On Some Further Generalizations of Strong Convergence in Probabilistic Metric Spaces Using Ideals

Abstract

Following the line of (Das et al., 2011, Savas and Das, 2011), we make a new approach in this paper to extend the notion of strong convergence and more general strong statistical convergence (Şençimen and Pehlivan, 2008) using ideals and introduce the notion of strong ℐ- and ℐ^∗-statistical convergence and two related concepts, namely, strong ℐ-lacunary statistical convergence and strong ℐ-𝜆-statistical convergence in a probabilistic metric space endowed with strong topology. We mainly investigate their interrelationship and study some of their important properties.

1. Introduction and Background

The usual idea of convergence is not enough to understand behaviours of those sequences which are not convergent. One of the approaches to include more sequences under purview is to consider those sequences which are convergent when restricted to some “big set of natural numbers.” To accomplish this, the idea of convergence of real sequences was extended to statistical convergence by Fast [1], and it was further developed by several authors [2–4]. Recall that “asymptotic density” of a set 𝐴⊆ℕ is defined as 𝑑(𝐴)=lim_𝑛→∞1𝑛|{𝑘≤𝑛∶𝑘∈𝐴}|, (1) provided that the limit exists, where ℕ denotes the set of natural numbers and the vertical bar stands for cardinality of the enclosed set. The sequence {𝑥_𝑛}_𝑛∈ℕ of real is said to be statistically convergent to a real number 𝑥 if for each 𝜖>0, lim_𝑛→∞1𝑛𝑘≤𝑛∶𝑥_𝑘−𝑥≥𝜖=0. (2)

In another direction, a new type of convergence called lacunary statistical convergence was introduced and studied in [5]. More results related to this convergence can be found in [6]. The concept of 𝜆-statistical convergence was introduced by Mursaleen in [7] as a further extension of statistical convergence. Afterward, in [8], Karakaya et al. defined statistical convergence of sequences of functions in intuitionistic fuzzy normed spaces and also used the concepts of lacunary and 𝜆-statistical convergence for sequences of functions in these spaces in [9, 10]. It must be mentioned in this context that some of the above mentioned convergence methods have applications in number theory, measure theory, fourier analysis, optimization, and many branches of mathematics.

The concepts of ℐ and ℐ^∗-convergence were introduced and investigated by Kostyrko et al. [11] as further generalizations of statistical convergence. They were much more general than other approaches as they were based on the very general notion of ideals of ℕ. In recent years, a lot of investigations have been done on ideal convergence and in some particular new approaches were made in [12, 13] to generalize the above mentioned convergences (for works on ideal convergence, see e.g., the papers [12–17] where many more references can be found).

On the other hand, the idea of probabilistic metric space was first introduced by Menger [18] in the name of “statistical metric space.” In this theory, the concept of distance is probabilistic rather than deterministic. More precisely, the distance between two points 𝑝, 𝑞 is defined as a distribution function 𝐹_𝑝𝑞 instead of a nonnegative real number. For a positive number 𝑡, 𝐹_𝑝𝑞(𝑡) is interpreted as the probability that the distance between the points 𝑝 and 𝑞 is less than 𝑡. The theory of probabilistic metric spaces was brought to prominence by path breaking works of Schweizer et al. [19–22] and Tardiff [23] among others. Detailed theory of probabilistic metric space can be found in the famous book written by Schweizer and Sklar [24]. Several topologies can be defined on this space, but the topology that was found to be most useful is the “strong topology.” Şençimen and Pehlivan [25] noted in their paper that as the strong topology is first countable and Hausdorff, it can be completely specified in terms of strong convergence of sequences. Since probabilistic metric spaces have many applications in applied mathematics, in order to provide a more general framework for applications, they have recently studied the statistical convergence and then strong ideal convergence in probabilistic metric spaces [25, 26] and also carried out further investigations in [27, 28].

In this paper, first of all, we introduce the notion of strong ℐ-statistical convergence in probabilistic metric spaces which happens to be more general than strong statistical convergence. We also introduce the concepts of strong ℐ-lacunary statistical convergence and strong ℐ-𝜆-statistical convergence in probabilistic metric spaces and investigate some of their important properties, in particular their relations with strong ℐ-statistical convergence. As these concepts are more general than the concept of strong statistical convergence, we believe that they can extend the general framework introduced by [25] which in the future may enhance the applicability of strong convergence in probabilistic metric spaces.

2. Preliminaries

First, we recall some basic concepts related to the probabilistic metric spaces (in short PM spaces) (see [24]).

Definition 1. A nondecreasing function 𝐹∶ℝ→[0,1] defined on ℝ with 𝐹(−∞)=0 and 𝐹(∞)=1, where ℝ=[−∞,∞], is called a distribution function.
The set of all left continuous distribution functions over (−∞,∞) is denoted by Δ.
We consider the relation “≤” on Δ defined by 𝐹≤𝐺 if and only if 𝐹(𝑥)≤𝐺(𝑥) for all 𝑥∈ℝ. It can be easily verified that the relation “≤” is a partial order on Δ.

Definition 2. For any 𝑎∈ℝ, the unit step function at 𝑎 is denoted by 𝜀_𝑎 and is defined to be a function in Δ given by 𝜀_𝑎(𝑥)=0,−∞≤𝑥≤𝑎1,𝑎<𝑥≤∞. (3)

Definition 3. A sequence {𝐹_𝑛}_𝑛∈ℕ of distribution functions converges weakly to a distribution function 𝐹, and one writes 𝐹−→_𝑛^𝑤𝐹 if and only if the sequence {𝐹_𝑛(𝑥)}_𝑛∈ℕ converges to 𝐹(𝑥) at each continuity point 𝑥 of 𝐹.

Definition 4. The distance between 𝐹 and 𝐺 in Δ is denoted by 𝑑_𝐿(𝐹,𝐺) and is defined as the infimum of all numbers ℎ∈(0,1] such that the inequalities 𝐹()(𝑥)()𝐺()(𝑥)()𝑥−ℎ−ℎ≤𝐺≤𝐹𝑥+ℎ+ℎ,𝑥−ℎ−ℎ≤𝐹≤𝐺𝑥+ℎ+ℎ (4) hold for every 𝑥∈(−1/ℎ,1/ℎ).

Here, we are interested in the subset of Δ consisting of those elements 𝐹 that satisfy 𝐹(0)=0.

Definition 5. A distance distribution function is a nondecreasing function 𝐹 defined on ℝ⁺=[0,∞] that satisfies 𝐹(0)=0 and 𝐹(∞)=1 and is left continuous on (0,∞).

The set of all distance distribution functions is denoted by Δ⁺.

The function 𝑑_𝐿 is clearly a metric on Δ⁺. The metric space (Δ⁺,𝑑_𝐿) is compact and hence complete (see [29]).

Theorem 6 (see [24]). Let 𝐹∈Δ⁺ be given. Then, for any 𝑡>0, 𝐹(𝑡)>1−𝑡 if and only if 𝑑_𝐿(𝐹,𝜀₀)<𝑡. >

Note 7. Geometrically, 𝑑_𝐿(𝐹,𝜀₀) is the abscissa of the point of intersection of the line 𝑦=1−𝑥 and the graph of 𝐹 (if necessary we add vertical line segment at the point of discontinuity).

Definition 8. A triangle function is a binary operation 𝜏 on Δ⁺, 𝜏∶Δ⁺×Δ⁺→Δ⁺ which is commutative, associative, and nondecreasing in each place and has 𝜀₀ as identity.

Definition 9. A PM space is a triplet (𝑋,𝔉,𝜏) where 𝑋 is a nonempty set, 𝔉 is a function from 𝑋×𝑋 into Δ⁺, and 𝜏 is a triangle function. The following conditions for a PM space are satisfied for all 𝑥,𝑦,𝑧∈𝑋:(1)𝔉(𝑥,𝑥)=𝜀₀,(2)𝔉(𝑥,𝑦)≠𝜀₀ if 𝑥≠𝑦,(3)𝔉(𝑥,𝑦)=𝔉(𝑦,𝑥),(4)𝔉(𝑥,𝑧)≥𝜏(𝔉(𝑥,𝑦),𝔉(𝑦,𝑧)).
In the sequel, we shall denote 𝔉(𝑥,𝑦) by 𝐹_𝑥𝑦 and its value at 𝑡 by 𝐹_𝑥𝑦(𝑡). Throughout this paper 𝑋 shall represent the PM space (𝑋,𝔉,𝜏).

Definition 10. Let 𝑋 be a PM space. For 𝑥∈𝑋 and 𝑡>0, the strong 𝑡-neighbourhood of 𝑥 is defined as the set 𝒩_𝑥(𝑡)=𝑦∈𝑋∶𝐹_𝑥𝑦(𝑡)>1−𝑡. (5)
The collection 𝔑_𝑥={𝒩_𝑥(𝑡)∶𝑡>0} is called the strong neighbourhood system at 𝑥, and the union 𝔑=⋃_𝑥∈𝑋𝔑_𝑥 is called the strong neighbourhood system for 𝑋.

By Theorem 6, we can write 𝒩_𝑥(𝑡)=𝑦∈𝑆∶𝑑_𝐿𝐹_𝑥𝑦,𝜀₀<𝑡. (6)

If 𝜏 is continuous then the strong neighbourhood system 𝔑 determines a Hausdorff topology for 𝑋. This topology is called the strong topology for 𝑋.

Definition 11. Let 𝑋 be a PM space. Then, for any 𝑡>0 the subset 𝔘(𝑡) of 𝑋×𝑋 is given by 𝔘(𝑡)={(𝑥,𝑦)∶𝐹_𝑥𝑦(𝑡)>1−𝑡} and it is called the strong 𝑡-vicinity.

Theorem 12 (see [24]). Let 𝑋 be a PM space, and, 𝜏 be continuous. Then, for any 𝑡>0, there exists a 𝜂>0 such that 𝔘(𝜂)∘𝔘(𝜂)⊂𝔘(𝑡), where 𝔘(𝜂)∘𝔘(𝜂)={(𝑥,𝑧)∶𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑦,(𝑥,𝑦)𝑎𝑛𝑑(𝑦,𝑧)∈𝔘(𝑡)}.

Note 13. Under the hypothesis of Theorem 12, we can say that for any 𝑡>0 there is a 𝜂>0 such that 𝐹_𝑥𝑧(𝑡)>1−𝑡 whenever 𝐹_𝑥𝑦(𝜂)>1−𝜂 and 𝐹_𝑦𝑧(𝜂)>1−𝜂. Equivalently, it can be written as follows: for any 𝑡>0 there is an 𝜂>0 such that 𝑑_𝐿(𝐹_𝑥𝑧,𝜀₀)<𝑡, whenever 𝑑_𝐿(𝐹_𝑥𝑦,𝜀₀)<𝜂 and 𝑑_𝐿(𝐹_𝑦𝑧,𝜀₀)<𝜂.
If 𝜏 is continuous in a PM space 𝑋, then the strong neighbourhood system 𝔑 determines a Kuratowski closure operation. It is termed as the strong closure. For any subset 𝐴 of 𝑋, the strong closure of 𝐴 is denoted by 𝜅(𝐴) and is defined as 𝜅(𝐴)=𝑥∈𝑋∶forany𝑡>0,thereisa𝑦∈𝐴suchthat𝐹_𝑥𝑦(𝑡)>1−𝑡. (7)

Remark 14. Throughout the rest of the paper, we always assume that in a PM space 𝑋, the triangle function 𝜏 is continuous and 𝑋 is endowed with strong topology.

Definition 15. Let 𝑋 be a PM space. A sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑋 is said to be strongly convergent to a point 𝜉∈𝑋 if for any 𝑡>0 there exists a natural number 𝑁 such that 𝑥_𝑛∈𝔑_𝜉(𝑡) whenever 𝑛≥𝑁. One writes 𝑥_𝑛→𝜉 or lim_𝑛→∞𝑥_𝑛=𝜉.
Similarly, a sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑋 is called a strong Cauchy sequence if for any 𝑡>0 there exists a natural number 𝑁 such that (𝑥_𝑚,𝑥_𝑛)∈𝔘(𝑡) whenever 𝑚,𝑛≥𝑁.

Next, we recall some of the basic concepts related to the theory of ℐ-convergence, and we refer to [11] for more details.

Definition 16. Let 𝑋 be any nonempty set. Then, the family ℐ⊆𝒫(𝑋) is called an ideal in 𝑋 if(1)𝐴,𝐵∈ℐ imply 𝐴∪𝐵∈ℐ,(2)if 𝐴∈ℐ and 𝐵⊆𝐴 then 𝐵∈ℐ.

Definition 17. Let 𝑋 be any nonempty set. The family ℱ⊆𝒫(𝑋) is called a filter in 𝑋 if(1)∅∉ℱ,(2)𝐴,𝐵∈ℱ imply 𝐴∩𝐵∈ℱ,(3)if 𝐴∈ℱ and 𝐴⊆𝐵 then 𝐵∈ℱ.
If ℐ is an ideal in 𝑋, then ℱ(ℐ)={𝑋⧵𝐴∶𝐴∈ℐ} is a filter in 𝑋, which is called the filter associated with the ideal ℐ. An ideal ℐ in 𝑋 is called proper if and only if 𝑋∉ℐ. ℐ is called nontrivial if ℐ≠{∅}. An ideal is called an admissible ideal if it is proper and contains {𝑥} for all 𝑥∈𝑋. In other words, it is called an admissible ideal if it is proper and contains all of its finite subsets.

Definition 18. An admissible ideal ℐ is said to satisfy the condition (AP) if for every countable family of mutually disjoint sets {𝐴₁,𝐴₂,…} belonging to ℐ there exists a countable family of sets {𝐵₁,𝐵₂,…} such that 𝐴_𝑗Δ𝐵_𝑗 is a finite set for every 𝑗∈ℕ and 𝐵=⋃^∞_𝑗=1𝐵_𝑗∈ℐ.

Throughout the paper ℐ stands for a nontrivial admissible ideal of ℕ, and ℱ(ℐ) is the filter associated with the ideal ℐ of ℕ.

3. Strong ℐ- and ℐ^∗-Statistical Convergence in PM Space

In this section, we extend the concept of strong statistical convergence in PM spaces [25] via ideals and prove some associated results.

Definition 19. A sequence {𝑥_𝑛}_𝑛∈ℕ in a PM space 𝑋 is said to be strong statistically convergent to 𝑥 in 𝑋 if for 𝜖>0, lim_𝑛→∞1𝑛𝑘≤𝑛∶𝑥_𝑛∉𝒩_𝑥(𝜖)=0. (8)

Definition 20. A sequence {𝑥_𝑛}_𝑛∈ℕ of real numbers is said to be strong ℐ-statistically convergent to 𝑥 if for each 𝜖>0 and 𝛿>0, 1𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑥_𝑘−𝑥≥𝜖≥𝛿∈ℐ. (9) In this case, we write 𝑥_𝑛→𝑥(𝑆(ℐ)).

We now introduce the definition of strong ℐ-statistical convergence in PM space. So, consider the following [12].

Definition 21. A sequence {𝑥_𝑛}_𝑛∈ℕ in a PM space 𝑋 is said to be strong ℐ-statistically convergent to 𝑥 if for each 𝜖>0 and 𝛿>0, 1𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑥_𝑘∉𝒩_𝑥(𝜖)≥𝛿∈ℐ. (10) In this case, we write 𝑥_𝑛→𝑥(𝑆_PM(ℐ)) and the class of all strong ℐ-statistically convergent sequences is simply denoted by 𝑆_PM(ℐ).

Theorem 22. Let 𝑋 be a PM space, and, let 𝜏 be continuous. Then, the strong ℐ-statistical limit of a sequence in 𝑋 is unique.

Proof. If possible, let the sequence {𝑥_𝑛}_𝑛∈ℕ converge to two different limits 𝑥₁ and 𝑥₂ in 𝑋. Since 𝑥₁≠𝑥₂, we have 𝐹_𝑥1𝑥2≠𝜀₀ and so there exists a positive real number 𝑡 such that 𝑑_𝐿(𝐹_𝑥1𝑥2,𝜀₀)=𝑡. Choose 𝜂>0 such that 𝑑_𝐿(𝐹_𝑥𝑦,𝜀₀)<𝜂 and 𝑑_𝐿(𝐹_𝑦𝑧,𝜀₀)<𝜂 imply that 𝑑_𝐿(𝐹_𝑥𝑧,𝜀₀)<𝑡. Since 𝑥_𝑛→𝑥₁(𝑆_PM(ℐ)) and 𝑥_𝑛→𝑥₂(𝑆_PM(ℐ)) for this 𝜂>0 and 0<𝛿<1, 1𝐴=𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑥_𝑘∉𝒩_𝑥1𝜂<𝛿3(ℐ),1∈ℱ𝐵=𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑥_𝑘∉𝒩_𝑥2𝜂<𝛿3∈ℱ(ℐ). (11) Clearly, 𝐴∩𝐵≠∅. Let 𝑚∈𝐴∩𝐵. Hence, 1𝑚𝑘≤𝑚∶𝑥_𝑘∉𝒩_𝑥1𝜂<𝛿3,1𝑚𝑘≤𝑚∶𝑥_𝑘∉𝒩_𝑥2𝜂<𝛿3, (12) and consequently, 1𝑚𝑘≤𝑚∶𝑥_𝑘∉𝒩_𝑥1𝜂or𝑥_𝑘∉𝒩_𝑥2𝜂<𝛿<1. (13) Then, there exists some 𝑘≤𝑚 such that 𝑥_𝑘∈𝒩_𝑥1(𝜂) and 𝑥_𝑘∈𝒩_𝑥2(𝜂), that is, 𝑑_𝐿(𝐹_{𝑥𝑘𝑥1},𝜀₀)<𝜂 and 𝑑_𝐿(𝐹_{𝑥𝑘𝑥2},𝜀₀)<𝜂. Combining these two, we have 𝑑_𝐿(𝐹_𝑥1𝑥2,𝜀₀)<𝑡, which is a contradiction. Hence, the limit is unique.

Theorem 23. Let (𝑋,𝔉,𝜏) be a PM space. If {𝑥_𝑛}_𝑛∈ℕ and {𝑦_𝑛}_𝑛∈ℕ are two sequences in 𝑋 such that 𝑥_𝑛→𝑥(𝑆_𝑃𝑀(ℐ)) and 𝑦_𝑛→𝑦(𝑆_𝑃𝑀(ℐ)), then one has 𝑑_𝐿(𝐹_{𝑥𝑛𝑦𝑛},𝐹_𝑥𝑦)→0(𝑆(ℐ)).

Proof. For 𝑡>0 there exists a 𝜂(𝑡)>0 such that 𝑑_𝐿(𝐹_𝑥𝑦,𝐹_{𝑥′𝑦′})<𝑡 whenever 𝑥^′∈𝒩_𝑥(𝜂) and 𝑦^′∈𝒩_𝑦(𝜂). Now, assume that 𝑥_𝑛→𝑥(𝑆_PM(ℐ)) and 𝑦_𝑛→𝑦(𝑆_PM(ℐ)). Then, for any 𝑡>0, we have 𝑛∈ℕ∶𝑑_𝐿𝐹_{𝑥𝑛𝑦𝑛},𝐹_𝑥𝑦≥𝑡⊆𝑛∈ℕ∶𝑥_𝑛∉𝒩_𝑥𝜂∪𝑛∈ℕ∶𝑦_𝑛∉𝒩_𝑦𝜂. (14) Clearly, 1𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑑_𝐿𝐹_{𝑥𝑘𝑦𝑘},𝐹_𝑥𝑦⊆1≥𝑡≥𝛿𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝛿≥𝜂2∪1𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑑_𝐿𝐹_𝑦𝑘𝑦,𝜀₀≥𝛿≥𝜂2. (15) Since 𝑥_𝑛→𝑥(𝑆_PM(ℐ)) and 𝑦_𝑛→𝑦(𝑆_PM(ℐ)), we have 1𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝛿≥𝜂21∈ℐ,𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑑_𝐿𝐹_𝑦𝑘𝑦,𝜀₀≥𝛿≥𝜂2∈ℐ. (16) Therefore, union of these two sets also belongs to ℐ, and consequently {𝑛∈ℕ∶(1/𝑛)|{𝑘≤𝑛∶𝑑_𝐿(𝐹_{𝑥𝑘𝑦𝑘},𝐹_𝑥𝑦)≥𝑡}|≥𝛿}∈ℐ. This completes the proof.

Definition 24. A sequence {𝑥_𝑛}_𝑛∈ℕ in a PM space 𝑋 is said to be ℐ^∗-statistically convergent to 𝑥 if there exists a set 𝑀={𝑚₁<𝑚₂<⋯}∈ℱ(ℐ) such that for all 𝑡>0, lim_𝑘→∞1𝑚_𝑘𝑟≤𝑚_𝑘∶𝑥_𝑟∉𝒩_𝑥(𝑡)=0. (17) In this case, we write 𝑥_𝑛→𝑥(𝑆_PM(ℐ^∗)).

Theorem 25. If a sequence {𝑥_𝑛}_𝑛∈ℕ in a PM space 𝑋 is ℐ^∗-statistically convergent, then it is strong ℐ-statistically convergent. Moreover, if ℐ is an admissible ideal satisfying the property (AP), then strong ℐ-statistical convergence implies ℐ^∗-statistical convergence for any sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑋.

Proof. Let {𝑥_𝑛}_𝑛∈ℕ in 𝑋 be a sequence such that 𝑥_𝑛→𝑥(𝑆_PM(ℐ^∗)). Then, there exists a set 𝑀={𝑚₁<𝑚₂<⋯}∈ℱ(ℐ) such that lim_𝑘→∞1𝑚_𝑘𝑟≤𝑚_𝑘∶𝑥_𝑟∉𝒩_𝑥(𝑡)=0, (18) that is, for 𝛿>0 there exists a natural number 𝑘₀ such that (1/𝑚_𝑘)|{𝑟≤𝑚_𝑘∶𝑥_𝑟∉𝒩_𝑥(𝑡)}|<𝛿 for all 𝑘>𝑘₀. Now, we have 1𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑥_𝑘∉𝒩_𝑥(𝑡𝑚)≥𝛿⊆𝐻∪₁,𝑚₂,…,𝑚_𝑘0, (19) where 𝐻=ℕ⧵𝑀∈ℐ. Since ℐ is admissible, the set on the right hand side belongs to ℐ and consequently {𝑛∈ℕ∶(1/𝑛)|{𝑘≤𝑛∶𝑥_𝑘∉𝒩_𝑥(𝑡)}|≥𝛿}∈ℐ. Hence, 𝑥_𝑛→𝑥(𝑆_PM(ℐ)).
Next, suppose that 𝑥_𝑛→𝑥(𝑆_PM(ℐ)). Clearly, the sequence (1/𝑛)|{𝑘≤𝑛∶𝑥_𝑘∉𝒩_𝑥(𝑡)}| is ℐ-convergent to zero. Since the ideal ℐ has the property (AP), the sequence (1/𝑛)|{𝑘≤𝑛∶𝑥_𝑘∉𝒩_𝑥(𝑡)}| is ℐ^∗-convergent to zero. Therefore, 𝑥_𝑛→𝑥(𝑆_PM(ℐ^∗)).

Definition 26. A sequence {𝑥_𝑛}_𝑛∈ℕ in a PM space 𝑋 is said to be strong ℐ-statistically Cauchy if for every 𝜖>0, there exists a positive integer 𝑁=𝑁(𝜖)∈ℕ such that for any 𝛿>0, 1𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑥_𝑘∉𝒩_𝑥𝑁(𝜖)≥𝛿∈ℐ. (20)

Theorem 27. In a PM space 𝑋 every strong ℐ-statistically convergent sequence is strong ℐ-statistically Cauchy.

Proof. Since 𝑥_𝑛→𝑥(𝑆_PM(ℐ)), then for any 𝜖>0 and 0<𝛿<1, 1𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑥_𝑘∉𝒩_𝑥𝜖2<𝛿∈ℱ(ℐ). (21) It follows that 1𝑛𝑘≤𝑛∶𝑥_𝑘∉𝒩_𝑥𝜖2<𝛿<1. (22) Then, there exists a 𝑁≤𝑛 such that 𝑥_𝑁∈𝒩_𝑥(𝜖/2). Therefore, 𝑑_𝐿(𝐹_𝑥𝑁𝑥,𝜀₀)<𝜖/2, and so 𝑘≤𝑛∶𝑑_𝐿𝐹_{𝑥𝑘𝑥𝑁},𝜀₀⊆≥𝜖𝑘≤𝑛∶𝑑_𝐿𝐹_{𝑥𝑘𝑥𝑁},𝐹_𝑥𝑁𝑥≥𝜖2. (23) Consequently, 1𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑑_𝐿𝐹_{𝑥𝑘𝑥𝑁},𝜀₀⊆1≥𝜖≥𝛿𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑑_𝐿𝐹_{𝑥𝑘𝑥𝑁},𝐹_𝑥𝑁𝑥≥𝜖2≥𝛿∈ℐ. (24) This completes the proof.

Definition 28. A sequence {𝑥_𝑛}_𝑛∈ℕ in 𝑋 is said to be ℐ^∗-statistically Cauchy if for every 𝜖>0 there exists 𝑁∈ℕ and a set 𝑀={𝑚₁<𝑚₂<⋯}∈ℱ(ℐ) such that lim_𝑘→∞1𝑚_𝑘𝑟≤𝑚_𝑘∶𝑥_𝑟∉𝒩_𝑥𝑁(𝜖)=0. (25)

Theorem 29. In a PM space 𝑋, every ℐ^∗-statistically Cauchy sequence is strong ℐ-statistically Cauchy. Moreover, if ℐ is an admissible ideal satisfying the property (AP), then strong ℐ-statistically Cauchy sequence coincides with ℐ^∗-statistically Cauchy sequence.

Proof. The proof is straightforward, and so it has been omitted.

4. Strong ℐ-Lacunary Statistical Convergence in PM Space

In this section, we discuss some of the results associated with the lacunary statistical convergence and extend certain summability methods using this notion.

By a lacunary sequence, we mean an increasing integer sequence 𝜃={𝑘_𝑟} such that 𝑘₀=0 and ℎ_𝑟=𝑘_𝑟−𝑘_𝑟−1→∞ as 𝑟→∞. Throughout this paper, the interval determined by 𝜃 shall be denoted by 𝐼_𝑟=(𝑘_𝑟−1,𝑘_𝑟], and the ratio 𝑘_𝑟/𝑘_𝑟−1 shall be denoted by 𝑞_𝑟 [5]. The lacunary sequence 𝛽={𝑙_𝑟} is said to be the lacunary refinement of the lacunary sequence 𝜃={𝑘_𝑟} if {𝑘_𝑟}⊆{𝑙_𝑟} [30].

Definition 30. Let 𝜃={𝑘_𝑟} be a lacunary sequence. A sequence {𝑥_𝑛}_𝑛∈ℕ in a PM space 𝑋 is said to be strong ℐ-lacunary statistically convergent to 𝑥 if for every 𝜖>0 and 𝛿>0, 1𝑟∈ℕ∶ℎ_𝑟𝑘∈𝐼_𝑟∶𝑥_𝑘∉𝒩_𝑥(𝜖)≥𝛿∈ℐ. (26) In this case, we write 𝑥_𝑛→𝑥(𝑆^𝜃_PM(ℐ)). The class of all strong ℐ-lacunary statistically convergent sequences is simply denoted by (𝑆^𝜃_PM(ℐ)).

Theorem 31. In a PM space 𝑋, the strong ℐ-lacunary statistical limit of a sequence is unique.

Proof. It is similar to the proof of Theorem 22, and therefore it has been omitted.

Theorem 32. For a sequence {𝑥_𝑛}_𝑛∈ℕ in a PM space 𝑋, the following conditions are equivalent:(1){𝑥_𝑛}_𝑛∈ℕ is strong ℐ-lacunary statistically convergent to 𝑥;(2)for all 𝜖>0, 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖∈ℐ. (27)

Proof. (1)⇒(2): First suppose that 𝑥_𝑛→𝑥(𝑆^𝜃_PM(ℐ)). Since 𝑑_𝐿(𝐹_𝑥𝑘𝑥,𝜀₀)≤1 for all 𝑘∈ℕ then for 𝜖>0, we have 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀=1ℎ_{𝑟𝑘∈𝐼𝑟𝑑𝐿𝐹𝑥𝑘𝑙,𝜀0≥𝜖/2}𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀+1ℎ_{𝑟𝑘∈𝐼𝑟𝑑𝐿𝐹𝑥𝑘𝑥,𝜀0<𝜖/2}𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≤1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖2+𝜖2. (28) Consequently, we have 1𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑑_𝑙𝐹_𝑥𝑘𝑥,𝜀₀⊆1≥𝜖𝑟∈ℕ∶ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖≥𝜀2∈ℐ. (29)(2)⇒(1): Next suppose that condition (2) holds. Now, for 𝜖>0, we have _{𝑘∈𝐼𝑟}𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥_{𝑘∈𝐼𝑟𝑑𝐿𝐹𝑥𝑘𝑥,𝜀0≥𝜖}𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖𝑘∈𝐼_𝑟∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖. (30) Therefore, 1𝜖ℎ_{𝑟𝑘∈𝐼𝑟}𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖. (31) Then, for any 𝛿>0, we have 1𝑟∈ℕ∶ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀⊆1≥𝜖≥𝛿𝑟∈ℕ∶ℎ_{𝑟𝑘∈𝐼𝑟}𝑑_𝐿𝐹_𝑥𝑘𝑙,𝜀₀≥𝜀𝛿∈ℐ. (32) Therefore, 𝑥_𝑛→𝑥(𝑆^𝜃_PM(ℐ)). This completes the proof.

Theorem 33. For any lacunary sequence 𝜃={𝑘_𝑟}, strong ℐ-statistical convergence in a PM space implies strong ℐ-lacunary statistical convergence if and only if liminf_𝑟→∞𝑞_𝑟>1.

Proof. Suppose first that liminf_𝑟→∞𝑞_𝑟>1. Then, there exists a 𝛼>0 such that 𝑞_𝑟≥1+𝛼 for sufficiently large 𝑟. This implies that ℎ_𝑟/𝑘_𝑟≥𝛼/(1+𝛼). Since 𝑥_𝑘→𝑥(𝑆_PM(ℐ)), for every 𝜖>0 and for sufficiently large 𝑟, we have 1𝑘_𝑟𝑘≤𝑘_𝑟∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥1≥𝜖𝑘_𝑟𝑘∈𝐼_𝑟∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝛼≥𝜖⋅11+𝛼ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖. (33) Then, for any 𝛿>0, we have 1𝑟∈ℕ∶ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀⊆1≥𝜖≥𝛿𝑟∈ℕ∶𝑘_𝑟𝑘≤𝑘_𝑟∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖≥𝛿𝛼1+𝛼∈ℐ. (34) This proves the result.

Let ℐ be an admissible ideal of ℕ, and let 𝑓∶ℕ→ℕ be any function. Define ℐ^𝑓={𝐴⊆ℕ∶𝑓(𝐴)∈ℐ}. We can easily show that ℐ^𝑓 is also an ideal of ℕ. First, we note that ∅=𝑓(∅)∈ℐ and so ∅∈ℐ^𝑓. If 𝐴,𝐵∈ℐ^𝑓, then 𝑓(𝐴∪𝐵)=𝑓(𝐴)∪𝑓(𝐵)∈ℐ which implies that 𝐴∪𝐵∈ℐ^𝑓. Also if 𝐵∈ℐ^𝑓 and 𝐴⊆𝐵, then 𝑓(𝐴)⊆𝑓(𝐵) and 𝑓(𝐵)∈ℐ which in turn implies that 𝐴∈ℐ^𝑓. Therefore, ℐ^𝑓 is an ideal of ℕ.

Theorem 34. Let 𝛽={𝑙_𝑟} be a lacunary refinement of the lacunary sequence 𝜃={𝑘_𝑟}. Let 𝐼_𝑟=(𝑘_𝑟−1,𝑘_𝑟] and 𝐽_𝑟=(𝑙_𝑟−1,𝑙_𝑟], 𝑟=1,2,…. If there exists a 𝛿>0 such that 𝐽_𝑗𝐼_𝑖≥𝛿𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝐽_𝑗⊆𝐼_𝑖,𝑖=𝑓𝑗𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑓∶ℕ→ℕ, (35) then 𝑥_𝑘→𝑥(𝑆^𝜃_𝑃𝑀(ℐ)) implies that 𝑥_𝑘→𝑥(𝑆^𝛽_𝑃𝑀(ℐ^𝑓)).

Proof. Let 𝑥_𝑘→𝑥(𝑆^𝜃_PM(ℐ)). Therefore, 1𝑟∈ℕ∶𝐼_𝑟𝑘∈𝐼_𝑟∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖≥𝛿∈ℐ. (36) For 𝜖>0and for every 𝐽_𝑗, we can find 𝐼_𝑖such that 𝐽_𝑗⊆𝐼_𝑖. Thus, we obtain 1𝐽_𝑗𝑘∈𝐽_𝑗∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≤1≥𝜖𝛿1𝐼_𝑖𝑘∈𝐼_𝑖∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖. (37) For 𝜂>0, we have 1𝐼_𝑖𝑘∈𝐼_𝑖∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝛿≥𝜖𝐽_𝑗𝑘∈𝐽_𝑗∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖≥𝜂𝛿. (38) Since 𝑥_𝑘→𝑥(𝑆^𝜃_PM(ℐ)), there exists some 𝑗∈ℕ such that 𝐴=𝑓𝑗1∈ℕ∶𝐽_𝑗𝑘∈𝐽_𝑗∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖≥𝜂∈ℐ. (39) Therefore, 1𝑗∈ℕ∶𝐽_𝑗𝑘∈𝐽_𝑗∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖≥𝜂∈ℐ^𝑓. (40) This completes the proof.

5. Strong ℐ-𝜆-Statistical Convergence in PM Space

In this section, we introduce the concepts of strong ℐ-𝜆-statistical convergence and [𝑉,𝜆](ℐ)-summability in a PM space.

Let 𝜆={𝜆_𝑛}_𝑛∈ℕ be a nondecreasing sequence of positive numbers tending to ∞ such that 𝜆_𝑛+1≤𝜆_𝑛+1, 𝜆₁=1. The collection of all such sequences 𝜆 is denoted by 𝒟. The generalised de la Vallée-Poussin mean is defined for the sequence {𝑥_𝑛}_𝑛∈ℕ of reals by 𝑡_𝑛(𝑥)=1𝜆_{𝑛𝑘∈𝑄𝑛}𝑥_𝑘, (41) where 𝑄_𝑛=[𝑛−𝜆_𝑛+1,𝑛]. A sequence {𝑥_𝑛}_𝑛∈ℕ of reals is said to be [𝑉,𝜆](ℐ) summable to 𝑥 if ℐ-lim 𝑡_𝑛(𝑥)=𝑥, that is, for 𝜖>0, {𝑛∈ℕ∶|𝑡_𝑛(𝑥)−𝑥|≥𝜖}∈ℐ [13, 31].

We now introduce the main definitions for this section.

Definition 35. Any sequence {𝑥_𝑛}_𝑛∈ℕ in a PM space 𝑋 is said to be strong ℐ-𝜆-statistically convergent to 𝑥 if for every 𝜖>0 and 𝛿>0, 1𝑛∈ℕ∶𝜆_𝑛𝑘∈𝑄_𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖≥𝛿∈ℐ. (42) In this case, we write 𝑥_𝑛→𝑥(𝑆^𝜆_PM(ℐ)). The collection of all such sequences is simply denoted by (𝑆^𝜆_PM(ℐ)).

Definition 36. Any sequence {𝑥_𝑛}_𝑛∈ℕ in a PM space 𝑋 is said to be [𝑉,𝜆](ℐ)-summable to 𝑥 in 𝑋 if for every 𝛿>0, 1𝑛∈ℕ∶𝜆_{𝑛𝑘∈𝑄𝑛}𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝛿∈ℐ. (43) In this case, we write 𝑥_𝑛→𝑥[𝑉,𝜆]_PM(ℐ). The collection of all such sequences is simply denoted by [𝑉,𝜆]_PM(ℐ).

Theorem 37. For any sequence {𝑥_𝑛}_𝑛∈ℕ in a PM space 𝑋, the following conditions are equivalent.(1)The sequence {𝑥_𝑛}_𝑛∈ℕ is strong ℐ-𝜆-statistically convergent to 𝑥 in 𝑋.(2)The sequence {𝑥_𝑛}_𝑛∈ℕ is [𝑉,𝜆](ℐ) summable to 𝑥 in 𝑋.

Proof. It is similar to the proof of Theorem 32, and therefore it has been omitted.

Example 38. Consider that 𝑋=ℝ with the usual metric 𝑑 and 𝐺(𝑥)=1−𝑒^−𝑥. Clearly, 𝐺∈Δ⁺. We define 𝔉∶𝑋×𝑋→Δ⁺ by 𝔉(𝑥,𝑦)(𝑡)=𝐹_𝑥𝑦(𝑡)=𝐺(𝑡/𝑑(𝑥,𝑦))=1−𝑒^{−𝑡/|𝑥−𝑦|} for all 𝑥,𝑦∈𝑋. Let us define 𝐺(𝑥/0)=𝐺(∞)=1 for 𝑥>0 and 𝐺(0/0)=𝐺(0)=0. Clearly, (𝑋,𝔉,𝜏) is a PM space where 𝜏 is the continuous triangle function. Recall that for any 𝑡>0, 𝐹(𝑡)>1−𝑡 if and only if 𝑑_𝐿(𝐹,𝜀₀)<𝑡 [24].
Let ℐ be an admissible ideal of ℕ and 𝜆_𝑛∈𝒟. Take a fixed 𝐴∈ℐ and define a sequence {𝑥_𝑘}_𝑘∈ℕ in 𝑋 by 𝑥_𝑘=𝑘𝑢,for𝑛−𝜆_𝑛+1≤𝑘≤𝑛,𝑛∉𝐴𝑘𝑢,for𝑛−𝜆_𝑛+1≤𝑘≤𝑛,𝑛∈𝐴, (44) where 𝑢 is a fixed element in 𝑋 with |𝑢|=1. Clearly, 𝐹_𝑥𝑘0(𝑡)=1−𝑒^{−𝑡/|𝑥k|}=1−𝑒^−𝑡/𝑘. Now, for every 𝜖>0(0<𝜖<1), there exists a 𝑘₀∈ℕ such that 𝐹_𝑥𝑘0(𝜀)≤1−𝜖 for all 𝑘≥𝑘₀. Therefore, for every 𝛿>0, 1𝜆_𝑛𝑘∈𝑄_𝑛∶𝑑_𝐿𝐹_𝑥𝑘0,𝜀₀≥𝜖=𝜆_𝑛𝜆_𝑛⟶0, (45) as 𝑛→∞ and 𝑛∉𝐴. Consequently, for every 𝛿>0, 1𝑛∈ℕ∶𝜆_𝑛𝑘∈𝑄_𝑛∶𝑑_𝐿𝐹_𝑥𝑘0,𝜀₀{},≥𝜖≥𝛿⊆𝐴∪1,2,…,𝑚 (46) for some 𝑚∈ℕ. Since ℐ is admissible, the expression on the right hand side belongs to ℐ. It follows that 𝑥_𝑘→0(𝑆^𝜆_PM(ℐ)).

Theorem 39. If liminf_𝑛→∞(𝜆_𝑛/𝑛)>0, then 𝑆_𝑃𝑀(ℐ)⊆𝑆^𝜆_𝑃𝑀(ℐ).

Proof. For 𝜖>0, we have 1𝑛𝑘≤𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥1≥𝜖𝑛𝑘∈𝑄_𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀=𝜆≥𝜖_𝑛𝑛1𝜆_𝑛𝑘∈𝑄_𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖. (47) If liminf_𝑛→∞(𝜆_𝑛/𝑛)=𝑎>0, then from definition {𝑛∈ℕ∶𝜆_𝑛/𝑛<𝑎/2} is finite. Thus, for 𝛿>0, 1𝑛∈ℕ∶𝜆_𝑛𝑘∈𝑄_𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀⊆1≥𝜖≥𝛿𝑛∈ℕ∶𝑛𝑘∈𝑄_𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝑎≥𝜖2𝛿∪𝜆𝑛∈ℕ∶^𝑛𝑛<𝑎2. (48) Since ℐ is admissible, the set on the right hand side belongs to ℐ, and this completes the proof.

Theorem 40. Let 𝜆∈𝒟 be such that lim_𝑛→∞(𝜆_𝑛/𝑛)=1. Then, 𝑆^𝜆_𝑃𝑀(ℐ)⊂𝑆_𝑃𝑀(ℐ).

Proof. Let 𝛿>0 be given. Since lim_𝑛→∞(𝜆_𝑛/𝑛)=1, we can choose 𝑚∈ℕ such that |𝜆_𝑛/𝑛−1|<𝛿/2 for all𝑛≥𝑚. Now, observe that for 𝜖>0, 1𝑛𝑘≤𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀=1≥𝜖𝑛𝑘≤𝑛−𝜆_𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀+1≥𝜖𝑛𝑘∈𝑄_𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≤≥𝜖𝑛−𝜆_𝑛𝑛+1𝑛𝑘≤𝑄_𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀𝛿≥𝜖≤1−1−2+1𝑛𝑘∈𝑄_𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀=𝛿≥𝜖2+1𝑛𝑘≤𝑄_𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝜖, (49) for all 𝑛≥𝑚. Hence, 1𝑛∈ℕ∶𝑛𝑘≤𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀⊆1≥𝜖≥𝛿𝑛∈ℕ∶𝑛𝑘∈𝑄_𝑛∶𝑑_𝐿𝐹_𝑥𝑘𝑥,𝜀₀≥𝛿≥𝜖2∪{1,2,…,𝑚}. (50) If 𝑥_𝑘→𝑥(𝑆^𝜆_PM(ℐ)), then the set on the right hand side belongs to ℐ and so the left hand side also belongs to ℐ. This shows that 𝑥_𝑘→𝑥(𝑆_PM(ℐ)).

Remark 41. Consider the sequence {𝜆_𝑛}_𝑛∈ℕ where 𝜆_𝑛=1 for 𝑛=1,2,…,10 and 𝜆_𝑛=𝑛−10 for all 𝑛≥10. Construct the sequence as in Example 38. For ℐ=ℐ_𝑑 (the ideal of density zero sets in ℕ) and 𝐴={1²,2²,3²,…}, the sequence {𝑥_𝑘}_𝑘∈ℕ is an example of a sequence which is strong ℐ-statistically convergent but is not statistically convergent in a PM space.