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

and Applied Analysis 3 If τ is continuous then the strong neighbourhood system N determines a Hausdorff topology for X. This topology is called the strong topology forX. Definition 11. Let X be a PM space. Then, for any t > 0 the subsetU(t) ofX×X is given byU(t) = {(x, y) : F xy (t) > 1−t} and it is called the strong t-vicinity. Theorem 12 (see [24]). LetX be a PM space, and, τ be continuous. Then, for any t > 0, there exists a η > 0 such thatU(η) ∘ U(η) ⊂ U(t), whereU(η) ∘U(η) = {(x, z) : for some y, (x, y) and (y, z) ∈ U(t)}. Note 13. Under the hypothesis ofTheorem 12, we can say that for any t > 0 there is a η > 0 such that F xz (t) > 1−twhenever 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 , ε 0 ) < t, whenever d L (F xy , ε 0 ) < η and d L (F yz , ε 0 ) < 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 subsetA ofX, the strong closure ofA is denoted by κ(A) and is defined as κ (A) = {x ∈ X : for any t > 0, there is a y ∈ A such that F 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 andX is endowed with strong topology. Definition 15. LetX be a PM space. A sequence {x n } n∈N inX 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) wheneverm, n ≥ N. Next, we recall some of the basic concepts related to the theory ofI-convergence, andwe refer to [11] formore details. Definition 16. Let X be any nonempty set. Then, the family I ⊆ P(X) is called an ideal inX 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 inX if (1) 0 ∉ F, (2) A, B ∈ F imply A ∩ B ∈ F, (3) if A ∈ F and A ⊆ B then B ∈ F. IfI is an ideal in X, thenF(I) = {X \ A : A ∈ I} is a filter in X, which is called the filter associated with the ideal I. An idealI in X is called proper if and only if X ∉ I.I is called nontrivial ifI ̸ = {0}. 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 {A 1 , A 2 , . . .} belonging toI there exists a countable family of sets {B 1 , B 2 , . . .} such that A j ΔB j is a finite set for every j ∈ N and B = ⋃∞ j=1 B j ∈ I. Throughout the paperI stands for a nontrivial admissible ideal ofN, andF(I) is the filter associated with the ideal I of N. 3. Strong Iand 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 inX if for ε > 0,


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][3][4].Recall that "asymptotic density" of a set  ⊆ N is defined as 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 {  } ∈N of real is said to be statistically convergent to a real number  if for each  > 0, 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 -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 -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][13][14][15][16][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 ,  is defined as a distribution function   instead of a nonnegative real number.For a positive number ,   () is interpreted as the probability that the distance between the points  and  is less than .The theory of probabilistic metric spaces was brought to prominence by path breaking works of Schweizer et al. [19][20][21][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." S ¸enc ¸imen 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 Ilacunary statistical convergence and strong I--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.

Preliminaries
First, we recall some basic concepts related to the probabilistic metric spaces (in short PM spaces) (see [24]).
The set of all left continuous distribution functions over (−∞, ∞) is denoted by Δ.
We consider the relation "≤" on Δ defined by  ≤  if and only if () ≤ () for all  ∈ R. It can be easily verified that the relation "≤" is a partial order on Δ. Definition 2. For any  ∈ R, the unit step function at  is denoted by   and is defined to be a function in Δ given by Definition 4. The distance between  and  in Δ is denoted by   (, ) and is defined as the infimum of all numbers ℎ ∈ (0, 1] such that the inequalities hold for every  ∈ (−1/ℎ, 1/ℎ).
The set of all distance distribution functions is denoted by The function   is clearly a metric on Δ + .The metric space (Δ + ,   ) is compact and hence complete (see [29]).
Theorem 6 (see [24]).Let  ∈ Δ + be given.Then, for any  > 0, () > 1 −  if and only if   (,  0 ) < .> Note 7. Geometrically,   (,  0 ) is the abscissa of the point of intersection of the line  = 1 −  and the graph of  (if necessary we add vertical line segment at the point of discontinuity).Definition 8.A triangle function is a binary operation  on Δ + ,  : Δ + × Δ + → Δ + which is commutative, associative, and nondecreasing in each place and has  0 as identity.Definition 9. A PM space is a triplet (, F, ) where  is a nonempty set, F is a function from  ×  into Δ + , and  is a triangle function.The following conditions for a PM space are satisfied for all , ,  ∈ : F(, ) ≥ (F(, ), F(, )).
In the sequel, we shall denote F(, ) by   and its value at  by   ().Throughout this paper  shall represent the PM space (, F, ).Definition 10.Let  be a PM space.For  ∈  and  > 0, the strong -neighbourhood of  is defined as the set The collection N  = {N  () :  > 0} is called the strong neighbourhood system at , and the union N = ⋃ ∈ N  is called the strong neighbourhood system for .
By Theorem 6, we can write If  is continuous then the strong neighbourhood system N determines a Hausdorff topology for .This topology is called the strong topology for .Definition 11.Let  be a PM space.Then, for any  > 0 the subset U() of × is given by U() = {(, ) :   () > 1−} and it is called the strong -vicinity.
If  is continuous in a PM space , then the strong neighbourhood system N determines a Kuratowski closure operation.It is termed as the strong closure.For any subset  of , the strong closure of  is denoted by () and is defined as Remark 14.Throughout the rest of the paper, we always assume that in a PM space , the triangle function  is continuous and  is endowed with strong topology.Definition 15.Let  be a PM space.A sequence {  } ∈N in  is said to be strongly convergent to a point  ∈  if for any  > 0 there exists a natural number  such that   ∈ N  () whenever  ≥ .One writes   →  or lim  → ∞   = .
Similarly, a sequence {  } ∈N in  is called a strong Cauchy sequence if for any  > 0 there exists a natural number  such that (  ,   ) ∈ U() whenever ,  ≥ .
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  be any nonempty set.Then, the family Definition 17.Let  be any nonempty set.The family 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.

Strong Iand 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.
We now introduce the definition of strong I-statistical convergence in PM space.So, consider the following [12].
In this case, we write   →  ( PM (I)) and the class of all strong I-statistically convergent sequences is simply denoted by  PM (I).

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

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  = {  } such that  0 = 0 and ℎ  =   −  −1 → ∞ as  → ∞.Throughout this paper, the interval determined by  shall be denoted by   = ( −1 ,   ], and the ratio   / −1 shall be denoted by   [5].The lacunary sequence  = {  } is said to be the lacunary refinement of the lacunary sequence Definition 30.Let  = {  } be a lacunary sequence.A sequence {  } ∈N in a PM space  is said to be strong Ilacunary statistically convergent to  if for every  > 0 and  > 0, In this case, we write   → (  PM (I)).The class of all strong I-lacunary statistically convergent sequences is simply denoted by (  PM (I)).
Theorem 31.In a PM space , 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 {  } ∈N in a PM space , the following conditions are equivalent: (1) {  } ∈N is strong I-lacunary statistically convergent to ; (2) for all  > 0, Proof.
Theorem 33.For any lacunary sequence  = {  }, strong Istatistical convergence in a PM space implies strong I-lacunary statistical convergence if and only if lim inf  → ∞   > 1.
Let I be an admissible ideal of N, and let  : N → N be any function.Define I  = { ⊆ N : () ∈ I}.We can easily show that I  is also an ideal of N. First, we note that 0 = (0) ∈ I and so 0 ∈ I  .If ,  ∈ I  , then ( ∪ ) = () ∪ () ∈ I which implies that  ∪  ∈ I  .Also if  ∈ I  and  ⊆ , then () ⊆ () and () ∈ I which in turn implies that  ∈ I  .Therefore, I  is an ideal of N.
This completes the proof.

Strong I-𝜆-Statistical Convergence in PM Space
In this section, we introduce the concepts of strong I-statistical convergence and [, ](I)-summability in a PM space.
Let  = {  } ∈N be a nondecreasing sequence of positive numbers tending to ∞ such that  +1 ≤   + 1,  1 = 1.The collection of all such sequences  is denoted by D. The generalised de la Vallée-Poussin mean is defined for the sequence {  } ∈N of reals by where Proof.It is similar to the proof of Theorem 32, and therefore it has been omitted.
Let I be an admissible ideal of N and   ∈ D. Take a fixed  ∈ I and define a sequence {  } ∈N in  by where  is a fixed element in  with || = 1.Clearly,    0 () = 1 −  −/| k | = 1 −  −/ .Now, for every  > 0 (0 <  < 1), there exists a  0 ∈ N such that    0 () ≤ 1 −  for all  ≥  0 .Therefore, for every  > 0, If   → (  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   →  ( PM (I)).

)
Definition 3. A sequence {  } ∈N of distribution functions converges weakly to a distribution function , and one writes     →  if and only if the sequence {  ()} ∈N converges to () at each continuity point  of .
which is called the filter associated with the ideal I.An ideal I in  is called proper if and only if  ∉ I.I is called nontrivial if I ̸ = {0}.An ideal is called an admissible ideal if it is proper and contains {} for all  ∈ .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 { 1 ,  2 , . ..} belonging to I there exists a countable family of sets { 1 ,  2 , . ..} such that   Δ  is a finite set for every  ∈ N and  = ⋃ ∞ =1   ∈ I.
Definition 35.Any sequence {  } ∈N in a PM space  is said to be strong I--statistically convergent to  if for every  > 0 and  > 0, { ∈ N : 1        { ∈   :   (    ,  0 ) ≥ }      ≥ } ∈ I. (42) In this case, we write   →  (  PM (I)).The collection of all such sequences is simply denoted by (  PM (I)).Any sequence {  } ∈N in a PM space  is said to be [, ](I)-summable to  in  if for every  > 0, In this case, we write   → [, ] PM (I).The collection of all such sequences is simply denoted by [, ] PM (I).For any sequence {  } ∈N in a PM space , the following conditions are equivalent.The sequence {  } ∈N is strong I--statistically convergent to  in .(2) The sequence {  } ∈N is [, ](I) summable to  in .