Ideal Convergence of Random Variables

Fast [1] and Steinhaus [2] independently introduced the notion of statistical convergence for sequences of real numbers, which is a generalization of the concept of convergence. The concept of statistical convergence is a very useful functional tool for studying the convergence problems of numerical sequences through the concept of density. Later on, several generalizations and applications of this concept have been presented by various authors (see [3–10] and references therein). Kostyrko et al. [11] presented a generalization of the concept of statistical convergence with the help of ideal I of subsets of the set of natural numbers N and further studied in [12–16]. Menger [17] presented an interesting and important generalization of the concept of a metric space under the name of statistical metric space by using probability distribution function, which is now called a probabilistic metric space. By using the concept of Menger, Šerstnev [18] introduced the concept of probabilistic normed space (for random normed space, see [19]), which is an important generalization of deterministic results of linear normed spaces. Afterward, Alsina et al. [20] presented a new definition of probabilistic normed space which includes the definition of Šerstnev as a special case. The concept of ideal convergence for single and double sequence of real numbers in probabilistic normed space was introduced and studied by Mursaleen and Mohiuddine [21, 22]. In the recent past, Mursaleen and Alotaibi [23] and Mohiuddine et al. [24] studied the notion of ideal convergence for single and double sequences in random 2normed spaces, respectively. For more detail and related concept, we refer to [25–33] and references therein.


Introduction
Fast [1] and Steinhaus [2] independently introduced the notion of statistical convergence for sequences of real numbers, which is a generalization of the concept of convergence.The concept of statistical convergence is a very useful functional tool for studying the convergence problems of numerical sequences through the concept of density.Later on, several generalizations and applications of this concept have been presented by various authors (see [3][4][5][6][7][8][9][10] and references therein).Kostyrko et al. [11] presented a generalization of the concept of statistical convergence with the help of ideal  of subsets of the set of natural numbers N and further studied in [12][13][14][15][16].
Menger [17] presented an interesting and important generalization of the concept of a metric space under the name of statistical metric space by using probability distribution function, which is now called a probabilistic metric space.By using the concept of Menger, Šerstnev [18] introduced the concept of probabilistic normed space (for random normed space, see [19]), which is an important generalization of deterministic results of linear normed spaces.Afterward, Alsina et al. [20] presented a new definition of probabilistic normed space which includes the definition of Šerstnev as a special case.
The concept of ideal convergence for single and double sequence of real numbers in probabilistic normed space was introduced and studied by Mursaleen and Mohiuddine [21,22].In the recent past, Mursaleen and Alotaibi [23] and Mohiuddine et al. [24] studied the notion of ideal convergence for single and double sequences in random 2normed spaces, respectively.For more detail and related concept, we refer to [25][26][27][28][29][30][31][32][33] and references therein.

Basic Definitions and Notations
The notion of statistical convergence depends on the density (asymptotic or natural) of subsets of N. A subset  of N is said to have natural density () if A sequence  = (  ) is said to be statistically convergent [1] to ℓ if for every  > 0 In this case, we write  − lim  = ℓ or   → ℓ(), and  denotes the set of all statistically convergent sequences.An ideal is defined as a hereditary and additive family of subsets of a nonempty arbitrary set ; here, in our study, it suffices to take  as a family of subsets of N, positive integers; that is,  ⊂ 2 N , such that  ∈ ,  ∪  ∈  for each ,  ∈ , and each subset of an element of  is an element of .A nonempty family of sets F ⊂ 2  [11]).Recall that a sequence  = (  ) of points in R is said to be -convergent to a real number ℓ if { ∈ N : |  − ℓ| ≥ } ∈  for every  > 0 (see [11]).In this case, we write  − lim   = ℓ.Now, we recall some notations and basic definitions that we are going to use in this paper.
We use the notion and terminology of [34].Thus, Δ + is the space of probability distribution functions  that are left continuous on R + = (0, +∞), (0) = 0, and (+∞) = 1.The space Δ + is partially ordered by the usual pointwise ordering of functions and has both a maximal element  0 and a minimal element  ∞ ; these are given, respectively, by There is a natural topology on Δ + that is induced by the modified Lévy metric   (see, [34,35]); that is, for all ,  ∈ Δ + and ℎ ∈ (0, 1], where [, ; ℎ] denote the condition Convergence with respect to this metric is equivalent to weak convergence of distribution functions, that is (  ) in Δ + converges weakly to  in Δ + (written as     → ) if and only if (  ()) converges to () at every point of continuity of the limit function .Consequently, we have Moreover, the metric space (Δ + ,   ) is compact.
Particular and relevant triangle functions are the functions   ,   * and those of the form Π  which, for any continuous -norm  and any  > 0, are given by Definition 3. A probabilistic normed space (or briefly, PN space) is a quadruple (, ], ,  * ), where  is a real linear space,  and  * are continuous triangle functions such that  ≤  * , and the mapping ] :  → Δ + called the probabilistic norm, for all  and  in , satisfies the following conditions: (PN1) ]  =  0 if and only if  =  ( is the null vector in ); If a PN space (, ], ,  * ) satisfies the following condition: ( Š) for all  ∈ , for all  ∈ R \ {0}, for all  > 0, ]  () = ]  (/||), then it is called a Šerstnev space; the condition ( Š) implies that the best-possible selection for  * is  * =   , which satisfies a stricter version of (PN4); namely, A Šerstnev space is denoted by (, ], ), since the role of  * is placed by a fixed triangle function   , which satisfies (PN2).
A PN space  is endowed with the strong topology (briefly S-topology) generated by the strong neighborhood system {N  () :  > 0}, where determines a first countable and Hausdorff topology on  (see [34]), and it is metrizable.
The following lemma is an immediate consequence of the definition of neighborhood of zero and (7).Lemma 4. In a PN space (, ], ,  * ), for each  ∈ , one has A sequence (  ) of elements in  converges to , the null element of , in the strong topology (briefly S-topology) (written   → ) if and only if lim That is, for every  > 0, there is an integer  = () ∈ N such that   (]   , ) <  for all  ≥ , where   is defined in (4).In terms of neighborhood, we have   →  provided that for any  > 0 there is an () ∈ N such that   ∈ N  () (i.e., ]   () > 1−) whenever  ≥ .In this case, we write   S  →  or S − lim    = .Thus, the S-topology can be completely specified by means of S-convergence of sequences.
A sequence (  ) is said to be S-Cauchy if for any  > 0, there exists an integer () ∈ N such that   −   ∈ N  () whenever ,  ≥ ().
Example 9. Let  0 =  0 (Ω, A, ), the linear space of (equivalence classes of) random variable  : Ω → R. Let ] :  → Δ + be defined, for every  ∈  0 and for every  ∈ R, by Then, the couple ( 0 , ]) is an EN space.It is a PN space under the triangle function   and   (see [34]).

Ideal Convergence of Random Variables
Throughout the paper, we denote  as an admissible ideal of subsets of N, unless otherwise stated.In this section, we begin with the definition of ideal convergence of probability distribution functions.Definition 10.Let  ⊂ 2 N , and let (Δ + ,   ) be a Lévy metric space.A sequence (  ) in Δ + is said to be -convergent (weakly) to  ∈ Δ + if and only if for every  > 0, the set or In this case, we write   W   →  or W − lim   = .By ( 7) and ( 19), the following lemma can be easily verified.
Lemma 11.Let (Δ + ,   ) be a Lévy metric space and (  ) a sequence in Δ + .Then, for every  > 0, the following statements are equivalent: Definition 12. Let (, ], ,  * ) be a PN space.A sequence (  ) in  is said to be -convergent to  in the strong topology (or strong-I-convergent) if and only if for every  > 0, the set or In this case, we write   S   →  or S − lim   = , where  is called the S-limit of (  ).In terms of neighborhoods, we have The following lemma is an immediate consequence of the above definition.Lemma 13.Let (, ], ,  * ) be a PN space and (  ) a sequence in .Then, for every  > 0, the following statements are equivalent: Theorem 14.Let (, ], ,  * ) be a PN space, and if a sequence (  ) in  is S-convergent, then S − lim   is unique.
The next theorem gives the algebraic characterization of S-convergence in PN space.
Nis a filter on N if and only if  ∉ F,  ∩  ∈ F for each ,  ∈ F, and any superset of an element of F is in F.An ideal  is called nontrivial if  ̸ =  and N ∉ .Clearly,  is a nontrivial ideal if and only if F = F() = {N −  :  ∈ } is a filter in N, called the filter associated with the ideal .A nontrivial ideal  is called admissible if and only if {{} :  ∈ N} ⊂ .A nontrivial ideal  is maximal if there cannot exist any nontrivial ideal  ̸ =  containing  as a subset.Further details on ideals can be found in Kostyrko et al. (see