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On the convergence rates in the asymmetric SLLN for independent and nonidentically distributed random fields
Journal of Inequalities and Applications volume 2014, Article number: 220 (2014)
The purpose of this paper is to obtain the convergence rates in strong laws of large numbers for nonidentically distributed and independent random fields such that different indices have different powers in the normalization.
Let , where d is a positive integer, denote positive integer d-dimensional lattice points. For the elements of we use bold symbols m, n and so forth. Further, we will assume the usual partial ordering for elements of , i.e. for and , if and only if for all . The strict inequality is defined as follows: if and only if and . We assume that means . We also use for .
Most of the results concerning limit theorems, especially for random fields, are obtained for identically distributed random variables.
Kuczmaszewska and Łagodowski  considered the concept of weak boundedness as follows.
Definition 1.1 (Kuczmaszewska and Łagodowski )
The random variables are said to be weakly bounded by the random variable ξ if there exist some constants , and such that for every and ,
Clearly, the regular cover implies weak boundedness. If only the right-hand side inequality is satisfied we say that the random field and the random variable ξ satisfy the weak mean dominating condition.
Gut  proved Marcinkiewicz laws and convergence rates in the law of large numbers for i.i.d. random fields and Gut  also showed convergence rates for probabilities of moderate deviations for independent random fields.
Recently, the limit theorems for independent random fields such that different indices have different powers in the normalization are investigated. For example, Thanh  provided the strong law of large numbers for independent random fields and Gut and Stadtmüller  proved the Marcinkiewicz-Zygmund law for i.i.d. random fields and Gut and Stadtmüller  also obtained the Hsu-Robbins-Erdős-Spitzer-Baum-Katz theorem for i.i.d. random fields.
It is clear that independent random fields imply martingale difference random fields. Hence, from Theorem 4.1 in Kuczmaszewska and Łagodowski  we obtained the following convergence rates in the strong law of large numbers for independent random fields with nonidentical distribution.
Theorem 1.2 Let be the a field of independent random variables with for . Assume, for , and ,
for all . Then
From Corollary 4.1 in Kuczmaszewska and Łagodowski  we also obtain the following corollary.
Corollary 1.3 Let be a field of independent random variables with for . Then (1.2), (1.3), and
Kuczmaszewska and Łagodowski  investigated the following convergence rate for independent random field weakly dominated by the random variable ξ.
Theorem 1.4 (Kuczmaszewska and Łagodowski )
Let be a field of independent random variables weakly dominated by the random variable ξ and for . Moreover, we assume that for ,
where . Then, for and , (1.5) holds.
In this paper we generalize the above results (Theorem 1.2, Corollary 1.3, and Theorem 1.4) to the case where different indices have different powers in the normalization.
In Section 1 every coordinate , , is raised to the same power α. The main point of this section is to allow for different powers for different coordinates. In order to continue we therefore define , where, w.l.o.g., we assume that the coordinates are arranged in nondecreasing order, such that is the smallest one and the largest one. We further let p denote the number of α’s which are equal to the smallest one, that is, .
As is easily seen the domain of interest concerning the α’s becomes
where the boundary takes us into the realm of the central limit theorem and the boundary 1 corresponds to the Kolmogorov strong law.
For ease of notation, we use the notation and .
Lemma 2.1 (Gut and Stadtmüller )
Let ξ be a random variable. Let and , where with . Then, for any and all ,
The following lemma appeared in Lemma 3.1 of Thanh .
Lemma 2.2 (Thanh )
Let be a field of independent random variables with . Then there exists a constant C depending only on p and d such that
Remark In the case , the independent hypothesis and the hypothesis that are superfluous, and C is given by . In the case , C is given by . In the case , Lemma 2.2 was proved by Wichura  and C is given by .
Theorem 2.3 Let be a field of independent random variables with for and let , where and . Assume, for and ,
for all . Then we obtain
Proof Let , and . Then
It follows from (2.2) and (2.4) that
hold. It remains to prove that .
By the Markov inequality and Lemma 2.2, for some positive constant C we obtain
since . Therefore, by (2.3) and (2.8) we obtain
Hence, by (2.6), (2.7), and (2.9) the result (2.5) follows. □
Remark Theorem 2.3 generalizes Theorem 1.2 to the case where different indices have different powers in the normalization. That is, Theorem 1.2 is a special case of Theorem 2.3 with .
Corollary 2.4 Let be a field of independent random variables such that for . Let , and , where with and . Assume (2.2), (2.3), and
Then (2.5) holds.
Proof It is easy to see that (2.10) implies (2.4). Thus, by Theorem 2.3 we get (2.5). □
Remark The following corollary shows that the assumption (2.4) is natural and it reduces to the known one.
In the case of a weakly dominated random field, we have the following theorem.
Corollary 2.5 Let be a field of independent random variables weakly dominated by the random variable ξ and such that for . Let , where and . Assume that for
Then (2.5) holds.
Proof We are going to prove Corollary 2.5 by using Corollary 2.4. It follows from Lemma 2.1 and (2.11) that, for , we have
which yields (2.2).
Furthermore, we get
It follows from (2.11) and Lemma 2.1 that
Let and let .
To show define
with differences .
(See Gut and Stadtmüller .)
Then we have
Thus, by (2.13), (2.14), and (2.15) we established (2.3).
It remains to obtain (2.10). Since , we get
which yields (2.10). Therefore, by (2.12)-(2.16) and Corollary 2.4 we get (2.5). □
Kuczmaszewska A, Łagodowski ZA: Convergence rates in the SLLN for some classes of dependent random fields. J. Math. Anal. Appl. 2011, 380: 571–584. 10.1016/j.jmaa.2011.03.042
Gut A: Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. Ann. Probab. 1978, 6: 469–482. 10.1214/aop/1176995531
Gut A: Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices. Ann. Probab. 1980, 8: 298–313. 10.1214/aop/1176994778
Thanh L: On the strong law of large numbers for d -dimensional arrays of random variables. Electron. Commun. Probab. 2007, 12: 434–441.
Gut A, Stadtmüller U: An asymmetric Marcinkiewicz-Zygmund LLN for random fields. Stat. Probab. Lett. 2009, 79: 1016–1020. 10.1016/j.spl.2008.12.006
Gut A, Stadtmüller U: On the Hsu-Robbins-Erdős-Spitzer-Baum-Katz theorem for random fields. J. Math. Anal. Appl. 2012, 387: 447–463. 10.1016/j.jmaa.2011.09.010
Wichura MJ: Inequalities with applications to the weak convergence of random processes with multi-dimensional time parameters. Ann. Math. Stat. 1969, 60: 681–687.
The author would like to sincerely thank the anonymous referees for their careful reading of the manuscript and valuable suggestions.
The author declares that she has no competing interests.
The author read and approved the final manuscript.
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Ko, M.H. On the convergence rates in the asymmetric SLLN for independent and nonidentically distributed random fields. J Inequal Appl 2014, 220 (2014). https://doi.org/10.1186/1029-242X-2014-220
- convergence rate
- nonidentical distribution
- random field; strong law of large numbers