On Cluster Points , Continuity , and Boundedness Associated with the Generalized Statistical Convergence in Probabilistic Normed Spaces

and Applied Analysis 3 Theorem 11. Let (X, η, τ, τ∗) be a probabilistic normed space and letF be the function fromX × X to Δ+ defined by F (x, y) = η (x − y) = N x−y . (7) Then, (X, η, τ, τ∗) is a probabilistic metric space (briefly PM space). Definition 12. Let X be a PN space. For x ∈ X and t > 0, the strong t-neighbourhood of p is defined as the set N p (t) = {q ∈ X : d L (N p−q , ε 0 ) < t} . (8) Since τ is continuous, strong neighbourhood systemN = {N p (t) : t > 0, p ∈ X} that determines a Hausdorff and first countable topology for X. This topology is called the strong topology forX. Remark 13. Throughout the rest of this paper, we always assume that in a PN space X the triangle function τ is continuous andX is endowed with strong topology. Definition 14. A sequence {p n } n∈N in the PN space X is said to be strongly convergent to a point p in X and one writes p n → p or lim n→∞ p n = p if for any t > 0 there exists a natural numberN such that p n ∈ N p (t) whenever n ≥ N. Definition 15. Given a nonempty set A in the PN spaceX, its probabilistic radiusR A is defined by R A (x) = { l −1 φ A (x) , if x ∈ [0,∞) ,


Introduction
The idea of convergence of real sequences had been extended to statistical convergence by Fast [1] and basic ideas were further developed in [2][3][4][5].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 reals is said to be statistically convergent to a real number  if, for each  > 0, The concepts of I and I * -convergence, two important generalizations of statistical convergence, were introduced and investigated by Kostyrko et al. [6].The ideas were based on the notion of ideal I of N. Subsequently, a lot of investigations have been done on ideal convergence (see [7][8][9][10][11][12][13][14][15][16][17] where many more references both on ideal as well as statistical convergence can be found).Very recently, ideals were used in a different way to generalize the notion of statistical convergence [18,19] and certain new and summability methods were introduced and their basic properties were investigated.More recently these ideas were extended to double sequences in [20].
On the other hand, the idea of probabilistic metric space was first introduced by Menger [21] in the name of "statistical metric space." Probabilistic normed space (briefly PN space) is a generalisation of an ordinary normed linear space.In a PN space, the norms of the vectors are represented by the distribution functions instead of nonnegative real numbers.Detailed theory of these spaces can be found in the famous book written by Schweizer and Sklar [22] and the monogram [23].One can also see the papers [22,[24][25][26][27][28][29][30][31][32][33][34][35] where the basic ideas were established.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 have very recently extended the notion of strong convergence to strong statistical convergence in probabilistic metric spaces [36] and carried out further investigations on statistical continuity and statistical -boundedness in PN spaces [37,38].These were followed by the studies of strong ideal convergence in PM and PN spaces in [10,13,39], studies of lacunary statistical

Preliminaries
First, we recall some of the basic concepts related to the theory of probabilistic metric and normed spaces (see [22,23,[31][32][33][34][35][36][37][38][39]42] for more details).The set of all left continuous distribution functions over (−∞, ∞) is denoted by Δ.One considers the relation "≤" on Δ defined by  ≤  if and only if () ≤ () for all  ∈ R. It can be easily verified that the relation "≤" is a partially 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).
In the sequel, we will denote F(, ) by   and its value at  by   ().Definition 10.A probabilistic normed space (briefly a PN space) is a quadruple (, , ,  * ), where  is a real linear space,  and  * are continuous triangle functions with  ≤  * , and  is a mapping (the probabilistic norm) from  into the space of distribution functions Δ + such that, writing   for () for all ,  in , the following conditions hold: (N1)   =  0 if and only if  = , the null vector in ; A Menger PN space under  is a PN space (, , ,  * ) in which  =   and  * =   * for some continuous -norm  and its -conorm  * .It is denoted by (, , ).
Throughout the text,  will represent the PN space (, , ,  * ).Theorem 11.Let (, , ,  * ) be a probabilistic normed space and let F be the function from  ×  to Δ + defined by Then, (, , ,  * ) is a probabilistic metric space (briefly PM space).
Definition 12. Let  be a PN space.For  ∈  and  > 0, the strong -neighbourhood of  is defined as the set Since  is continuous, strong neighbourhood system N = {N  () :  > 0,  ∈ } that determines a Hausdorff and first countable topology for .This topology is called the strong topology for .
Remark 13.Throughout the rest of this paper, we always assume that in a PN space  the triangle function  is continuous and  is endowed with strong topology.Definition 14.A sequence {  } ∈N in the PN space  is said to be strongly convergent to a point  in  and one writes   →  or lim  → ∞   =  if for any  > 0 there exists a natural number  such that   ∈ N  () whenever  ≥ .
Definition 15.Given a nonempty set  in the PN space , its probabilistic radius R  is defined by where  −1 () denotes the left limit of the function  at the point  and   () = inf{  () :  ∈ }.
Definition 16.A nonempty set  in a PN space  is said to be (1) certainly bounded if R  ( 0 ) = 1 for some  0 ∈ (0, ∞); (2) perhaps bounded if R  () < 1 for every  ∈ (0, ∞) and  −1 R  (+∞) = 1; (3) perhaps unbounded if R  ( 0 ) > 0 for some  0 ∈ (0, ∞) and  −1 R  (+∞) ∈ (0, 1); Moreover,  is said to be distributionally bounded (bounded) if either (1) or (2) holds; that is, if In the following, we now recall some of the basic concepts related to ideals.Definition 17.Let  be any nonempty set.A nonempty family (2)  ∈ I and  ⊆  imply  ∈ I. Definition 18.Let  be any nonempty set.A nonempty family Throughout the paper, I stands for a nontrivial admissible ideal of N and F(I) is the filter associated with the ideal Definition 20 (see [18]).A sequence of real numbers {  } ∈N is said to be I-statistically convergent to  if, for each  > 0 and  > 0, In this case, we write   →  ((I)).
Definition 21 (see [41]).A sequence {  } ∈N in a PM space (, F, ) is said to be strong I-statistically convergent to  if, for each  > 0 and  > 0, In this case, we write   →  ( PM (I)) and the class of all strong I-statistically convergent sequences is simply denoted by  PM (I).
Definition 22 (see [41]).A sequence {  } ∈N in a PM space (, F, ) is said to be strong I-statistically Cauchy if, for every  > 0, there exists a positive integer  = () such that, for any  > 0,

Strong I-Statistical Limit Points and Strong I-Statistical Cluster Points in Probabilistic Normed Spaces
In this section, we extend the notions of strong statistical limit points and strong statistical cluster points in PN spaces using ideals.Let (, , ,  * ) be a PN space.
Definition 23 (see [36]).Let {  } ∈N be a sequence in .We say that a point  ∈  is a strong limit point of {  } ∈N provided that there exists a subsequence of {  } ∈N that strongly converges to .We denote the set of all strong limit points of {  } ∈N by   (  ).
Definition 24.Let {  } ∈N be a sequence in  and let then we say that {   } ∈N is an I-statistical thin subsequence of {  } ∈N .If, for some  > 0, then In this sequel, we will abbreviate the subsequence and the subsequence {   } ∈N strongly converges to .We denote the set of all strong I-statistical limit points of {  } ∈N by Λ (I) (  ).
Theorem 29.For any sequence {  } ∈N in , the set Γ (I) (  ) of strong I-statistical cluster points of {  } ∈N is strongly closed.

Strong I-Statistical Continuity in Probabilistic Normed Spaces
In this section, we introduce the notion of the strong Istatistical continuity and investigate the same for a probabilistic norm, vector addition operation, and scalar multiplication.
If  is strongly I-statistically continuous at each point of a set  ⊆ , then  is said to be strongly I-statistically continuous on .
Proof.Proof of this result immediately follows from Theorems 32 and 33.
We now investigate the strong I-statistical continuity properties of scalar multiplication given by M(, ) =  for all  ∈ R and  ∈ .
Theorem 36.The mapping M is strongly I-statistically continuous in its second place; that is, for a fixed  ∈ R, scalar multiplication is a strongly I-statistically continuous mapping from  to .
However, in general, the mapping M needs not to be strongly I-statistically continuous in its first place.
It can be easily shown that   → 0 ( PM (I)) but   (    ,  0 )   0 ( PM (I)).This example shows that the mapping from R into  defined by   →  is not strongly I-statistically continuous for any fixed  ∈ ; that is, the mapping M is not strongly I-statistically continuous in its first place.
A triangle function  * is called Archimedean if  * admits no idempotents other than  0 and  ∞ .More details on Archimedean triangle function can be found in the book [22].If  * is Archimedean, then we can establish the following lemmas.
Theorem 42.Let (, , ,  * ) be a PN space such that  * is Archimedean and   ̸ =  ∞ for all  ∈ .Then, the scalar multiplication is a jointly strong I-statistically continuous mapping from R ×  endowed with the natural product topology onto .Furthermore, the mapping   : R ×  → Δ + given by   (, ) = () =   for any  ∈ R and any  ∈  is also jointly strong I-statistically continuous.
In this section, we generalize the above definition for sequences in a PN space and introduce the concept of a strongly I-statistically -bounded sequence.
Clearly, in this case, R {   :∈N} = R {}  ∈ D + .Note that a -bounded sequence is always strongly I-statistically bounded, but the converse is not generally true.

) 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 .

Theorem 45 .
A sequence {  } ∈N in the PN space  is strongly I-statistically -bounded if and only if there exists a set  = { 1 <  2 < ⋅ ⋅ ⋅ } with { ∈ N : (1/)|{ ≤  :  ∉ }| ≥ } ∈ I for any  > 0 and a distribution function  ∈ D + such that     ≥ .Proof.The proof of the theorem immediately follows from Theorem 2.1 of [29] and Definition 44.Example 46.Let us consider the simple space (R, | ⋅ |, , ), where | ⋅ | denotes the usual norm on R;  ∈ Δ + ,  ̸ =  0 ,  ∞ , 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.Triangle functions can be constructed through leftcontinuous -norms.If  is such a -norm, then is a triangle function, where  ∈ R + .If, moreover,  is continuous, then   is uniformly continuous on (Δ + ,   ).If  * is a continuous -conorm, then   * (, ) () = inf { * 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  ∈ .Definition 19.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.