Convergence in fractional probability space and 0-1 Kolmogorov theorem

In this study, we define the fractional random variable. The concept of convergence in fractional probability, almost surely convergence and some related theorems and examples are studied with the purpose of expanding the fractional probability theory parallel to the classical one. It is shown that almost surely convergence in the fractional probability space does not lead to the convergence in fractional probability. And, some valuable features related to fractional probability theory such as Cauchy function in fractional probability are discussed. We proved that a fractional random variable converges in fractional probability if it is Cauchy in fractional probability. Finally, the well-known 0-1 Kolmogorov theorem is proved in a fractional probability space