A study of boundedness in probabilistic normed spaces(II)

It was shown in Lafuerza-Guillén, Rodríguez-Lallena and Sempi (1999)that uniform boundedness in a Serstnev PN space (V,\un,\tau,\tau^*), named boundedness in the present setting, of a subset A in V with respect to the strong topology is equivalent to the fact that the probabilistic radius R_A of A is an element of D^+.Here we extend the equivalence just mentioned to a larger class of PN spaces, namely those PN spaces that are topological vector spaces (briefly TV spaces), but are not Serstnev PN spaces. We present a characterization of those PN spaces, whether they are TV spaces or not,in which the equivalence holds. Then, a characterization of the Archimedeanmity of triangle function \tau^* of type \tau_{T,L} is given. This work is a partial solution to a problema of comparing the concepts of distributional boundedness (D-bounded in short) and that of boundedness in the sense of associated strong topology

Finite products of probabilistic normed spaces

We consider finite products of probabilistic normed spaces. As is to be expected, the dominance relation plays a central role

Probabilistic norms and convergence of random variables

We prove that probabilistic norms of suitable Probabilistic Normed spaces induce convergence in probability,L^p convergence and almost sure convergence

On ideal convergence of double sequences in probabilistic normed spaces

The notion of ideal convergence is a generalization of statistical convergence wich has been intensively investigated in last few years. For an admisible ideal f in NxN, the aim of the present paper is to introduce the concepts of f-convergence and f^*-convergence for double sequences on probabilistic normed spaces (PN spaces for short). We give some relations related to these notions and find conditions on the ideal f for which both the notions coincide. We also define f-Cauchy and f^*-Cauchy double sequences on PN spaces and show that f-convergent double sequences are f-Cauchy on these spaces. We establish example which shows that our method of convergence for double sequences on PN spaces is more general

A study of boundedness in probabilistic normed spaces

It was shown in Lafuerza-Guillén, Rodríguez- Lallena and Sempi (1999) that uniform boundedness in a Serstnev PN space (V,\un, \tau,\tau^*), (named boundeness in the present setting) of a subset A in V with respect to the strong topology is equivalent to the fact that the probabilistic radius R_A of A is an element of D^+. Here we extend the equivalence just mentioned to a larger class of PN spaces, namely those PN spaces that are topological vector spaces(briefly TV spaces), but are not Serstnev PN spaces. We present a characterization of those PN spaces, whether they are TV spaces or not, in which the equivalence holds. Then a charaterization of the Archimedeanity of triangle functions \tau^* of type \tau_{T,L} is given.This work is a partial solution to a problema of comparing the concepts of distributional boundedness (D-bounded in short) and that of boundedness in the sense of associated strong topology

Total boundedness in probabilistic normed spaces

In this paper, we study total boundedness in probabilistic normed space and we give criterion for total boundedness and D-boundedness in these spaces. Also we show that in general a totally bounded set is not D-bounded

Statistical convergence in strong topology of probabilistic normed spaces

Following the concept of statistical convergence, we define and study statistical analog concepts of convergence and Cauchy's sequence on a probabilistic normed space that is endowed with a strong topology. Some important properties of statistical convergence have been extended in this new setting

Probabilistic norms and statistical convergence of random variables

The paper extends certain stochastic convergence of sequences of <B>R<SUP>k</SUP></B> -valued random variables (namely, the convergence in probability, in L<SUP>p</SUP> and almost surely) to the context of E-valued random variables

Translation-invariant generalized topologies induced by probabilistic norms

One considers probabilistic normed spaces as defined by Alsina,Schweizer and Sklar, but with non necessarily continuous triangle functions. Such spaces are endowed with a generalized topology that is Fréchet-separated, translation-invariant and countably generated by radial and circled 0-neighborhoods. Conversely, we show that such generalized topologies are probabilistically normable