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On strong law for blockwise M-orthogonal random fields
Journal of Inequalities and Applications volume 2013, Article number: 380 (2013)
We consider M-orthogonal random fields. Using a lemma from summability theory, we prove strong law of large numbers for blockwise M-orthogonal random fields under various moment conditions, thereby generalizing some results in the literature from independent random fields.
Recently, Móricz et al. (cf. ) using the summability theory proved a strong law of large numbers for blockwise M-dependent random variables under moment conditions. Huan and Quang (cf. ) established the Doob’s inequality for martingale difference arrays and provided a sufficient condition, so that the strong law of large numbers would hold for an arbitrary array of random elements without imposing any geometric condition on the Banach space. Quang et al. (cf. ) provided conditions to obtain the almost sure convergence for a double array of blockwise M-dependent random elements , taking values in a real separable Rademacher-type p (), and they also demonstrated that some of the well-known theorems in the literature were special cases of their results.
Let , where d is a positive integer, denote the positive integer d-dimensional lattice points. Motivated by the results above, in this paper, we are going to study strong law of large numbers for M-orthogonal random fields with . The notation , where and , means that , , means .
Definition 1 The sequence is called a sequence of M-orthogonal random variables if
for all k and l with .
A somewhat weaker dependence condition is given by the following definition.
Definition 2 For given sequences of natural numbers () (as ), we say is blockwise M-orthogonal with respect to blocks if for all , the random variables are M-orthogonal for indices .
The latter definition is a generalization of the corresponding definition for the one-dimensional case. It allows the random variables in the different blocks be strongly dependent. The particular case , , , () is especially interesting.
In order to prove our main results, we shall state the following two lemmas, and it will be shown that they play a key role in the proof.
Lemma 1 (cf. )
Let be a random field with M-orthogonal, centered random variables, if for all , then we have
Next, we consider sequences of real or complex numbers. We say that is boundedly convergent to S if and if for any there exists some such that for all () (Pringsheim convergence). We write shortly (bd).
Lemma 2 (cf. )
Let be positive, strictly increasing unbounded functions on , and let , be strictly increasing sequences of integers with , . Consider the following relations for array sequences as :
Then relation (1.3) implies relation (1.4), provided
and relation (1.4) implies relation (1.3), provided
Consequently, under condition (1.5) and (1.6) the two relations (1.3) and (1.4) are equivalent.
2 The main results and proofs
With the preliminaries accounted for, we can formulate and prove the main results of this paper.
Theorem 1 Let be a random field with centered and integrable random variables being blockwise M-orthogonal with respect to the blocks . Let be as in Lemma 2 satisfying (1.5) and (1.6) with , . If, in addition,
Proof By virtue of Lemma 2, it suffices to show that
At first, we prove (2.1). Applying the Chebyshev’s inequality gives
where we used (1.5) and (1.6) yielding with , , respectively.
Applying Lemma 1 (note that the random variables are within the blocks M-orthogonal), we obtain
where C is a constant, which may differ from line to line. From the Borel-Cantelli lemma it follows that
In order to prove the bounded convergence it remains to show that
Using the same arguments as above with , we obtain that for almost all ω, there exist only finite many such that
These complete the proof. □
Corollary 1 Let be a random field being blockwise M-orthogonal as in Theorem 1 with centered and integrable random variables. If
This follows from Theorem 1 choosing for , the identity function.
In particular, a strong law of large numbers holds for a blockwise M-orthogonal random field with bounded moment for any a condition, which is just a little bit stronger than the necessary moment condition in the i.i.d. case. Next, choose
in Theorem 1, then we obtain the following corollaries, which are related to the Marcienkiewicz laws in .
Corollary 2 Let be a random field as in Theorem 1. If
Corollary 3 Let be a random field with blockwise M-orthogonal, centered random variables satisfying for all , then for any , we have
This follows from our Theorem 1, using , .
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The authors are grateful to the referees for carefully reading the paper and for offering some comments, which helped to improve the paper. The second author is supported by the NNSF of China (No. 11071104), the NSSF of China (13BJY011), PNSF of AnHui (1308085QF113) and GIF of AnHui University of Technology (D2011025).
The authors declare that they have no competing interests.
WZ and XW carried out the design of the study and performed the analysis, WZ drafted the manuscript. All authors read and approved the final manuscript.
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Wu-ling, X., Zhong-zhi, W. On strong law for blockwise M-orthogonal random fields. J Inequal Appl 2013, 380 (2013). https://doi.org/10.1186/1029-242X-2013-380
- random fields
- limit theorem