Modes of convergence of random variables and algebraic genericity

Important probabilistic problems require to find the limit of a sequence of random variables. However, this limit can be understood in different ways and various kinds of convergence can be defined. Among the many types of convergence of sequences of random variables, we can highlight, for example, that convergence in L^p -sense implies convergence in probability, which, in turn, implies convergence in distribution, besides that all these implications are strict. In this paper, the relationship between several types of convergence of sequences of random variables will be analyzed from the perspective of lineability theory.