Convergence Types. Almost Sure Convergence. L p ‐Convergence. Convergence in Probability

The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and it is a very important application to statistics and stochastic processes. In this chapter, the authors will talk about a notion of convergence defined purely by the distribution of random variables. Before they introduce notions taking advantage of the structure of the probability space, they would like to recall the more traditional real analysis types of convergence. In real analysis there is a notion of convergence which is applicable to probability theory. That concept is looking at the convergence of integrals of functions instead of the convergence of the functions. They have seen that convergence almost sure (a.s.) and convergence in Lp are generally not compatible. However, they will give an integrability condition that will make all the convergence types equivalent.