Research Article | Open Access
Ideal Convergence of Random Variables
The aim of this paper is to introduce and study the notion of -convergence of random variables via probabilistic norms. Furthermore, we introduce -convergence in space and establish some interesting results.
Fast  and Steinhaus  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.  presented a generalization of the concept of statistical convergence with the help of ideal of subsets of the set of natural numbers and further studied in [12–16].
Menger  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  introduced the concept of probabilistic normed space (for random normed space, see ), which is an important generalization of deterministic results of linear normed spaces. Afterward, Alsina et al.  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  and Mohiuddine et al.  studied the notion of ideal convergence for single and double sequences in random 2-normed spaces, respectively. For more detail and related concept, we refer to [25–33] and references therein.
2. Basic Definitions and Notations
The notion of statistical convergence depends on the density (asymptotic or natural) of subsets of . A subset of is said to have natural density if
A sequence is said to be statistically convergent  to if for every In this case, we write or , 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 , positive integers; that is, , such that , for each , and each subset of an element of is an element of . A nonempty family of sets is a filter on if and only if , for each , and any superset of an element of is in . An ideal is called nontrivial if and . Clearly, is a nontrivial ideal if and only if is a filter in , called the filter associated with the ideal . A nontrivial ideal is called admissible if and only if . 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 ). Recall that a sequence of points in is said to be -convergent to a real number if for every (see ). In this case, we write .
Now, we recall some notations and basic definitions that we are going to use in this paper.
We use the notion and terminology of . Thus, is the space of probability distribution functions that are left continuous on , , and . The space is partially ordered by the usual pointwise ordering of functions and has both a maximal element and a minimal element ; these are given, respectively, by There is a natural topology on that is induced by the modified Lévy metric (see, [34, 35]); that is, for all and , where denote the condition Convergence with respect to this metric is equivalent to weak convergence of distribution functions, that is in converges weakly to in (written as ) if and only if converges to at every point of continuity of the limit function . Consequently, we have
Moreover, the metric space is compact.
Definition 1. A triangular norm (or briefly, -norm) is a binary operation that satisfies the following conditions (see ):(TN1) for all ( is commutative), (TN2) for all ( is associative), (TN3) for all whenever ( is nondecreasing), (TN4) for every ( satisfies the boundary condition). is a continuous t-conorm, namely, a continuous binary operation on that is related to a continuous -norm through .
Notice that by virtue of its commutativity, any -norm is nondecreasing in each place. Some examples of -norms and its -conorms are , , and and , , and .
Definition 2. A triangle function is a binary operation on , namely, a function that is associative, commutative, and nondecreasing and which has as unit; that is, for all , one has:(1); (2); (3) whenever ;(4).
Particular and relevant triangle functions are the functions and those of the form which, for any continuous -norm and any , 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 called the probabilistic norm, for all and in , satisfies the following conditions: (PN1) if and only if ( is the null vector in ); (PN2) for all ;(PN3); (PN4) for all , .
If a PN space satisfies the following condition: (Š) for all , for all , for all , ,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 -topology) generated by the strong neighborhood system , where determines a first countable and Hausdorff topology on (see ), 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 -topology) (written ) if and only if That is, for every , there is an integer such that for all , where is defined in (4). In terms of neighborhood, we have provided that for any there is an such that (i.e., ) whenever . In this case, we write or . Thus, the -topology can be completely specified by means of -convergence of sequences.
A sequence is said to be -Cauchy if for any , there exists an integer such that whenever .
Lemma 5 (see ). For any , any , and any , there exists a such that
Lemma 6 (see ). If , then for any
Lemma 7 (see ). For any , any , and any , there is a such that
We observe that, in view of Lemma 4 and (PN3), we have the following lemma.
Lemma 8. Let be a PN space. For all ,
An important class of PN spaces is that of -normed spaces (see ). Let be a probability space, a normed space, and a linear space of -valued random variables (possibly, the entire space). For every and for every , let be defined by then is an -normed space (briefly, EN space) with the base and target .
Example 9. Let , the linear space of (equivalence classes of) random variable . Let be defined, for every and for every , by
Then, the couple is an EN space. It is a PN space under the triangle function and (see ).
3. Ideal Convergence of Random Variables
Throughout the paper, we denote as an admissible ideal of subsets of , unless otherwise stated. In this section, we begin with the definition of ideal convergence of probability distribution functions.
Definition 10. Let , and let be a Lévy metric space. A sequence in is said to be -convergent (weakly) to if and only if for every , the set or In this case, we write or .
Lemma 11. Let be a Lévy metric space and a sequence in . Then, for every , the following statements are equivalent: (i), (ii), (iii),(iv).
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 , the set
In this case, we write or , where is called the -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 , the following statements are equivalent: (i), (ii), (iii).
Theorem 14. Let be a PN space, and if a sequence in is -convergent, then is unique.
Proof. Suppose that and with . Then, for , define the following sets:
Since , using Lemma 13, we have . Also, using , we get . Let Then, for all . This implies that its complement is a nonempty set in for all . Now, if , then . Let . Then, , and the uniform continuity of implies that there exists a such that whenever . Now, let , and then . Thus, by (16), we have
Hence, . Since is arbitrary, we get , which yields ; that is, . Thus, this completes the proof.
The next theorem gives the algebraic characterization of -convergence in PN space.
Theorem 15. Let be a PN space and and two sequences in .(a)If and , then .(b)If and , then .
Proof. (a) Let , and let be a sequence in such that . Then, from (15), we have
for any . Since , we have
Thus, we have for each
This shows that .
(b) Let and be two sequences in such that and . Then, for , define the following sets: Now, we can write and hence, By uniform continuity of , we can say that for any there exists a such that whenever and , where . Now, let . Then, we can find a such that that is, whenever and , that is, and . Thus, we have for each .
Then, for each , we have Since is admissible, from (37), we have Hence, .
Similarly, we can show that .
Theorem 16. Let be a PN space, and let be a sequence in . If , then .
Proof. Let ; then for every there exists an integer such that
Therefore, the set But, with being admissible, we have . Hence, .
Theorem 17. Sequential method is regular.
Proof. The proof follows from the fact that is admissible and from Theorem 16.
Theorem 18. Let be a PN space. A sequence in is -convergent to if and only if there exists a subset such that and .
Proof. Suppose that . Then, for , we define the following set:
Since , it follows that .
Now, for , we observe that and
We show that, for , . Suppose that, for is not -convergent to . Then, there exists some such that for infinitely many terms . Let and , . Then, we have . Also, implies that , which contradicts (42) as . Hence, .
Converse part is easy and can be omitted.
4. Ideal Convergence in Probability and in Space
Let be a sequence of random variables defined on a probability space taking values in a separable normed space , where is the norm. Then, we say that a sequence converges in probability or converges in measure to if for every ,
Equivalently, for any , there is an integer such that
In this case, we write .
Now, we give the definition of ideal convergence in probability as follows.
Definition 19. A sequence of random variables is said to be -convergent in probability to , if, for every , the set
In this case, we write or .
Example 20. Let and a Lebesgue measure on . Define the sequence of random variables for as follows: For any , we have It means that is -convergent in probability to zero. That is, .
Theorem 21. Let be a sequence of (equivalence classes of) -valued random variables. Then, the following are equivalent: (i); (ii); (iii) in the Šerstnev space .
Proof. By definition, it is clear that (ii) and (iii) are equivalent, and it suffices to establish the equivalence of (i) and (ii).
Let . We note that if and only if . But . Therefore, for every , we have . By (17), implies that . By the property of -topology, we have that is,
Since , therefore .
Thus, ; hence .
Theorem 22. Let for and . If the probabilistic norm is defined by then for every , and for every .
With the help of Theorem 22, one can characterize ideal convergence in .
Theorem 23. Let be a sequence of (equivalence classes of) E-valued random variables in . Then, the following statements are equivalent.
If ,(i); (ii) the sequence of the pth moments of the probabilistic norms I-converges to 0.
If ,(iii); (iv)for every , .
Proof. (i)(ii) We note that for every . But
(iii)(iv) Suppose that , that is, , and let , then, for every , we have
This implies that
(iv) (iii) For , suppose that , and therefore which implies that .
The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
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