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Journal of Probability and Statistics
Volume 2011 (2011), Article ID 701952, 16 pages
Weighted Strong Law of Large Numbers for Random Variables Indexed by a Sector
1Institute of Mathematics, Marie Curie-Skłodowska University, Plac Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland
2Faculty of Mathematics and Computer Science, University of Łódź, Ulica Banacha 22, 90-238 Łódź, Poland
Received 13 May 2011; Revised 30 September 2011; Accepted 22 October 2011
Academic Editor: Nikolaos E. Limnios
Copyright © 2011 Przemysław Matuła and Michał Seweryn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We find necessary and sufficient conditions for the weighted strong law of large numbers for independent random variables with multidimensional indices belonging to some sector.
1. Introduction and the Notation
Let , , be a -dimensional lattice. The points of this lattice will be denoted by , , and so forth. The set is partially ordered by the relation if and only if for every we have . We will also write if for every and for at least one we have . Let be an infinite subset of ; moreover, assume that a nonnegative, increasing real function is given, and set , , . Assume that , for each and as . Moreover, put and . We aim to study the convergence of sequences indexed by lattice points. For this means let us recall that may have different meanings; in other words, the term “ tends to infinity” may be understood as (equivalently ) or , we will be using the first meaning. Thus, for a field of real numbers indexed by positive lattice points, we write if and only if for every there exist a such that for each we have .
Our setting is an extension of the one investigated by Klesov and Rychlik  or Indlekofer and Klesov , that is, of the so called “sectorial convergence”. Let , , be nondecreasing nonnegative real functions such that , , . Let us denote by a -dimensional sector defined by these functions in the following way: Moreover, for , let us consider the Dirichlet divisors for the sector defined by and set For we extend the function by defining the step function where denotes the integer part of .
We consider a modified version of “sectorial convergence” in which we say that a field of real numbers indexed by lattice points in converge in the set to , and write if and only if for every the inequality holds for all but finite number of . It was Gut (see ), who for the first time considered sectorial convergence for random fields, with the sector defined as and , , where . For recent results in this area, see [1, 2, 4]. Our aim is to extend the results of  in the spirit considered by Lagodowski and Matuła in .
We will be studying necessary and sufficient conditions for the weighted strong law of large numbers (WSLLN for short) for random fields of independent random variables for a general class of weights (defined by Feller in  and Jajte in ). The case of such summability methods in the multi-index setting was considered in .
Let us recall the definition of the class of transformations considered by Lagodowski and Matuła (see ). Let be nonnegative real functions; moreover, let be increasing with range , and set . We will say that the functions satisfy the Feller-Jajte condition if the following two conditions are satisfied.(A1)There exists such that the function is increasing in the interval with range and .(A2)There exists a constant such that
The first assumption is technical, and it is required in order for the inverse function to exist. In our present work we will consider a modification of this class. Instead of (A2) we will consider the following condition.(A2’)There exists a constant such that
Making use of the well-known asymptotics (see [7, 8]) for and , that is, the relations where are nonnegative constants, we see that, in the case of the sector considered by Gut, the conditions (A2) and (A2’) coincide. It is worth noting that the number does not exceed .
Let be a field of independent random variables. The aim of the paper is to find the necessary and sufficient conditions for the almost sure convergence of where the summation is extended over all or , and the centering constants are either the moments or truncated moments of . Our main results, the necessary and sufficient conditions for the WSLLN for independent random fields, may also be seen as an extension of the previous results of [2, 4].
2. Main Results
Let be a field of independent random variables; moreover, let and the functions be as before. For simplicity we impose some regularity condition on the function ; namely, Note—similarly as in —that, if (2.1) holds, then is equivalent to Furthermore, Let us observe that (2.1) is satisfied for standard sectors and if , for some and . Let the functions satisfy the conditions (A1) and (A2’). We will also be using a well-known truncation technique with In the next theorem we give sufficient conditions for the WSLLN of the form: In the first theorem we will not assume that the random variables have the same distribution. Instead we will use the notion of weak domination.
Definition 2.1. A random field is said to be weakly dominated on the average in the set by the random variable if there exists a constant such that, for every and , where and obviously .
This condition, to the best of our knowledge, was introduced in , where it is also discussed that this condition is independent of the notion of weak mean domination (see also ). In the rest of our work we will write “weakly dominated on the average” without indicating the set on which we consider the domination condition (unless it causes any confusion). The price to pay for weakening the condition of identically distributed random variables to the weakly dominated on the average is to only be able to prove sufficient conditions. In some cases we will also consider a narrower class of summability methods; that is, for a given set , apart from the conditions (A1) and (A2’), we will assume that(A3)there exists a constant such that, for each , With such preparations we can formulate our first main result.
Theorem 2.2. Let be a field of independent random variables weakly dominated on the average in the set by the random variable . Moreover, let (A1), (A2’), and (2.1) be satisfied. If then If additionally as and the function satisfies the condition , then
Proof. From the Kolmogorov-type maximal inequality and the strong law of large numbers which is due to Christofides and Serfling (see [11, Corollary 2.5 and Theorem 2.8]), it follows that the sufficient condition for the convergence of series of independent random fields is analogous to the one-dimensional case. Therefore, it suffices to prove that the series
is convergent. We have
which is equivalent to since the function satisfies (2.1). It is clear that , and from the relation (A2’) we have
which again is equivalent to by (2.1). Therefore, the first part of the theorem is proved. In order to prove the second part of the theorem, let us observe that
The first summand converges to 0 by the first part of the theorem applied to the random variables defined by (2.4). The convergence of the third one follows from the Borel-Cantelli lemma since
It remains to prove that the second summand converges to 0. In order to prove this let us put , and we will prove that the series
is absolutely convergent. We have
Now, since , the second series on the r.h.s. of the above inequality is convergent by the same argument as in the proof of the first part, whereas for the second series it is true that which is convergent again by .
Under the assumption that the random variables are identically distributed, one may obtain a necessary and sufficient condition for the WSLLN.
Proof. In view of Theorem 2.2, it suffices to prove the necessity of (2.8).
Similarly as in  we have from the Lebesgue dominated convergence criterion. Now—again analogously to —for write where and with or for . (To see this one should simply put for and apply a well-known summation technique to the whole .) From the relations (2.21) and (2.9), we easily obtain that for details see [4, page 20]. Thus, from the above and (2.20), we obtain the necessity of condition (2.9) via the Borel-Cantelli lemma.
Remark 2.4. In both of the above theorems, we do not need to assume that the function is regularly varying in the sense of (2.1) nor that the function is invertible (we may omit (A1)). If we omit both of these assumptions, then, instead of the condition , one ought to write and, instead of , with some minor technical changes in the proofs.
3. WSLLN in the Case
In this section we aim to extend the results of the papers [1, 2], which are “sectorial SLLN” for fields of i.i.d. random variables in the case . In this section we will study the weighted strong law of large numbers. Here we will use a less general definition of a sector than in the previous section. In this case it is possible to obtain the necessary and sufficient conditions for a stronger form of the WSLLN; in other words, we will set and consider the almost sure convergence Let us note that the main difference lies in the set over which we sum up the random variables. In the former section we were performing the summation only over the indices which belong to the sector, and now we sum up over all indices in and the difference between the present setting and the classical SLLN for random fields is in the very definition of convergence. In order to be able to prove our main results, we have to adopt some techniques form  to the setting of the WSLLN, but, for the sake of brevity, we will not include all the justifications here, and instead we refer to proper lines of the proofs in  or in . In what follows, we will assume that the sector satisfies the conditions: These conditions were originally introduced in . Let us observe that from (3.5) it follows that and , for . Thus, according to (3.4), . From this inequality it follows that , that is, that the nonlinear sector contains a standard sector , where . Moreover, we assume that the functions satisfy the assumption (A1) and (A2’). Since we are dealing with a different problem now, then we have to use a modification of the definition of weakly mean domination on the average.
Let be an arbitrary, nonempty sector, defined by the functions and . Let us divide the set into three parts:
A random field is said to be weakly dominated on the average by the random variable if (2.6) holds for .
Let us now state the main result of this section.
Theorem 3.1. Let the functions , the sector , and the function be as above and satisfy (2.1), (3.3)–(3.5). Moreover, let the functions satisfy the conditions (A1) and (A2’). If a field of independent random variables is weakly dominated on the average by the random variable and , then Assume additionally that as and the function satisfies the condition . If , then
Proof. Let us begin with the justification of the first assertion. Assume that , then . Let us consider the truncated random variables as defined in (2.3). From the Borel-Cantelli lemma, we have that, since , then
and this of course means that it suffices to prove that
Now let us divide the partial sums into three terms as in [2, 12]; in other, words let us consider
where and are defined in (3.6). It is clear (see [2, 12] for details) that the families
are linearly ordered by inclusion, thus may be enumerated in an ascending order. We also write
for . By the above remark the partial sums may be seen as subsequences of cumulative sums of weakly dominated random variables. Thus, we may use the sufficient condition for the the Feller-Jajte WSLLN for weakly dominated random variables (see Section 2) to conclude that the condition is sufficient for . and . Therefore, it remains to prove that ., which by the results of Klesov [12, 13] is implied by the fact that
The first summand is bounded by ; the second is bounded since . The fact that, under the above assumptions on the functions and , the convergence of the last summand follows from the assumption follows by the same lines as in the proofs from the preceding section.
The proof of the second assertion of the theorem is much more the same as in the proof of Theorem 2.2 from the former section. Let us first consider the division of the partial sums of the random field (weakly dominated on the average by the random variable ) into three terms as given in (3.21). Let us notice that by applying the beforehand defined truncation technique we obtain again that where . We will only give the proof for , remembering that for may be seen as a subsequence of a sequence of weakly mean dominated (see ) random variables (indexed by natural numbers), and; therefore, the same techniques may be used to prove the convergence of and to zero (the same as for the convergence of ) with the change that . Now from the Borel-Cantelli lemma we infer that, since then Now, the almost sure convergence to zero of the first term on the r.h.s. of (3.15) follows by the same lines as in the first part of the proof. Therefore, it remains to prove that which follows from the proof of the Theorem 2.2 in the preceding section.
As before, below we show that under the assumption that the random variables are i.i.d. the sufficient conditions in the above proofs become necessary.
Theorem 3.2. For , the sector , the function , the functions as above, and the field of i.i.d. random variables, the following conditions are equivalent:(1), (2).
Proof. As obviously i.i.d. random variables are weakly dominated on the average for each sector in , then only the necessity needs justification. Let us, therefore, assume that . From this it immediately follows that where Therefore, from the Feller-Jajte WSLLN (see [5, 6]), we infer that . As in (3.6) divide the set into three subsets , , and—as before—write Since, as we have noted, the last two sums may be seen as subsequences of a sequence of i.i.d. random variables with the same distribution as , then from the fact that we obtain and as ; thus, as . As in the proof of Theorem 2.3 in the preceding section we have as , and in turn this implies that (for details see ). Since the are independent, then the Borel-Cantelli lemma implies that and the necessary part of the Theorem is proved.
The aim of the present section is to show some examples of functions for which the condition (A2’) is satisfied. In the case and for , Klesov and Rychlik proved that (see [1, Lemma 2]) where the sector is defined by the functions and such that We will extend this result to more general classes of functions satisfying some additional technical conditions. We will apply the theory of regularly varying functions (we refer the reader to  for details), and in order to use integrals and sums interchangeably we impose the following conditions:
Furthermore, let be differentiable with the derivative such that
Proof. We will make use of the Abel transform: From Lemma 4 of , we have where and is a slowly varying function (even belonging to the Zygmund class). Since is regularly varying with index , we have for some slowly varying function . Thus, from (4.8) and (4.9) we get since and is slowly varying. By mean value theorem and (4.4)–(4.5), we get where . Let us note that the function is regularly varying with index ; therefore, by Theorem 1.5.11 in  with , we get In consequence, for some , from this it follows that Now, from (4.7) and (4.10), the conclusion follows.
The above proof was essentially based on the inequality (4.8) and may be repeated in higher dimensions for any sector with such asymptotics for . This is the case for nonlinear sectors such that , for some and . Therefore, we may state the following proposition (in view of the remark following (3.5), the first inclusion holds automatically).
Remark 4.3. It is well known (see ) that a function on , regularly varying with index , may be represented in the form , where is slowly varying and admits the representation: where and as . To prove (4.6) it suffices to consider the case , and it is easy to see that (4.3)–(4.5) are satisfied if is positive, continuous, and nonincreasing function which tends to 0 as .
The authors are grateful to the referees for careful reading of the paper. They would also like to thank Professor Adam Paszkiewicz for helpful discussion.
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