                                      holds.   Theorem 4.2: Let              and       Then for any     the following inequality holds                                              where               ..   Proof: Let                  . Then                                                                                          By the maximal inequality for for submartingale         [12] we have, that                                                    Applying the contracting property of the conditional expectation operator in   , we obtain                                                                   Now applying the inequality                               , we have                                                            .  The following theorems can be proved analogously to theorems 4.1 and 4.2.   Theorem 4.3: Let         . Then for      the following inequality holds                                       where               .   Theorem 4.4: Let        . Then for any     and      the following inequality holds                                              where               .    MULTIPARAMETER WEIGHTED MARTINGALE ERGODIC THEOREMS   In this section we define the  -dimensional case of martingale ergodic averages and consider convergence theorems for such averages.  Let              be the class of weights and                                                                                                                                       Middle-East J. Sci. Res., 13 (Mathematical Applications in Engineering): 70-76, 2013 75  For                       means that    for any                Definition 5.1: [9] The sequence              is called  -Besicovich if for every     there is a sequence of trigonometric polynomials in d variables,      , such that                 ∑                         This class is denoted by       The sequence  (k) is called  -bounded Besicovitch if               . If                then        is called a bounded Besicovich sequence.  Let            denote a family of d linear operators in   . We consider averages              ∑              where                             In [9] it is proved the following   Theorem 5.2: Let                   denoted d Dunford-Schwartz operators. Then        converges a.e. for every            and all bounded Besicovich sequences     .   Let, as in the Section 2          be a monotone sequence of -subalgebras of   and        We put                                 Theorem 5.3: Let      ,      ,      be a bounded Besicovich sequence,                        as      Then                   →     in    .  The proof of Theorem 5.3 is analogous to the proof of Theorem 2.1 1).   Theorem 5.4: Let             ,            bounded Besicovich sequence,                ,                        as     Then   1)                                     2)                                          Proof 1) Let                     Then                                                 and                                                                                                   (1) Therefore                                          By the dominant inequality,                                 Since the conditional expatiation operators is contracting in     we have                                                                                               (2) Thus  Middle-East J. Sci. Res., 13 (Mathematical Applications in Engineering): 70-76, 2013 76                                                                            2) By the inequality (1) we have                                                                                           According to the maximal inequality                                     Applying inequality (2), we obtain                                                    Since                              we obtain                                            ACKNOWLEDGEMENT  This work has been supported by IIUM-RMC through project EDW B 11-185-0663.  REFERENCES  1. Kachurovskii, A.G., 2007. General theorems unifing ergodic averages and martingales. Prosedings of the Steklov Institute of Mathematics, 256: 160-187.  2. Kachurovskii, A.G., 1998. Martingale ergodic theorem. Math. Notes, 64: 266-269.  3. Kachurovskii,   A.G.,  2001.  Convergence  of  averages  in  the  ergodic  theorem  for  groups   . J. Math. Sci., 107 (5): 4231-4236.  4. Jerson, M., 1959. Martingale formulation of ergodic theorems. Proc. Am. Math. Soc., 10 (4): 531-539.  5. Rota,  G.C.,  1961.  Une  theorie  unifiee  des  martingales  et  des moyennes ergodiques. C.R. Acad. Sci. Paris. 252 (14): 2064- 2066.  6. Ionescu  Tulcea,  A.  and  C.  Ionescu  Tulcea,  1963.  Abstract  ergodic  theorems. Trans. Amer. Math. Soc., 107 (1): 107-124.  7. Vershik, A.M., 1982. Amenability and approximation of infinite groups. Sel. Math. Sov., 2 (4): 311-330.  8. Baxter, J.R. and J.H. Olsen, 1983. Weighted and subsequential ergodic theorems. Can. J. Math., 35: 145-166. 9. Jones, R.L. and J.H. Olsen, 1994. Multiparametr weighted ergodic theorems. Can. J. Math., 46 (2): 343-356. 10. Lin, M., 2007. Veber M.Weighted ergodic theorems and strong laws of large nambers. Ergod. Th. and Dynam. Sys., 27: 511-543.  11. Krendel, U., 1985. Ergodic Theorems. Walter de Grugwer. Berlin, New York, pp: 357. 12. Doob, J.L., 1956. Stochastic Processes, Wiley, New York, 1953, Inostrannaya Literatura, Moscow.  13. Shiryaev, A.N., 1995. Probability. Nauka, Moscow, 1980, Springer, New York.  

Weight martingale-ergodic and ergodic-martingale theorems

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Weight martingale-ergodic and ergodic-martingale theorems

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