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Journal of Probability and Statistics
Volume 2011 (2011), Article ID 701952, 16 pages
http://dx.doi.org/10.1155/2011/701952
Research Article

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.

Abstract

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 𝑑, 𝑑1, be a 𝑑-dimensional lattice. The points of this lattice will be denoted by 𝑚=(𝑚1,,𝑚𝑑), 𝑛=(𝑛1,,𝑛𝑑), and so forth. The set 𝑑 is partially ordered by the relation 𝑚𝑛 if and only if for every 𝑖=1,,𝑑 we have 𝑚𝑖𝑛𝑖. We will also write 𝑚<𝑛 if for every 𝑖=1,,𝑑,𝑚𝑖𝑛𝑖 and for at least one 𝑖0 we have 𝑚𝑖0<𝑛𝑖0. Let 𝑊 be an infinite subset of 𝑑; moreover, assume that a nonnegative, increasing real function 𝛿𝑊+ is given, and set 𝑇𝑊,𝛿(𝑚)=card{𝑘𝑊𝛿(𝑘)𝑚}, 𝜏𝑊,𝛿(𝑚)=𝑇𝑊,𝛿(𝑚)𝑇𝑊,𝛿(𝑚1), 𝑚. Assume that 𝑇𝑊,𝛿(𝑚)<, for each 𝑚 and 𝑇𝑊,𝛿(𝑚) as 𝑚. Moreover, put |𝑛|=𝑑𝑖=1𝑛𝑖 and 𝑛=max1𝑖𝑑|𝑛𝑖|. We aim to study the convergence of sequences indexed by lattice points. For this means let us recall that 𝑛 may have different meanings; in other words, the term “𝑛 tends to infinity” may be understood as |𝑛| (equivalently 𝑛) or min1𝑖𝑑(𝑛𝑖), we will be using the first meaning. Thus, for a field (𝑎𝑛)𝑛𝑑 of real numbers indexed by positive lattice points, we write 𝑎𝑛𝑎,𝑛 if and only if for every 𝜀>0 there exist a 𝑛0𝑑 such that for each 𝑛𝑛0 we have |𝑎𝑛𝑎|<𝜀.

Our setting is an extension of the one investigated by Klesov and Rychlik [1] or Indlekofer and Klesov [2], that is, of the so called “sectorial convergence”. Let 𝑓𝑖,𝑗,𝐹𝑖,𝑗++, 1𝑖<𝑗𝑑, be nondecreasing nonnegative real functions such that 𝑓𝑖,𝑗(𝑥)𝑥𝐹𝑖,𝑗(𝑥), 1𝑖<𝑗𝑑, 𝑥+. Let us denote by 𝐴𝑓,𝐹 a 𝑑-dimensional sector defined by these functions in the following way:𝐴𝑓,𝐹𝑛=𝑑𝑓𝑖,𝑗𝑛𝑗𝑛𝑖𝐹𝑖,𝑗𝑛𝑗,1𝑖<𝑗𝑑.(1.1) Moreover, for 𝑚=1,2,, let us consider the Dirichlet divisors 𝜏𝑓,𝐹(𝑚) for the sector 𝐴𝑓,𝐹 defined by𝜏𝑓,𝐹𝑛(𝑚)=card𝐴𝑓,𝐹||𝑛||=𝑚,(1.2) and set𝑇𝑓,𝐹(𝑚)=𝑚𝑘=1𝜏𝑓,𝐹𝑛(𝑘)=card𝐴𝑓,𝐹||𝑛||𝑚.(1.3) For 𝑥+ we extend the function 𝑇𝑓,𝐹 by defining the step function 𝑇𝑓,𝐹(𝑥)=𝑇𝑓,𝐹([𝑥]) where [𝑥] denotes the integer part of 𝑥.

We consider a modified version of “sectorial convergence” in which we say that a field of real numbers (𝑎𝑛) indexed by lattice points in 𝑑 converge in the set 𝑊 to 𝑎, and write lim𝑊𝑎𝑛=𝑎 if and only if for every 𝜀>0 the inequality |𝑎𝑛𝑎|<𝜀 holds for all but finite number of 𝑛𝑊. It was Gut (see [3]), who for the first time considered sectorial convergence for random fields, with the sector defined as𝐴𝜃𝑛=𝑑𝜃𝑛𝑗𝑛𝑖𝑛𝑗𝜃,1𝑖,𝑗𝑑,𝑖𝑗,(1.4) and 𝜏𝜃(𝑚)=card{𝑛𝐴𝜃|𝑛|=𝑚},𝑇𝜃(𝑚)=𝑚𝑘=1𝜏𝜃(𝑘), 𝑚=1,2,,𝜃[0,1), where 𝐴0=𝑑. For recent results in this area, see [1, 2, 4]. Our aim is to extend the results of [2] in the spirit considered by Lagodowski and Matuła in [4].

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 [5] and Jajte in [6]). The case of such summability methods in the multi-index setting was considered in [4].

Let us recall the definition of the class of transformations considered by Lagodowski and Matuła (see [4]). Let 𝑔,++ be nonnegative real functions; moreover, let 𝑔 be increasing with range (0,), and set 𝜙(𝑥)=𝑔(𝑥)(𝑥),𝑥>0. We will say that the functions 𝑔, satisfy the Feller-Jajte condition if the following two conditions are satisfied.(A1)There exists 𝑝>0 such that the function 𝜙 is increasing in the interval (𝑝,) with range (0,) and lim𝑛𝜙(𝑛)=.(A2)There exists a constant 𝑎>0 such that𝑘=𝑠𝜏𝜃(𝑘)𝜙2𝑠(𝑘)𝑎𝜙2(𝑠),𝑠>𝑝,𝜃(0,1),𝑘=𝑠𝜏0(𝑘)𝜙2(𝑘)𝑎𝑠log𝑑1𝑠𝜙2(𝑠),𝑠>𝑝.(1.5)

The first assumption is technical, and it is required in order for the inverse function 𝜙1 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 𝑎>0 such that 𝑘=𝑠𝜏𝑊,𝛿(𝑘)𝜙2𝑇(𝑘)𝑎𝑊,𝛿(𝑠)𝜙2(𝑠),𝑠>𝑝.(1.6)

Making use of the well-known asymptotics (see [7, 8]) for 𝜏𝜃 and 𝑇𝜃, that is, the relations𝑐1𝑘𝑇𝜃(𝑘)𝑐2𝑐𝑘,𝜃(0,1),1𝑘log𝑑1𝑘𝑇0(𝑘)𝑐2𝑘log𝑑1𝑘,(1.7) where 𝑐1,𝑐2 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 𝜏𝑊,𝛿(𝑘) does not exceed 𝜏0(𝑘).

Let (𝑋𝑛)𝑛𝑑 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 oflim𝑊1𝑔𝛿𝑛𝑋𝑘𝑎𝑘𝛿𝑘=0,(1.8) where the summation is extended over all 𝑘𝑛 or 𝑊𝑘𝑛, and the centering constants 𝑎𝑘 are either the moments or truncated moments of 𝑋𝑘. 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 (𝑋𝑛)𝑛𝑑 be a field of independent random variables; moreover, let 𝑊𝑑 and the functions 𝛿𝑊+,𝑇𝑊,𝛿,𝜏𝑊,𝛿 be as before. For simplicity we impose some regularity condition on the function 𝑇𝑊,𝛿; namely,𝑇𝑊,𝛿(𝑘+1)𝑇𝑊,𝛿(𝑘)𝐵,forsome𝐵>0andevery𝑘.(2.1) Note—similarly as in [2]—that, if (2.1) holds, then 𝐸𝑇𝑊,𝛿(|𝑋|)< is equivalent to𝑛𝑊𝑃||𝑋||𝑛𝛿<.(2.2) Furthermore,𝑛𝑊𝑃||𝑋||𝑛𝛿=𝑘=1𝜏𝑊,𝛿(||𝑋||=𝑘)𝑃𝑘𝑘=1𝑇𝑊,𝛿(||𝑋||.𝑘)𝑃𝑘<𝑘+1(2.3) Let us observe that (2.1) is satisfied for standard sectors 𝐴𝜃 and if 𝐴𝜃1𝑊𝐴𝜃2, for some 𝜃1 and 𝜃2. Let the functions 𝑔, satisfy the conditions (A1) and (A2’). We will also be using a well-known truncation technique with𝑌𝑛=𝑋𝑛𝐼||𝑋𝑛||𝛿𝑛𝜙,𝑚𝑛=𝐸𝑌𝑛.(2.4) In the next theorem we give sufficient conditions for the WSLLN of the form:lim𝑊1𝑔𝛿𝑛𝑊𝑘𝑛𝑋𝑘𝑚𝑘𝛿𝑘=0,almostsurely.(2.5) In the first theorem we will not assume that the random variables 𝑋𝑛 have the same distribution. Instead we will use the notion of weak domination.

Definition 2.1. A random field (𝑋𝑛)𝑛𝑑 is said to be weakly dominated on the average in the set 𝐴 by the random variable 𝑌 if there exists a constant 𝐶>0 such that, for every 𝑘 and 𝑡0, 𝑛𝑛𝐴𝛿=𝑘𝑃||𝑋𝑛||𝑡𝐶𝜏𝐴,𝛿||𝑌||(𝑘)𝑃𝑡,(2.6) where 𝜏𝐴,𝛿(𝑘)=card{𝑛𝐴𝛿(𝑛)=𝑘} and obviously 𝑇𝐴,𝛿(𝑘)=𝜏𝐴,𝛿(1)+𝜏𝐴,𝛿(2)++𝜏𝐴,𝛿(𝑘).

This condition, to the best of our knowledge, was introduced in [9], where it is also discussed that this condition is independent of the notion of weak mean domination (see also [10]). In the rest of our work we will write “weakly dominated on the average” without indicating the set 𝐴 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 𝑊, apart from the conditions (A1) and (A2’), we will assume that(A3)there exists a constant 𝑎>0 such that, for each 𝑘>1,𝑘𝑛=1𝜏𝑊,𝛿(𝑛)𝜙𝑇(𝑛)𝑎𝑊,𝛿(𝑘)𝜙.(𝑘)(2.7) With such preparations we can formulate our first main result.

Theorem 2.2. Let (𝑋𝑛)𝑛𝑑 be a field of independent random variables weakly dominated on the average in the set 𝑊 by the random variable 𝑌. Moreover, let (A1), (A2’), and (2.1) be satisfied. If 𝐸𝑇𝑊,𝛿𝜙1||𝑌||<,(2.8) then lim𝑊1𝑔𝛿𝑛𝑊𝑘𝑛𝑋𝑘𝑚𝑘𝛿𝑘=0,almostsurely.(2.9) If additionally 𝜙(𝑛)/𝑛0 as 𝑛 and the function 𝜙 satisfies the condition (𝐴3), then lim𝑊1𝑔𝛿𝑛𝑊𝑘𝑛𝑋𝑘𝑋𝐸𝑘𝛿𝑘=0,almostsurely.(2.10)

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 𝑛𝑊𝐸𝑌𝑛𝜙𝛿𝑛2,(2.11) is convergent. We have 𝑛𝑊𝐸𝑌𝑛2𝜙2𝛿𝑛=𝑛𝑊1𝜙2𝛿𝑛0𝑃𝑌2𝑛𝑡𝑑𝑡𝑛𝑊1𝜙2𝛿𝑛𝜙2(𝛿(𝑛0))𝑃𝑋2𝑛=𝑡𝑑𝑡𝑘=11𝜙2(𝑘)𝜙20(𝑘)𝑛𝑛𝑊𝛿=𝑘𝑃𝑋2𝑛𝑡𝑑𝑡𝐶𝑘=11𝜙2𝜏(𝑘)𝑊,𝛿(𝑘)𝜙20(𝑘)𝑃𝑡𝑌2<𝜙2(𝑘)𝑑𝑡+𝐶𝑘=11𝜙2𝜏(𝑘)𝑊,𝛿(𝑘)𝜙20(𝑘)𝑃𝜙2(𝑘)𝑌2𝑑𝑡=𝐶𝑘=11𝜙2𝜏(𝑘)𝑊,𝛿(𝑘)𝜙20(𝑘)𝑃𝑡𝑌2<𝜙2(𝑘)𝑑𝑡+𝐶𝑘=1𝜏𝑊,𝛿𝜙(𝑘)𝑃2(𝑘)𝑌2𝑑𝑡.(2.12) Now 𝑘=1𝜏𝑊,𝛿𝜙(𝑘)𝑃2(𝑘)𝑌2𝑚=1𝑃𝜙2(𝑚)𝑌2𝜙2(𝑚+1)𝑚𝑘=1𝜏𝑊,𝛿=(𝑘)𝑚=1𝑇𝑊,𝛿𝜙(𝑚)𝑃2(𝑚)𝑌2𝜙2,(𝑚+1)(2.13) which is equivalent to 𝐸𝑇𝑊,𝛿(𝜙1(|𝑌|))< since the function 𝑇𝑊,𝛿 satisfies (2.1). It is clear that 𝜙20(𝑘)𝑃(𝑡𝑌2<𝜙2(𝑘))𝑑𝑡=𝐸𝑌2𝐼{|𝑌|𝜙(𝑘)}, and from the relation (A2’) we have 𝑘=11𝜙2𝜏(𝑘)𝑊,𝛿(𝑘)𝜙20(𝑘)𝑃𝑡𝑌2<𝜙2(=𝑘)𝑑𝑡𝑘=11𝜙2𝜏(𝑘)𝑊,𝛿(𝑘)𝐸𝑌2𝐼{|𝑌|<𝜙(𝑘)}=𝑚=1𝐸𝑌2𝐼{𝜙(𝑚1)|𝑌|<𝜙(𝑚)}𝑘=𝑚𝜏𝑊,𝛿(𝑘)𝜙2(𝑘)𝑎𝑚=1𝐸𝑌2𝐼{𝜙(𝑚1)|𝑌|<𝜙(𝑚)}𝑇𝑊,𝛿(𝑚)𝜙2(𝑚)𝑎𝑚=1𝑇𝑊,𝛿𝜙||𝑌||,(𝑚)𝑃(𝑚1)<𝜙(𝑚)(2.14) which again is equivalent to 𝐸𝑇𝑊,𝛿(𝜙1(|𝑌|))< 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 1𝑔𝛿𝑛𝑊𝑘𝑛𝑋𝑘𝐸𝑋𝑘𝛿𝑘=1𝑔𝛿𝑛𝑊𝑘𝑛𝑌𝑘𝐸𝑌𝑘𝛿𝑘1𝑔𝛿𝑛𝑊𝑘𝑛𝐸𝑋𝑘𝐸𝑌𝑘𝛿𝑘+1𝑔𝛿𝑛𝑊𝑘𝑛𝑋𝑘𝑌𝑘𝛿𝑘.(2.15) The first summand converges to 0 by the first part of the theorem applied to the random variables 𝑌𝑘 defined by (2.4). The convergence of the third one follows from the Borel-Cantelli lemma since 𝑛𝑊𝑃𝑋𝑛𝑌𝑛𝑘=1𝑇𝑊,𝛿(||𝑌||𝑘)𝑃𝜙(𝑘1)<𝜙(𝑘)<.(2.16) It remains to prove that the second summand converges to 0. In order to prove this let us put 𝑍𝑛=𝑋𝑛𝑌𝑛, and we will prove that the series 𝑛𝑊𝐸𝑍𝑛𝜙𝛿𝑛(2.17) is absolutely convergent. We have 𝑛𝑊||||𝐸𝑍𝑛𝜙𝛿𝑛||||𝑛𝑊1𝜙𝛿𝑛0𝑃||𝑍𝑛||=𝑡𝑑𝑡𝑛𝑊1𝜙𝛿𝑛𝜙𝛿𝑛𝑃||𝑋𝑛||+𝑡𝑑𝑡𝑛𝑊1𝜙𝛿𝑛𝜙𝛿𝑛𝑃||𝑋𝑛||𝛿𝑛𝜙=𝑚=11𝜙(𝑚)𝑛𝑛𝑊𝛿=𝑚𝜙(𝑚)𝑃||𝑋𝑛||+𝑡𝑑𝑡𝑚=11𝜙(𝑚)𝑛𝑛𝑊𝛿=𝑚𝜙𝛿𝑛𝑃||𝑋𝑛||𝛿𝑛𝜙𝐶𝑚=11𝜏𝜙(𝑚)𝑊,𝛿(𝑚)𝜙(𝑚)𝑃||𝑌||𝑡𝑑𝑡+𝐶𝑚=1𝜏𝑊,𝛿||𝑌||.(𝑚)𝑃𝜙(𝑚)(2.18)
Now, since 𝐸𝑇𝑊,𝛿(𝜙1(|𝑌|))<, 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 𝑚=11𝜏𝜙(𝑚)𝑊,𝛿(𝑚)𝜙(𝑚)𝑃||𝑌||=𝑡𝑑𝑡𝑚=1𝜏𝑊,𝛿(𝑚)𝜙(𝑚)𝑘=𝑚𝜙(𝑘+1)𝜙(𝑘)𝑃||𝑌||𝑡𝑑𝑡𝑘=1𝜙(𝑘+1)𝜙(𝑘)𝑃||𝑌||𝑡𝑑𝑡𝑘𝑚=1𝜏𝑊,𝛿(𝑚)𝜙(𝑚)𝑎𝑘=1𝑇𝑊,𝛿(𝑘)𝜙(𝑘)𝜙(𝑘+1)𝜙(𝑘)𝑃||𝑌||𝑡𝑑𝑡𝑎𝑘=1𝑇𝑊,𝛿||𝑌||,(𝑘)𝑃𝜙(𝑘)(2.19) which is convergent again by 𝐸𝑇𝑊,𝛿(𝜙1(|𝑌|))<.

Under the assumption that the random variables are identically distributed, one may obtain a necessary and sufficient condition for the WSLLN.

Theorem 2.3. Let (𝑋𝑛)𝑛𝑑 be a field of i.i.d. random variables with the same distribution as the random variable 𝑌. Moreover, let (A1), (A2’), and (2.1) be satisfied. Then (2.8) is equivalent to (2.9).

Proof. In view of Theorem 2.2, it suffices to prove the necessity of (2.8).
Similarly as in [4] we have lim𝑊𝑚𝑛𝜙𝛿𝑛=0,almostsurely,(2.20) from the Lebesgue dominated convergence criterion. Now—again analogously to [4]—for 𝑛𝑊 write 𝑋𝑛𝑚𝑛𝜙𝛿𝑛=1𝑔𝛿𝑛𝑎{0,1}𝑑(1)𝑑𝑑𝑖=1𝑎𝑖𝑆𝑛𝑎,(2.21) where 𝑆𝑛=𝑘𝑛,𝑘𝑊((𝑋𝑘𝑚𝑘)/(𝛿(𝑘))) and 𝑎=(𝑎1,,𝑎𝑑) with 𝑎𝑖=0 or 𝑎𝑖=1 for 𝑖=1,,𝑑. (To see this one should simply put 𝑋𝑛=0 for 𝑛𝑊 and apply a well-known summation technique to the whole 𝑑.) From the relations (2.21) and (2.9), we easily obtain that lim𝑊𝑋𝑛𝑚𝑛𝜙𝛿𝑛=0,almostsurely,(2.22) 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 𝑇𝑊,𝛿 is regularly varying in the sense of (2.1) nor that the function 𝜙 is invertible (we may omit (A1)). If we omit both of these assumptions, then, instead of the condition 𝐸(𝑇𝑊,𝛿(𝜙1(|𝑌|)))<, one ought to write 𝑘=1𝑇𝑊,𝛿(||𝑌||𝑘)𝑃𝜙(𝑘1)<𝜙(𝑘)<,(2.23) and, instead of 𝐸(𝜙1(|𝑌|))<, 𝑘=1𝑃||𝑌||𝜙(𝑘1)<𝜙(𝑘)<,(2.24) with some minor technical changes in the proofs.

3. WSLLN in the Case 𝑑=2

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 𝑑=2. 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 set𝑆𝑛=𝑘𝑛𝑋𝑘𝑋𝐸𝑘||𝑘||(3.1) and consider the almost sure convergencelim𝐴𝑓,𝐹𝑆𝑛𝑔||𝑛||.(3.2) 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 𝑑 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 [2] 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 [2] or in [1]. In what follows, we will assume that the sector 𝐴𝑓,𝐹 satisfies the conditions:𝑓𝑓isincreasing,(3.3)𝑓(𝑥)𝑥𝐹(𝑥),𝑓(1)𝑥<𝐹(1),(3.4)(𝑥)𝑥isnonincreasing,𝐹(𝑥)𝑥isnondecreasing.(3.5) These conditions were originally introduced in [1]. Let us observe that from (3.5) it follows that 𝑓(𝑥)/𝑥𝑓(1) and 𝐹(𝑥)/𝑥𝐹(1), for 𝑥1. Thus, according to (3.4), 𝑓(𝑥)𝑥𝑓(1)<𝑥𝐹(1)𝐹(𝑥). From this inequality it follows that 𝐴𝜃𝐴𝑓,𝐹, that is, that the nonlinear sector 𝐴𝑓,𝐹 contains a standard sector 𝐴𝜃, where 𝜃=max{𝑓(1);1/𝐹(1)}. Moreover, we assume that the functions 𝑔, 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 𝐴𝑓,𝐹 be an arbitrary, nonempty sector, defined by the functions 𝑓 and 𝐹. Let us divide the set 2 into three parts:𝐴𝑓,𝐹,𝐴𝑓𝑛=21𝑛𝑖𝑛<𝑓2,𝐴𝐹𝑛=2𝑛2𝑛>𝐹1.(3.6)

A random field (𝑋𝑛)𝑛𝑑 is said to be weakly dominated on the average by the random variable 𝑌 if (2.6) holds for 𝐴=𝐴𝑓,𝐹,𝐴𝑓and𝐴𝐹.

Let us now state the main result of this section.

Theorem 3.1. Let the functions 𝑓,𝐹, the sector 𝐴𝑓,𝐹, and the function 𝑇𝑓,𝐹 be as above and satisfy (2.1), (3.3)–(3.5). Moreover, let the functions 𝑔, satisfy the conditions (A1) and (A2’). If a field (𝑋𝑛)𝑛2 of independent random variables is weakly dominated on the average by the random variable 𝑌 and 𝐸(𝑇𝑓,𝐹(𝜙1(|𝑌|)))<, then lim𝐴𝑓,𝐹𝑆𝑛𝑔||𝑛||=0,almostsurely.(3.7) Assume additionally that 𝜙(𝑛)/𝑛0 as 𝑛 and the function 𝜙 satisfies the condition (𝐴3). If 𝐸(𝑇𝑓,𝐹(𝜙1(|𝑌|)))<, then lim𝐴𝑓,𝐹1𝑔||𝑛||𝑘𝑛𝑋𝑘𝑋𝐸𝑘||𝑘||=0,almostsurely.(3.8)

Proof. Let us begin with the justification of the first assertion. Assume that 𝐸𝑇𝑓,𝐹(𝜙1(|𝑌|))<, then 𝐸𝜙1(|𝑌|)<. Let us consider the truncated random variables (𝑌𝑛)𝑛2 as defined in (2.3). From the Borel-Cantelli lemma, we have that, since 𝐸𝜙1(|𝑌|)<, then 𝑃||𝑋𝑛𝑌𝑛||0i.o.𝑛2=0,(3.9) and this of course means that it suffices to prove that lim𝐴𝑓,𝐹1𝑔||𝑛||𝑛𝑑𝑛𝑘𝑌𝑛𝑚𝑛||𝑛||=0,almostsurely.(3.10) Now let us divide the partial sums into three terms as in [2, 12]; in other, words let us consider 𝐴𝑓𝑛𝑑𝑛𝑘,𝐴𝐹𝑛𝑑𝑛𝑘,𝐴𝑓,𝐹𝑛𝑑𝑛𝑘,(3.11) where 𝐴𝑓 and 𝐴𝐹 are defined in (3.6). It is clear (see [2, 12] for details) that the families 𝐴𝑓𝑛𝑑𝑛𝑘𝑘𝑑,𝐴𝐹𝑛𝑑𝑛𝑘𝑘𝑑(3.12) are linearly ordered by inclusion, thus may be enumerated in an ascending order. We also write 𝑆𝐵𝑘=𝑛𝐵𝑑𝑛𝑘𝑌𝑘𝑚𝑛||𝑛||,(3.13) for 𝐵2. By the above remark the partial sums 𝑆𝐴𝑓(𝑘),𝑆𝐴𝐹(𝑘) 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 𝐸𝜙1(|𝑌|)< is sufficient for 𝑆𝐴𝑓(𝑘)0𝑎.𝑠. and 𝑆𝐴𝐹(𝑘)0𝑎.𝑠. Therefore, it remains to prove that 𝑆𝐴𝑓,𝐹(𝑘)0𝑎.𝑠., which by the results of Klesov [12, 13] is implied by the fact that 𝐴𝑓,𝐹𝑛2𝑛𝑘𝑃||𝑌𝑛||||𝑛>𝜙||<,𝐴𝑓,𝐹𝑛2𝑛𝑘𝐸𝑌𝑛𝑚𝑛𝜙||𝑛||<,𝐴𝑓,𝐹𝑛2𝑛𝑘𝑌Var𝑛𝑚𝑛𝜙||𝑛||<.(3.14) The first summand is bounded by 𝐸𝑇𝑓,𝐹(𝜙1(|𝑌|))<; the second is bounded since 𝐸𝑌𝑛=𝑚𝑛. The fact that, under the above assumptions on the functions 𝑔 and , the convergence of the last summand follows from the assumption 𝐸𝑇𝑓,𝐹(𝜙1(|𝑌|))< 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 (𝑋𝑛)𝑛𝑑 (weakly dominated on the average by the random variable 𝑌) into three terms as given in (3.21). Let us notice that by applying the beforehand defined truncation technique we obtain again that 𝑆𝐵𝑘=𝑛𝐵2𝑛𝑘𝑋𝑛𝐸𝑋𝑛||𝑛||=𝑛𝐵2𝑛𝑘𝑌𝑛𝐸𝑌𝑛||𝑛||𝑛𝐵2𝑛𝑘𝐸𝑋𝑛𝐸𝑌𝑛||𝑛||+𝑛𝐵2𝑛𝑘𝑋𝑛𝑌𝑛||𝑛||,(3.15) where 𝐵=𝐴𝑓,𝐹,𝐴𝑓,𝐴𝐹. We will only give the proof for 𝐵=𝐴𝑓,𝐹, remembering that for 𝐵=𝐴𝑓,𝐴𝐹𝑆𝐵(𝑘) may be seen as a subsequence of a sequence of weakly mean dominated (see [10]) random variables (indexed by natural numbers), and; therefore, the same techniques may be used to prove the convergence of 𝑆𝐴𝑓(𝑘) and 𝑆𝐴𝐹(𝑘) to zero (the same as for the convergence of 𝑆𝐴𝑓,𝐹(𝑘)) with the change that 𝑑=1. Now from the Borel-Cantelli lemma we infer that, since 𝑛𝐴𝑓,𝐹𝑃𝑋𝑛𝑌𝑛𝑘=1𝑇𝑓,𝐹(||𝑌||𝑘)𝑃𝜙(𝑘1)<𝜙(𝑘)<,(3.16) then 1𝑔||𝑛||𝐴𝑓,𝐹𝑛2𝑛𝑘𝑋𝑛𝑌𝑛||𝑛||0,almostsurely.(3.17) 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 lim𝑘1𝑔||𝑘||𝐴𝑓,𝐹𝑛2𝑛𝑘𝐸𝑋𝑛𝐸𝑌𝑛||𝑛||=0,(3.18) 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 𝑓,𝐹, the sector 𝐴𝑓,𝐹, the function 𝑇𝑓,𝐹, the functions 𝑔, as above, and the field (𝑋𝑛)𝑛2 of i.i.d. random variables, the following conditions are equivalent:(1)lim𝐴𝑓,𝐹(𝑆𝑛/𝑔(|𝑛|))=0,almostsurely, (2)𝐸(𝑇𝑓,𝐹(𝜙1(|𝑋1|)))<.

Proof. As obviously i.i.d. random variables are weakly dominated on the average for each sector in 2, then only the necessity needs justification. Let us, therefore, assume that lim𝐴𝑓,𝐹(𝑆𝑛/𝑔(|𝑛|))=0,a.s. From this it immediately follows that lim𝑘1𝑔𝑘2𝑛𝑘𝑋𝑛𝑚𝑛||𝑛||=0a.s.,(3.19) where 2𝑘=(𝑘,𝑘).(3.20) Therefore, from the Feller-Jajte WSLLN (see [5, 6]), we infer that 𝐸𝜙1(|𝑋1|)<. As in (3.6) divide the set {𝑛2𝑛𝑘} into three subsets 𝐴𝑓,𝐹, 𝐴𝑓, 𝐴𝐹 and—as before—write 𝑆𝑘=𝑛2𝑛𝑘𝑋𝑛𝑚𝑛||𝑛||=𝑆𝐴𝑓,𝐹𝑛+𝑆𝐴𝑓𝑘+𝑆𝐴𝐹𝑘.(3.21) 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 𝑋1, then from the fact that 𝐸𝜙1(|𝑋1|)< we obtain 𝑆𝐴𝑓(𝑘)0 and 𝑆𝐴𝐹(𝑘)0 as 𝑘; thus, 𝑆𝐴𝑓,𝐹(𝑘)0 as 𝑘. As in the proof of Theorem 2.3 in the preceding section we have 𝑚𝑛/𝜙(|𝑛|)0 as 𝑘, and in turn this implies that 𝑃||𝑋𝑛||||𝑛||||𝑛>𝑔||i.o.for𝑛𝐴𝑓,𝐹=0,(3.22) (for details see [4]). Since the 𝑋𝑛𝑠 are independent, then the Borel-Cantelli lemma implies that 𝐸𝑇𝑓,𝐹(𝜙1(|𝑋1|))< and the necessary part of the Theorem is proved.

4. Examples

The aim of the present section is to show some examples of functions Φ for which the condition (A2’) is satisfied. In the case 𝑑=2 and for Φ(𝑥)=𝑥, Klesov and Rychlik proved that (see [1, Lemma  2])𝑘=𝑛𝜏𝑓,𝐹(𝑘)𝑘2𝑇𝐶𝑓,𝐹(𝑛)𝑛2,(4.1) where the sector 𝐴𝑓,𝐹is defined by the functions 𝑓 and 𝐹 such that𝑓(𝑥)𝑥𝐹(𝑥),𝑓(𝑥)𝑥isnonincreasing,𝐹(𝑥)𝑥isnondecreasing.(4.2) We will extend this result to more general classes of functions Φ satisfying some additional technical conditions. We will apply the theory of regularly varying functions (we refer the reader to [14] for details), and in order to use integrals and sums interchangeably we impose the following conditions:[],Φispositive,nondecreasingon1,(4.3)Φ(𝑘+1)Φ(𝑘)𝐶1,forsome𝐶1>0andevery𝑘𝑁.(4.4)

Furthermore, let Φ be differentiable with the derivative such thatΦ(𝑦)Φ(𝑥)𝐶2,forsome𝐶2>0andevery1𝑥𝑦𝑥+1,𝑘Φ(𝑘)Φ(𝑘)𝐶3,forsome𝐶3>0andevery𝑘𝑁.(4.5)

Proposition 4.1. Let 𝑑=2 and the sector 𝐴𝑓,𝐹 be defined by the functions 𝑓 and 𝐹 satisfying (4.2). Assume that the function Φ is regularly varying with index 𝛿>1/2 and the conditions (4.3)–(4.5) are satisfied. Then 𝑘=𝑛𝜏𝑓,𝐹(𝑘)Φ2𝑇(𝑘)𝐶𝑓,𝐹(𝑛)Φ2(𝑛),forsome𝐶>0,𝑎𝑛𝑑every𝑛.(4.6)

Proof. We will make use of the Abel transform: 𝑘=𝑛𝜏𝑓,𝐹(𝑘)Φ2(𝑘)=lim𝑁𝑇𝑓,𝐹(𝑁)Φ2𝑇(𝑁+1)𝑓,𝐹(𝑛1)Φ2+(𝑛1)𝑁𝑘=𝑛1𝑇𝑓,𝐹1(𝑘)Φ21(𝑘)Φ2.(𝑘+1)(4.7) From Lemma  4 of [1], we have 𝑐1𝑚𝑙(𝑚)𝑇𝑓,𝐹(𝑚)𝑐2𝑚𝑙(𝑚),(4.8) where 𝑙(𝑚)=log𝑚log𝑥𝑚 and 𝑙(𝑡)=𝑙([𝑡]) is a slowly varying function (even belonging to the Zygmund class). Since Φ is regularly varying with index 𝛿, we have Φ(𝑁)=𝑁𝛿𝑙(𝑁),(4.9) for some slowly varying function 𝑙. Thus, from (4.8) and (4.9) we get 𝑇𝑓,𝐹(𝑁)Φ2𝑇(𝑁+1)𝑓,𝐹(𝑁)Φ2𝑐(𝑁)2𝑁𝑙(𝑁)𝑁2𝛿𝑙2(𝑁)0,(4.10) since 2𝛿>1 and 𝑙(𝑁)/𝑙(𝑁) is slowly varying. By mean value theorem and (4.4)–(4.5), we get 𝑁𝑘=𝑛1𝑇𝑓,𝐹1(𝑘)Φ21(𝑘)Φ2(𝑘+1)𝐶𝑘=𝑛1𝑇𝑓,𝐹Φ(𝑘)𝜃𝑘Φ3𝜃𝑘𝐶𝑘=𝑛1𝑙(𝑘)Φ2,(𝑘)(4.11) where 𝜃𝑘(𝑘,𝑘+1). Let us note that the function 𝑓(𝑡)=𝑙(𝑡)/Φ2(𝑡) is regularly varying with index 2𝛿<1; therefore, by Theorem  1.5.11 in [14] with 𝜎=0, we get 𝑦𝑓(𝑦)𝑦𝑓(𝑦)2𝛿1,as𝑦.(4.12) In consequence, for some 𝐶>0, 𝑦𝑙(𝑡)Φ2𝑙(𝑡)𝑑𝑡𝐶𝑦(𝑦)Φ2(𝑦),(4.13) from this it follows that 𝑘=𝑛1𝑙(𝑘)Φ2((𝑘)𝐶𝑛1)𝑙(𝑛1)Φ2𝐶(𝑛1)𝑐1𝑇𝑓,𝐹(𝑛1)Φ2𝐶(𝑛1)𝑐1𝐶1𝑇𝑓,𝐹(𝑛)Φ2.(𝑛)(4.14) 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 𝑇𝑓,𝐹(𝑛). This is the case for nonlinear sectors such that 𝐴𝜃1𝐴𝑓,𝐹𝐴𝜃2, for some 𝜃1 and 𝜃2. Therefore, we may state the following proposition (in view of the remark following (3.5), the first inclusion holds automatically).

Proposition 4.2. Let 𝐴𝑓,𝐹 be any sector in 𝑑,𝑑2 such that 𝐴𝜃1𝐴𝑓,𝐹𝐴𝜃2, for some 𝜃1 and 𝜃2. Assume that Φ is a regularly varying function with index 𝛿>1/2 satisfying the conditions (4.3)–(4.5). Then (4.6) holds.

Remark 4.3. It is well known (see [14]) that a function Φ(𝑥) on [1,), regularly varying with index 𝛿, may be represented in the form Φ(𝑥)=𝑥𝛿𝑙(𝑥), where 𝑙(𝑥) is slowly varying and admits the representation: 𝑙(𝑥)=𝑐(𝑥)exp𝑥1𝜀(𝑢)𝑢𝑑𝑢,(4.15) where 𝑐(𝑥)𝑐>0 and 𝜀(𝑢)0 as 𝑢. To prove (4.6) it suffices to consider the case 𝑐(𝑥)𝑐, and it is easy to see that (4.3)–(4.5) are satisfied if 𝜀(𝑢) is positive, continuous, and nonincreasing function which tends to 0 as 𝑢.

Acknowledgments

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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