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.
Keywords:Probabilistic normed space; Convergence in probability; Almost surely convergence; t4ht@.Serstnev space
Menger introduced probabilistic metric space in 1942 . The notion of probabilistic normed space was introduced by Šerstnev. 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 .
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 . 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 :
A generalized Menger space is a triple (S, F, T) where T is a t-norm, satisfying the following conditions:
For more details about this subject, we refer the reader to Hadzic and Pap’s study .
If and equality holds in (4), then is a Šerstnev probabilistic normed space . In fact, in Šerstnev probabilistic normed space, for each , and instead of second condition, we have:
Main results and discussion
Now, we recall some concepts and theorems about random variables and convergence from Lafuerza-Guillén and Sempi’s work . 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 , 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 .
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.
Real and complex valued random variables are examples of E-valued random variables.
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.
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
where and . Then, Lafuerza-Guillén and Sempi  proved the next theorem.
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.
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.
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 .
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.
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 Lp.
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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