Convergence in probability and almost surely convergence in probabilistic normed spaces

methods

Our purpose in this paper is researching about characteristics of convergent in probability and almost surely convergent in Šerstnev space. We prove that if two sequences of random variables are convergent in probability (almost surely), then, sum, product and scalar product of them are also convergent in probability (almost surely). Meanwhile, we will prove that each continuous function of every sequence convergent in probability sequence is convergent in probability too. Finally, we represent that for independent random variables, every almost surely convergent sequence is convergent in probability. In this paper, we conclude results in Šerstnev space are similar to probability space.

Menger introduced probabilistic metric space in 1942 [1]. The notion of probabilistic normed space was introduced by Šerstnev[2]. Alsina et al. generalized the definition of probabilistic normed space [3,4]. Lafuerza-Guillén and Sempi for probabilistic norms of probabilistic normed space induced the convergence in probability and almost surely convergence [5].

The structure of paper is as follows: the ‘Preliminaries’ section recalls some notions and known results in probabilistic metric space, probabilistic normed space and special case of probabilistic normed space which is called Šerstnev space. In the ‘Main results’ section, we prove some theorems which show convergence in probability for sum, product, scaler product and division of two sequences in Šerstnev space. We prove whether every continuous function of a sequence converges if it converges in probability. Also, we prove almost surely convergence for sum, product and scaler product in Šerstnev space. At the end, the relationship between almost surely convergence and convergence in probability is proved in Šerstnev space.

We recall some concepts from probabilistic metric space [6,7] and convergence concept. For more details, we refer the reader to Chung’s study [8]. A distance distribution function is a mapping which is nondecreasing, left continuous on and . The class of all distribution functions is denoted by . is the subset of containing all functions F which satisfy the condition . If S is a nonempty set, a mapping F from to is called a probabilistic distance on S, and is denoted by . The function is defined 0 if and 1 if .

A triangular norm (shorter t-norm) is a binary operation on the unit interval [0,1] which the following conditions are satisfied [9]:

1)

2)

3)

4)

Basic examples are t-norms (Lukasiewicz t-norm), and , defined by and

Definition 2.1

A generalized Menger space is a triple (S, F, T) where T is a t-norm, satisfying the following conditions:

1)

2)

3)

Definition 2.2

A triangle function is a binary operation on that is commutative, associative, nondecreasing in place and has as identity.

For more details about this subject, we refer the reader to Hadzic and Pap’s study [10].

Definition 2.3

A probabilistic normed space is a quadruple , where V is a real vector space, and are continuous triangle functions, and is a mapping from V into such that for every the following conditions hold:

1) ,

2)

3)

4) .

If and equality holds in (4), then is a Šerstnev probabilistic normed space [2]. In fact, in Šerstnev probabilistic normed space, for each , and instead of second condition, we have:

Now, we recall some concepts and theorems about random variables and convergence from Lafuerza-Guillén and Sempi’s work [5]. Let be a probability space. is called the linear space of equivalence classes of random variables. Suppose that be defined for all and for each by

the pair is called an equivalence normed space. In Lafuerza-Guillén et al.’s study [11], an equivalence relationship in the equivalence normed space was defined by if and only if .

Convergence in probability

We start this paragraph with a theorem from Lafuerza-Guillén and Sempi’s study [5].

Theorem 3.1

For a sequence of (equiv alence classes of) E-valued random variables , the following statements are equivalent: converges in probability to ( is the null element of S), when

the corresponding sequence of probability norms converges weakly to it means when

converges to in the strong topology of Šerstnev space

We will study the relation of two sequences of E-valued random variables in the probabilistic normed space, specially about their convergence in probability and almost surely. Note that, in probability space, we know that if two sequences of random variables are convergent in probability then the sequences also converge in probability. The same results hold for almost sure convergence. An interesting consequence in probability space is convergence in probability of all continuous functions on every convergent in probability sequence. Also, convergence almost surely implies convergence in probability.

Now, we show in the same way the consequence in the space which Lafuerza-Guillén and Sempi introduced means .

Real and complex valued random variables are examples of E-valued random variables.

Example 3.2

Let , and P is Lebesgue measure on [0,1]. Set is one if and is zero, otherwise. Therefore, , where . Since , we have where and

It means that is convergent in probability to zero. Then, is equal to one when and is equal to zero when . So, and . The probability of the above event tends to zero when . It means that .

Theorem 3.3

For two sequences of (equivalence class of) E-valued random variables and , and , if and converge in probability to , then we have the following statements:

(a) converges in probability to

(b) converges in probability to

(c) converges in probability to .

Proof

By theorem 0.4, the sequences and converge to in probability if and only if for every and . On the other hand, for each and

If , then the right-hand side of the above inequality tends to zero. Since for all and , the proof of (a) is complete. For proof of (b), we know in Šerstnev space;

proof of (c) is analogue to that of (a). In fact for every

Theorem 3.4

Let be a sequence of E-valued random variables and converges in probability to and are bounded) then the sequence converges in probability to .

Proof

Since and are bounded, then the right-hand side tends to zero when , and the proof is complete.

After showing convergence in probability for the sum of two convergent sequences, scalar product and product of two sequence, we will show that each continuous function of convergent in probability sequence is convergent in probability.

Theorem 3.5

Let be a sequence of E-valued random variable and is a continuous function from to . If converges in probability to X, then the sequence converges in probability to .

Proof

Since is continuous, the following relation is derived:

therefore,

or

but thus

Almost surely convergence

Let the family of all the sequences of (equivalence classes of) E-valued random variables. The set V is a real vector space with respect to the componentwise operations; specifically, if and are two sequences in V and if is a real number, then of s and s’ and the scalar product of and s are defined via

A mapping will be defined on V via

where and . Then, Lafuerza-Guillén and Sempi [5] proved the next theorem.

Theorem 3.6

The triple is a Šerstnev space.

If s is an element of V , which defined by of E-valued random variables such that for all , we can consider the relation of the n-shift of s, which is an element of V . They proved the next theorem, too.

Theorem 3.7

A sequence of E-valued random variable converges almost surely to , the null vector of S, if and only if, the sequence of the probabilistic norms of the n-shifts of s converges weakly to or equivalently, if and only if the sequence of the n-shifts of s converges to in the strong topology of

Like the theorem we showed for convergence in probability, we will prove for almost surely convergence.

Theorem 3.8

Let two sequences and of E-valued random variable converge a.s. to the null vector of S and , then the following statements are satisfied:(a) The sequence is converges a.s. to ,(b) the sequence is converges a.s. to ,(c) the sequence is converges a.s. to .

Proof

Let and then we have:

so that

Thus, the following relation for every and is reached;

If , then which gives (a).

As in the above proof for and ,

According to assumptions we have and the proof of (b) is complete.

Since , the proof of (c) is immediately derived.

Relation between almost surely convergence and convergence in probability

Now, let us turn to the relation between almost surely convergence and convergence in probability in this space.

Theorem 3.9

Suppose that is a sequence of E-valued independent random variable which converges almost surely to , then is convergent in probability to , too.

Proof

Suppose that converges almost surely to , so for every ,

Since are independent, then:

We know that if , then . This implies that:

Thus, , namely the series converges, and we obtain that . It means that . This proves the result.

We proved convergent in probability and almost surely convergent under algebraic operations on Šerstnev space are closed. In addition, every continuous function of each sequence convergent in probability sequence is convergent in probability. Also, if a sequence of independent random variables is almost surely convergent, then it is convergent in probability. There are some interesting problems that we have not solved in Šerstnev space, for example, proof of above theorems about another type of convergence like L^p.

The authors declare that they have no competing interests.

PA defined the definitions and wrote the introduction, preliminaries and abstract. AB proved the theorems. AB has approved the final manuscript. Also PA has verified the final manuscript. Both authors read and approved the final manuscript.

Authors thank the referee for good recommendations.

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