Convergence in probability and almost surely convergence in probabilistic normed spaces

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.