On complete convergence for weighted sums of martingale-difference random fields
© Ko; licensee Springer. 2013
Received: 21 June 2013
Accepted: 23 September 2013
Published: 7 November 2013
Let be an array of real numbers, and let be the martingale differences with respect to satisfying a.s., where denotes componentwise minimum, is a family of σ-algebras such that , , and X is any integrable random variable defined on the initial probability space. The aim of this paper is to obtain some results concerning complete convergence of weighted sums .
This result has been generalized and extended to the random fields of random variables. For example, Fazekas and Tómács  and Czerebak-Mrozowicz et al.  for fields of pairwise independent random variables, and Gut and Stadtmüller  for random fields of i.i.d. random variables.
Let be the set of positive integers. For fixed , set with coordinatewise partial order, ≤, i.e., for , if and only if , . For , let . For a field of real numbers, the limit superior is defined by and is denoted by .
Note that is equivalent to , which is weaker than the condition when .
Let be a field of random variables, and let be an array of real numbers. The weighted sums can play an important role in various applied and theoretical problems, such as those of the least squares estimators (see Kafles and Bhaskara Rao ) and M-estimates (see Rao and Zhao ) in linear models, the nonparametric regression estimators (see Priestley and Chao ), etc. So, the study of the limiting behavior of the weighted sums is very important and significant (see Chen and Hao ).
where denotes the componentwise minimum.
where and , are martingale differences with respect to (see Kuczmaszewska and Lagodowski ).
For the results concerning complete convergence for martingale arrays obtained in the one-dimensional case, we refer to Lagodowski and Rychlik , Elton , Lesigne and Volny , Stoica  and Ghosal and Chandra . Recently, complete convergence for martingale difference random fields was proved by Kuczmaszewska and Lagodowski .
The aim of this paper is to obtain some results concerning complete convergence of weighted sums , where is an array of real numbers, and is the martingale differences with respect to satisfying (1.1).
The following moment maximal inequality provides us a useful tool to prove the main results of this section (see Kuczmaszewska and Lagodowski ).
Let us denote . Now, we are ready to formulate the next result.
for all .
Proof Let us notice that the series is finite, then (2.2) always holds. Therefore, we consider only the case such that is divergent. Let , and .
by assumption (ii).
by assumption (ii)′. Thus, for all , and the proof of Theorem 2.2 is complete. □
then (2.2) holds.
Proof It is easy to see that (2.3) implies (iii). We omit details that prove it. □
The following corollary shows that assumption (iii) in Theorem 2.2 is natural, and in the case of independent random fields, it reduces to the known one.
then (2.2) holds.
Now, it is easy to see that (2.4) implies (iii) of Theorem 2.2. Thus, by Theorem 2.2, result (2.2) follows. □
Remark Theorem 2.2 and Corollary 2.4 are extensions of Theorem 4.1 and Corollary 4.1 in Kuczmaszewska and Lagodowski  to the weighted sums case, respectively.
then (2.2) holds.
which satisfies (i) of Theorem 2.2.
As the proof of Corollary 2.3, (2.6) implies (iii) of Theorem 2.2.
It remains to show that Theorem 2.2(ii) or (ii)′ is satisfied.
which satisfies Theorem 2.2(ii)′. Hence, the proof is complete. □
and Theorem 2.2(iii) hold, then (2.2) holds.
Hence, by (2.9) and (2.10), conditions (i) and (ii)′ in Theorem 2.2 are satisfied, respectively.
Hence, the proof is complete. □
If Theorem 2.2(iii) holds, then (2.2) holds.
Proof From (2.12), (2.8) follows. Hence, by Corollary 2.6, we obtain (2.2). □
Remark Linear random fields are of great importance in time series analysis. They arise in a wide variety of context. Applications to economics, engineering, and physical science are extremely broad (see Kim et al. ).
Let , where is a field of real numbers with , and is a field of the martingale difference random variables.
Hence, we can use the above results to investigate the complete convergence for linear random fields.
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