# Density by Moduli and Lacunary Statistical Convergence

Authors: Vinod K. Bhardwaj; Shweta Dhawan
Journal: Abstract and Applied Analysis
 (2016)

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## Abstract

We have introduced and studied a new concept off-lacunary statistical convergence, where f is an unbounded modulus. It is shown that, under certain conditions on a modulus f, the concepts of lacunary strong convergence with respect to a modulus f and f-lacunary statistical convergence are equivalent on bounded sequences. We further characterize those θ for which Sθf=Sf, where Sθf and Sf denote the sets of all f-lacunary statistically convergent sequences and f-statistically convergent sequences, respectively. A general description of inclusion between two arbitrary lacunary methods of f-statistical convergence is given. Finally, we give an Sθf-analog of the Cauchy criterion for convergence and a Tauberian theorem for Sθf-convergence is also proved.


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## Body

## 1. Introduction and Historical Background

Statistical convergence is a generalization of the usual notion of convergence. The idea of statistical convergence was given in the first edition (published in Warsaw in 1935) of the monograph of Zygmund [1], who called it “almost convergence.” Formally the concept of statistical convergence was introduced by Fast [2] and Steinhaus [3] independently in the year 1951 and later reintroduced by Schoenberg [4] in the year 1959.Although statistical convergence was introduced over nearly the last sixty years, it has become an active area of research in recent years. Statistical convergence has been studied by several authors [5–14].LetN denote the set of all natural numbers. A number sequence x=(xk) is said to be statistically convergent to the number l if for each ɛ>0 the set {k∈N:xk-l≥ɛ} has natural density zero, where the natural density of a subset K⊂N (chapter 11, see [15]) is defined by(1)dK=limn→∞⁡1nk≤n:k∈K,where {k≤n:k∈K} denotes the number of elements of K not exceeding n. Obviously we have d(K)=0 provided that K is a finite set of positive integers. If a sequence is statistically convergent to l, then we write it as S-lim⁡xk=l or xk→l(S). The set of all statistically convergent sequences is denoted by S.The notion of modulus function was introduced by Nakano [16] in 1953. Following Ruckle [17] and Maddox [18], we recall that a modulus f is a function from [0,∞) to [0,∞) such that(i)
f(x)=0 if and only if x=0,(ii)
f(x+y)≤f(x)+f(y) for x≥0,y≥0,(iii)
f is increasing,(iv)
f is continuous from the right at 0. Because of (ii), f(x)-f(y)≤fx-y so that, in view of (iv), f is continuous everywhere on [0,∞). A modulus may be unbounded or bounded. For example, f(x)=xp, where 0<p≤1, is unbounded, but f(x)=x/(1+x) is bounded.Connor [11], Öztürk and Bilgin [19], Ghosh and Srivastava [20], Bhardwaj and Singh [21–23], Çolak [24], Altin and Et [25], Aizpuru et al. [5], Bhardwaj and Dhawan [6], Bhardwaj et al. [26], and some others have used a modulus function to construct some sequence spaces.In the year 2014, Aizpuru et al. [5] defined a new concept of density with the help of an unbounded modulus function and, as a consequence, they obtained a new concept of nonmatrix convergence, namely, f-statistical convergence, which is intermediate between the ordinary convergence and the statistical convergence and agrees with the statistical convergence when the modulus function is the identity mapping.Quite recently, Bhardwaj and Dhawan [6] and Bhardwaj et al. [26] have introduced and studied the concepts of f-statistical convergence of order α and f-statistical boundedness, respectively, by using the approach of Aizpuru et al. [5].We now recall some definitions that will be needed in the sequel.Definition 1 (see [5]).
Letf be an unbounded modulus function. The f-density of a set A⊂N is defined by (2)dfA=limn→∞⁡fk≤n:k∈Afnin case this limit exists. Clearly, finite sets have zero f-density and df(N-A)=1-df(A) does not hold, in general. But if df(A)=0 then df(N-A)=1. Note that Aizpuru et al. [5] have used the notation df(A) to denote the f-density of a set A.Remark 2.
For any unbounded modulusf and A⊂N, df(A)=0 implies that d(A)=0. But converse need not be true in the sense that a set having zero natural density may have nonzero f-density with respect to some unbounded modulus f. For example, if we take f(x)=log⁡(x+1) and A={1,4,9,…}, then d(A)=0 but df(A)=1/2.Definition 3 (see [5]).
Letf be an unbounded modulus function. A number sequence x=(xk) is said to be f-statistically convergent to l or Sf-convergent to l, if, for each ɛ>0, (3)dfk∈N:xk-l≥ɛ=0,thatis,limn→∞⁡1fnfk≤n:xk-l≥ɛ=0,and one writes it as Sf-lim⁡xk=l or xk→l(Sf). The set of all f-statistically convergent sequences is denoted by Sf.In view of Definition3 and Remark 2, it follows that every f-statistically convergent sequence is statistically convergent, but a statistically convergent sequence need not be f-statistically convergent for every unbounded modulus f.By a lacunary sequenceθ=kr,r=0,1,2,…, where ko=0, we will mean an increasing sequence of nonnegative integers with kr-kr-1→∞ as r→∞. The intervals determined by θ will be denoted by Ir=(kr-1,kr], and we let hr=kr-kr-1. The ratio kr/kr-1, which also occurs frequently, will be denoted by qr.The space of all lacunary strongly convergent sequences,Nθ, was defined by Freedman et al. [27] as follows:(4)Nθ=x=xk:limr→∞⁡1hr∑k∈Irxk-l=0forsomenumberl.If x is lacunary strongly convergent to l, then we write Nθ-lim⁡xk=l or xk→l(Nθ).There is a strong connection [27] between Nθ and the space w of strongly Cesàro summable sequences, which is defined by (5)w=x=xk:limn→∞⁡1n∑k=1nxk-l=0forsomenumberl.In the special case, where θ=(2r), we have Nθ=w. In fact, for a lacunary sequence θ, Nθ=w if and only if 1<liminfrqr≤limsuprqr<∞ ([27], on page 511).Note that the spacew of strongly Cesàro summable sequences was denoted by σ1 in [27].In the year 1986, Maddox [28] extended the concept of strong Cesàro summability to that of strong Cesàro summability with respect to a modulus f. A sequence x=(xk) is said to be strongly Cesàro summable with respect to a modulus f to l if (6)limn→∞⁡1n∑k=1nfxk-l=0.The space of strongly Cesàro summable sequences, with respect to a modulus f, is denoted by w(f).Further, in the year 1994, the notion of lacunary strong convergence was extended to that of lacunary strong convergence with respect to a modulusf by Pehlivan and Fisher [29]. The space Nθ(f) of lacunary strongly convergent sequences with respect to a modulus f is defined as (7)Nθf=x=xk:limr→∞⁡1hr∑k∈Irfxk-l=0forsomenumberl.If x is lacunary strongly convergent with respect to a modulus f to l, then we write Nθ(f)-lim⁡xk or xk→l(Nθ(f)).Remark 4.
We will denote the spacesw(f) and Nθ(f) by wf and Nθf throughout the paper.Fridy and Orhan [12] introduced the concept of lacunary statistical convergence as follows.Definition 5.
Letθ=(kr) be a lacunary sequence. A number sequence x=(xk) is said to be lacunary statistically convergent to l or Sθ-convergent to l, if, for each ɛ>0, (8)limr→∞⁡1hrk∈Ir:xk-l≥ɛ=0.In this case, one writes Sθ-lim⁡xk=l or xk→l(Sθ). The set of all lacunary statistically convergent sequences is denoted by Sθ.Throughout this papers, l∞, and c will denote the spaces of all, bounded, and convergent sequences of complex numbers, respectively. Moreover, we will be concerned only with the sequences of scalars.Following a very recent and new approach of Aizpuru et al. [5], we first introduced the notion of fα-density and, consequently, obtained a new concept of nonmatrix convergence, namely, f-statistical convergence of order α [6], where f is an unbounded modulus function and α∈(0,1]. The fα-density of the subset A of N is defined as follows.Definition 6.
Letf be an unbounded modulus function and let α be any real number such that 0<α≤1. Then (9)dαfA=limn→∞⁡fk≤n:k∈Afnαin case this limit exists, where k≤n:k∈A denotes the number of elements of A not exceeding n.Now, in this paper we introduce a new concept off-lacunary statistical convergence, where f is an unbounded modulus, as follows.Definition 7.
Letf be an unbounded modulus and let θ=(kr) be a lacunary sequence. A sequence x=(xk) is said to be f-lacunary statistically convergent to l or Sθf-convergent to l, if, for each ɛ>0, (10)limr→∞⁡1fhrfk∈Ir:xk-l≥ɛ=0.In this case, one writes Sθf-lim⁡xk=l or xk→l(Sθf).For a given lacunary sequenceθ=(kr) and an unbounded modulus f, by Sθf we denote the set of all f-lacunary statistically convergent sequences.Remark 8.
In casef(x)=x, the concept of f-lacunary statistical convergence reduces to that of lacunary statistical convergence.In the next section it is shown that the concept off-lacunary statistical convergence is intermediate between the ordinary convergence and the lacunary statistical convergence. It is also proved that, under certain conditions on a modulus f, if a sequence is lacunary strongly convergent with respect to a modulus f then it is f-lacunary statistically convergent and that the concepts of lacunary strong convergence with respect to a modulus f and f-lacunary statistical convergence are equivalent on bounded sequences. In Section 3, we characterize those θ for which Sf=Sθf, where f is an unbounded modulus such that limt→∞(ft/t)>0 and there is a positive constant c such that f(xy)≥cf(x)f(y), for all x≥0,y≥0. Note that it was shown by Maddox [18] that, for any modulus f, limt→∞(ft/t) exists. Maddox [28] also showed that there exists an unbounded modulus f for which there is a positive constant c such that f(xy)≥cf(x)f(y), for all x≥0, y≥0. In Section 4, we first observe that it is possible for a sequence to have different Sθf-limits for different f and θ. In Theorem 24, we investigate certain conditions under which this situation cannot occur. In Section 5, a general description of inclusion between two arbitrary lacunary methods of f-statistical convergence is given. In the last section, f-lacunary statistical analog of Cauchy criterion for convergence is established. A Tauberian theorem for Sθf-convergence is also given.


## 2.f-Lacunary Statistical Convergence

We begin by establishing elementary connections between convergence,f-lacunary statistical convergence, and lacunary statistical convergence.Theorem 9.
Every convergent sequence isf-lacunary statistically convergent; that is, c⊂Sθf for any unbounded modulus f and lacunary sequence θ.Proof.
Letx=(xk) be any convergent sequence. Then, for each ϵ>0, the set {k∈N:xk-l≥ϵ} is finite. Suppose k∈N:xk-l≥ϵ=mo. Now, since {k∈Ir:xk-l≥ϵ}⊂{k∈N:xk-l≥ϵ} and f being modulus is increasing, therefore (11)fk∈Ir:xk-l≥ϵfhr≤fmofhr.Taking limit as r→∞, on both sides, we get (12)limr→∞⁡fk∈Ir:xk-l≥ϵfhr=0,asfhr⟶∞asr⟶∞.Remark 10.
The inclusionc⊂Sθf is strict as the sequence x=(xk) given in the necessity part of Lemma 19 is an example of a sequence which is Sθf-convergent but not convergent.Theorem 11.
Everyf-lacunary statistically convergent sequence is lacunary statistically convergent.Proof.
Supposex=(xk) is f-lacunary statistically convergent to l. Then by the definition of limit and the fact that f being modulus is subadditive, for every p∈N, there exists ro∈N such that, for r≥ro, we have (13)fk∈Ir:xk-l≥ϵ≤1pfhr≤1ppfhrp=fhrpand since f is increasing, we have (14)1hrk∈Ir:xk-l≥ϵ≤1p.Hence, x is lacunary statistically convergent to l.Remark 12.
It seems that the inclusionSθf⊂Sθ is strict. But right now we are not in a position to give an example of a sequence which is Sθ-convergent but not Sθf-convergent. So it is left as an open problem.Remark 13.
From Theorems9 and 11, we can say that the concept of f-lacunary statistical convergence is intermediate between the usual notion of convergence and the lacunary statistical convergence.We now establish a relationship betweenf-lacunary statistical convergence and lacunary strong convergence with respect to a modulus f. Fridy and Orhan [12] showed that on bounded sequences the concept of lacunary statistical convergence is equivalent to lacunary strong convergence. We now wish to find some condition on f, if any, so that the concept of f-lacunary statistical convergence becomes equivalent to lacunary strong convergence with respect to a modulus f.Theorem 14.
Letθ=(kr) be a lacunary sequence; then consider the following: (a)
For any unbounded modulusf for which limt→∞(ft/t)>0 and there is a positive constant c such that f(xy)≥cf(x)f(y), for all x≥0, y≥0,
(i)
xk→l(Nθf) implies xk→l(Sθf),(ii)
Nθf is a proper subset of Sθf.(b)
x∈l∞ and xk→l(Sθf) imply xk→l(Nθf), for any unbounded modulus f.(c)
Nθf∩l∞=Sθf∩l∞ for any unbounded modulus f for which limt→∞(ft/t)>0 and there is a positive constant c such that f(xy)≥cf(x)f(y) for all x≥0, y≥0.Proof.
(a) (i) For any sequencex=(xk) and ϵ>0, by the definition of a modulus function (ii) and (iii) we have (15)1hr∑k∈Irfxk-l≥1hrf∑k∈Irxk-l≥1hrf∑k∈Irxk-l≥ϵxk-l≥1hrfk∈Ir:xk-l≥ɛϵ≥chrfk∈Ir:xk-l≥ɛfϵ=chrfk∈Ir:xk-l≥ɛfhrfhrfϵfrom where it follows that x∈Sθf as x∈Nθf and limr→∞(fhr/hr)>0.
(ii) To show the strictness of inclusion, consider the sequencex=(xk) such that xk is to be 1,2,…,[hr] at the first [hr] integers in Ir, and xk=0 otherwise. Note that (xk) is not bounded. Also, for every ϵ>0, (16)1fhrfk∈Ir:xk-0≥ɛ=fhrfhr=fhrhr×hrfhr×hrhr⟶0asr⟶∞,becauselimr→∞⁡(f([hr])/[hr]), limr→∞⁡(fhr/hr) are positive and limr→∞⁡([hr]/hr)=0.
Thus,xk→0(Sθf). On the other hand, (17)1hr∑k∈Irfxk-0=f1+f2+⋯+fhrhr≥f1+2+⋯+hrhr=fhrhr+1/2hr≥cfhrfhr+1/2hr=c×fhrhr×fhr+1/2hr+1/2×hrhr+1/2hr>0asc,limr→∞⁡(f(hr)/hr), limr→∞⁡(fhr+1/2/hr+1/2), and limr→∞⁡([hr][hr]+1/2/hr) are positive. Hence xk↛0(Nθf).
(b) Suppose thatxk→l(Sθf) and x∈l∞; say xk-l≤H for all k∈N. Given ϵ>0, we have (18)1hr∑k∈Irfxk-l=1hr∑k∈Irxk-l≥ϵfxk-l+1hr∑k∈Irxk-l<ϵfxk-l≤1hrk∈Ir:xk-l≥ɛfH+1hrhrfϵ.Taking limit on both sides as r→∞, we get limr→∞⁡(1/hr)∑k∈Irf(|xk-l|)=0, in view of Theorem 11 and the fact that f is increasing.
(c) This is an immediate consequence of (a) and (b).Remark 15.
The example given in part (a) of the above theorem shows that the boundedness condition cannot be omitted from the hypothesis of part (b).Remark 16.
If we takef(x)=x in Theorem 14, we obtain Theorem 1 of Fridy and Orhan [12].


## 3.f-Lacunary Statistical Convergence versus f-Statistical Convergence

In this section we study the inclusionsSθf⊂Sf and Sf⊂Sθf under certain restrictions on θ and f.Lemma 17.
For any lacunary sequenceθ and unbounded modulus f for which limt→∞(ft/t)>0 and there is a positive constant c such that f(xy)≥cf(x)f(y), for all x≥0, y≥0, one has Sf⊂Sθf if and only if liminfrqr>1.Proof.
Sufficiency. If liminfrqr>1, then there exists δ>0 such that qr≥1+δ for sufficiently large r. Since hr=kr-kr-1, we have (19)hrkr≥δ1+δfor sufficiently large r. If xk→l(Sf), then, for given ϵ>0 and sufficiently large r, we have (20)1fkrfk≤kr:xk-l≥ɛ≥fk∈Ir:xk-l≥ɛfkr=fhrfkr×fk∈Ir:xk-l≥ɛfhr=fhrhrkrfkrhrkrfk∈Ir:xk-l≥ɛfhr≥fhrhrkrfkrδ1+δfk∈Ir:xk-l≥ɛfhr.This proves the sufficiency.
Necessity. Assume that liminfrqr=1. Proceeding as in Lemma 2.1 of [27], we can select a subsequence (kr(j)) of θ satisfying (21)krjkrj-1<1+1j,krj-1krj-1>j,whererj≥rj-1+2.Define a bounded sequence x=(xk) by(22)xk=1ifk∈Irj,forsomej=1,2,3,…;0otherwise.It is shown in Lemma 2.1 of [27] that x∉Nθ but x∈w. Thus in view of Theorem 3 of [29] and Theorem 14 we have x∉Sθf. On the other hand, it follows from Theorem 4 of [28] and Corollary 4.3 of [6] that x∈Sf. Hence Sf⊄Sθf. But this is a contradiction to the assumption that Sf⊂Sθf. This contradiction shows that our assumption is wrong. Hence liminfrqr>1.Remark 18.
The sequencex=(xk), constructed in the necessity part of the above lemma, is an example of f-statistically convergent sequence which is not f-lacunary statistically convergent.Lemma 19.
For any lacunary sequenceθ and unbounded modulus f for which limt→∞(ft/t)>0 and there is a positive constant c such that f(xy)≥cf(x)f(y), for all x≥0,y≥0, one has Sθf⊂Sf if and only if limsuprqr<∞.Proof.
Sufficiency. If limsuprqr<∞, then there is H>0 such that qr<H for all r. Now, suppose that xk→l(Sθf) and limr→∞(fhr/hr)=l′. Therefore, for given ϵ>0, there exists ro∈N such that for all r>ro(23)fhrhr<l′+ϵ,1fhrfk∈Ir:xk-l≥ɛ<ϵ.Let Nr=k∈Ir:xk-l≥ϵ. Using this notation, we have(24)fNrfhr<ϵ∀r>ro.Now, let M=max⁡{f(N1),f(N2),…f(Nro)} and let n be an integer such that kr-1<n≤kr, and then we can write (25)
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Necessity. Suppose that limsuprqr=∞. Following Lemma 2.2 of [27], we can select a subsequence (kr(j)) of lacunary sequence θ such that qr(j)>j. Define a bounded sequence x=(xk) by(26)xk=1ifkrj-1<k≤2krj-1,forsomej=1,2,3,…;0otherwise.It is shown in [27], on page 511, that x∈Nθ but x∉w. By Theorem 3 of [29] and Theorem 14, we conclude that x∈Sθf, but x∉Sf, in view of Theorem 2.1 of [10] and the fact that every f-statistically convergent sequence is statistically convergent. Hence Sθf⊄Sf. But this is a contradiction to the assumption that Sθf⊂Sf. This contradiction shows that limsuprqr<∞.Remark 20.
The sequencex=(xk), constructed in the necessity part of the above lemma, is an example of f-lacunary statistically convergent sequence which is not f-statistically convergent.Combining Lemmas17 and 19 we have the following.Theorem 21.
For any lacunary sequenceθ and unbounded modulus f for which limt→∞(ft/t)>0 and there is a positive constant c such that f(xy)≥cf(x)f(y), for all x≥0, y≥0, one has Sθf=Sf if and only if 1<liminfrqr≤limsuprqr<∞.Theorem 22.
For any lacunary sequenceθ and unbounded modulus f for which limt→∞(ft/t)>0 and there is a positive constant c such that f(xy)≥cf(x)f(y), for all x≥0, y≥0, one has (27)Sf=⋂liminfr⁡qr>1Sθf=⋃limsupr⁡qr<∞Sθf.Proof.
In view of Lemma17, we have Sf⊂⋂liminfr⁡qr>1Sθf. Suppose if possible x=(xk)∈⋂liminfr⁡qr>1Sθf but x∉Sf. We have (xk)∈Sθf for all θ=(kr) for which liminfrqr>1. If we take θ=(2r), then, in view of Theorem 21, we have Sθf=Sf and so x∈Sf, contrary to our assumption. Hence Sf=⋂liminfr⁡qr>1Sθf. The remaining part can be proved similarly and hence is omitted.Remark 23.
The sequencex=(xk) constructed in part (a) of Theorem 14 belongs to Sθf for every lacunary sequence θ, as well as unbounded modulus f for which limt→∞(ft/t)>0 and there is a positive constant c such that f(xy)≥cf(x)f(y) for all x≥0,y≥0. Hence ⋂liminfr⁡qr>1Sθf≠ϕ.


## 4. Uniqueness ofSθf-Limit

For any fixedθ, the Sθ-limit is unique; however, Fridy and Orhan [12] showed that it is possible for a sequence to have different Sθ-limits for different θ’s. They showed that this situation cannot occur if the sequence is statistically convergent. We now establish a similar result in case of Sθf-convergence. First we observe that it is possible for a sequence to have different Sθf-limits for different f and θ. To illustrate this, let us take f(x)=x, g(x)=2x, θ1=((2r)!), and θ2=((2r+1)!). Consider the sequence x given in [27], in the proof of Theorem 2.1, for which Nθ1-lim⁡xk=0 and Nθ2-lim⁡xk=1. Now, by applying Theorem 3 of [29] and Theorem 14 part (a), we see that Sθ1f-lim⁡xk=0 and Sθ2g-lim⁡xk=1.In the next theorem we investigate certain conditions under which this situation cannot occur.Theorem 24.
For any two lacunary sequencesθ1 and θ2, if x∈Sf∩Sθ1f and Sg∩Sθ2g, then Sθ1f-lim⁡xk=Sθ2g-lim⁡xk, where f and g are unbounded modulus functions for which(28)fx-fy=fx-y,gx-gy=gx-y,∀x≥0,y≥0.To prove this theorem we need the following lemma.Lemma 25.
For any lacunary sequenceθ, if x∈Sf∩Sθf, then Sθf-lim⁡xk=Sf-lim⁡xk, where f is an unbounded modulus function for which (29)fx-fy=fx-y,∀x≥0,y≥0.Proof.
SupposeSf-lim⁡xk=l and Sθf-lim⁡xk=l′, and l≠l′. Let ϵ>0 be such that 0<ϵ<l-l′/2. Using the definition of a modulus (iii) and (ii), we have (30)fk≤n:l-l′≥2ɛfn≤fk≤n:xk-l≥ɛfn+fk≤n:xk-l′≥ɛfn.Taking limit as n→∞ on both sides, we get(31)1≤0+limn→∞⁡fk≤n:xk-l′≥ɛfn≤1,hencelimn→∞fk≤n:xk-l′≥ɛfn=1.Now consider the kmth term of the sequence (f(n))-1f({k≤n:xk-l′≥ɛ}):(32)1fkmfk≤km:xk-l′≥ɛ=1fkmfk∈⋃r=1mIr:xk-l′≥ɛ=1fkmf∑r=1mk∈Ir:xk-l′≥ɛ≤1fkm∑r=1mfk∈Ir:xk-l′≥ɛ=1fkm∑r=1mfhr1fhrfk∈Ir:xk-l′≥ɛ.Also, in view of the choice of unbounded modulus f, we have(33)∑r=1mfhr=fh1+fh2+⋯+fhm=fk1-ko+fk2-k1+⋯+fkm-km-1=fk1-ko+fk2-k1+⋯+fkm-km-1=fk1-fko+fk2-fk1+⋯+fkm-fkm-1=fk1-fko+fk2-fk1+⋯+fkm-fkm-1=fkm.Now, using (33) in (32), we have(34)1fkmfk≤km:xk-l′≥ɛ≤1∑r=1mfhr∑r=1mfhrtr,where tr=(f(hr))-1f{k∈Ir:xk-l′≥ɛ}→0 because xk→l′(Sθf). Since θ is a lacunary sequence and f being modulus is increasing, the term on the right hand side of (34) is a regular weighted mean transformation of t=(tr), and therefore it, too, tends to zero as r→∞. Thus (35)1fkmfk≤km:xk-l′≥ɛ⟶0asm⟶∞.Also, since (f(km))-1f{k≤km:xk-l′≥ɛ} is a subsequence of sequence (f(n))-1f{k≤n:xk-l′≥ɛ}, we conclude that(36)fn-1fk≤n:xk-l′≥ɛ↛1.But this is a contradiction to (31). This contradiction shows that l=l′.Proof of Theorem24.
By Lemma25, we have(37)Sf-lim⁡xk=Sθ1f-lim⁡xk,Sg-lim⁡xk=Sθ2g-lim⁡xk.But, according to Corollary 2.2 of Aizpuru et al. [5], we have(38)Sf-lim⁡xk=Sg-lim⁡xk.Therefore, from (37) and (38) we have (39)Sθ1f-lim⁡xk=Sθ2g-lim⁡xk.If we takeg=f in Theorem 24 we have the following.Corollary 26.
For any two lacunary sequencesθ1 and θ2, if x∈Sf and x∈Sθ1f∩Sθ2f, then Sθ1f-lim⁡xk=Sθ2f-lim⁡xk, where f is an unbounded modulus function such that (40)fx-fy=fx-y,∀x≥0,y≥0.If we takef(x)=x in Corollary 26 we obtain the following result which contains Theorem 6 of Fridy and Orhan [12].Corollary 27.
For any two lacunary sequencesθ1 and θ2, if x∈S and x∈Sθ1∩Sθ2, then Sθ1-lim⁡xk=Sθ2lim⁡xk.If we takeθ1=θ2=θ in Theorem 24 we have the following.Corollary 28.
For any lacunary sequenceθ, if x∈Sf∩Sθf and Sg∩Sθg, then Sθf-lim⁡xk=Sθg-lim⁡xk, where f and g are unbounded modulus functions for which (41)fx-fy=fx-y,gx-gy=gx-y,∀x≥0,y≥0.


## 5. Inclusion between Two Lacunary Methods off-Statistical Convergence

Our first result shows that, for certainf, if β is a lacunary refinement of θ, then Sβf⊂Sθf. To establish this result, we first recall the definition of a lacunary refinement of lacunary sequence [27].Definition 29.
The lacunary sequenceβ′=(kr′) is called a lacunary refinement of the lacunary sequence θ=(kr) if (kr)⊂(kr′).Theorem 30.
Ifβ=(kr′) is a lacunary refinement of θ=(kr) and f is an unbounded modulus such that (42)fx-fy=fx-y,∀x≥0,y≥0,then x∈Sβf implies x∈Sθf.Proof.
Suppose eachIr of θ contains the points (kr,i′)i=1ν(r) of β so that (43)kr-1<kr,1′<kr,2′<⋯<kr,νr′=kr,whereIr,i′=kr,i-1′,kr,i′.Note that, for all r, ν(r)≥1 because (kr)⊂(kr′). Let xk→l(Sβf). Therefore, for each ϵ>0, we have (44)limr→∞1≤i≤νr⁡1fhr,i′fk∈Ir,i′:xk-l≥ɛ=0,where hr,i′=kr,i′-kr,i-1′ and hr,1′=kr,1′-kr-1, whence(45)limr→∞⁡∑Ir,i′⊂Ir1≤i≤νr1fhr,i′fk∈Ir,i′:xk-l≥ɛ=0.For each ϵ>0, we have(46)1fhrfk∈Ir:xk-l≥ɛ=1fhrfk∈⋃Ir,i′⊂Ir1≤i≤νrIr,i′:xk-l≥ɛ=1fhrf∑Ir,i′⊂Ir1≤i≤νrk∈Ir,i′:xk-l≥ɛ≤1fhr∑Ir,i′⊂Ir1≤i≤νrfk∈Ir,i′:xk-l≥ɛ=1fhr∑Ir,i′⊂Ir1≤i≤νrfhr,i′1fhr,i′fk∈Ir,i′:xk-l≥ɛ.Also, in view of the choice of unbounded modulus f and using the fact that β=(kr′) is increasing, we have(47)
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.Using (47) in (46), we have(48)1fhrfk∈Ir:xk-l≥ɛ≤1∑Ir,i′⊂Ir1≤i≤νrfhr,i′∑Ir,i′⊂Ir1≤i≤νrfhr,i′tr,i,where tr,i=(f(hr,i′))-1f{k∈Ir,i′:xk-l≥ɛ}. Since the term on the right hand side of (48) is a regular weighted mean transformation of (tr,i), which tends to zero as r→∞, therefore the term on the right hand side of (48) also tends to zero as r→∞. Thus,(49)1fhrfk∈Ir:xk-l≥ɛ⟶0asr⟶∞.Hence x∈Sθf.Our next result deals with the reverse inclusion.Theorem 31.
Letf be an unbounded modulus and β=(km′) is a lacunary refinement of the lacunary sequence θ=(kr). Let Ir=(kr-1,kr], hr=kr-kr-1, r=1,2,3,…, and Im′=(km-1′,km′], hm′=km′-km-1′, m=1,2,3,…. If there exists δ>0 such that (50)fhm′fhr≥δforeveryIm′⊂Ir,thenx∈Sθf implies x∈Sβf.Proof.
For anyϵ>0, and every Im′, we can find Ir such that Im′⊂Ir; then we have (51)1fhm′fk∈Im′:xk-l≥ɛ≤1fhm′fk∈Ir:xk-l≥ɛ=fhrfhm′1fhrfk∈Ir:xk-l≥ɛ≤1δ1fhrfk∈Ir:xk-l≥ɛfrom where it follows that Sθf⊂Sβf.In the next theorem we deal with a more general situation.Theorem 32.
Letf and g be any two unbounded modulus functions such that f(x)≤g(x), for all x∈[0,∞), and β=(km′) is a lacunary refinement of the lacunary sequence θ=(kr). Let Ir=(kr-1,kr], hr=kr-kr-1,r=1,2,3,…, and Im′=(km-1′,km′], hm′=km′-km-1′,m=1,2,3,…. If there exists 0<δ≤1 such that (52)fhm′ghr≥δforeveryIm′⊂Ir,thenx∈Sθg implies x∈Sβf.Proof.
For anyϵ>0, and every Im′, we can find Ir such that Im′⊂Ir; then we have (53)1fhm′fk∈Im′:xk-l≥ɛ≤1fhm′gk∈Im′:xk-l≥ɛ≤1fhm′gk∈Ir:xk-l≥ɛ=ghrfhm′1ghrgk∈Ir:xk-l≥ɛ≤1δ1ghrgk∈Ir:xk-l≥ɛ,from where it follows that Sθg⊂Sβf.In the next theorem we show that the inclusionSθf⊂Sβf is possible even if none of θ and β is refinement of the other.Theorem 33.
Letf be an unbounded modulus such that (54)fx-fy=fx-y,∀x≥0,y≥0.Suppose β=(km′) and θ=(kr) are two lacunary sequences. Let Ir=(kr-1,kr], hr=kr-kr-1,r=1,2,3,…, Im′=(km-1′,km′], hm′=km′-km-1′,m=1,2,3,…, and Irm=Ir∩Im′,r,m=1,2,3,…. If there exists δ>0 such that(55)fρrmfhr≥δforeveryr,m=1,2,3,…,provided ρrm>0, where ρrm denotes the length of the interval Irm, then x∈Sθf implies Sβf.The proof is similar to Theorem2 of Li [14] and hence is omitted.Remark 34.
If the condition in Theorem33 is replaced by f(ρrm)/f(hm′)≥δ for every r,m=1,2,…, provided ρrm>0, where ρrm denotes the length of the interval Irm=Ir∩Im′,r,m=1,2,3,…, it can be seen that x∈Sβf implies x∈Sθf.Combining Remark34 and Theorem 33, we get the following.Theorem 35.
Letf be an unbounded modulus such that(56)fx-fy=fx-y,∀x≥0,y≥0.Suppose β=(km′) and θ=(kr) are two lacunary sequences. Let Ir=(kr-1,kr], hr=kr-kr-1,r=1,2,3,…, Im′=(km-1′,km′], hm′=km′-km-1′,m=1,2,3,…, and Irm=Ir∩Im′,r,m=1,2,3,…. If there exists δ>0 such that(57)fρrmfhr+hm′≥δforeveryr,m=1,2,3,…,provided ρrm>0, where ρrm denotes the length of the interval Irm, then Sθf=Sβf.


## 6. TheSθf-Cauchy Criterion and a Tauberian Theorem

We now introduce theSθf-analog of the Cauchy criterion, which turns out to be equivalent to the Sθf-convergence.Definition 36.
Letθ=(kr) be a lacunary sequence and let f be an unbounded modulus. The sequence x=(xk) is said to be f-lacunary statistically Cauchy or Sθf-Cauchy sequence if there is a subsequence (xk′(r)) of x such that k′(r)∈Ir, for each r, limrxk′(r)=l, and, for every ϵ>0, (58)limr→∞⁡1fhrfk∈Ir:xk-xk′r≥ɛ=0.Theorem 37.
The sequencex is Sθf-convergent if and only if x is Sθf-Cauchy sequence, where f is an unbounded modulus and θ=(kr) is a lacunary sequence.The proof is similar to proof of Theorem2 of Fridy and Orhan [13] and hence is omitted.Corollary 38.
EverySθf-convergent sequence has a convergent subsequence.This result leads to the following Tauberian theorem. We state the following result, which can be established following the technique of Theorem3 of Fridy and Orhan [13].Theorem 39.
IfSθf-lim⁡xk=l and limr→∞hrmax⁡{Δxi:i∈Ir}=0 then limkxk=l.



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