In Section 7.2, we discussed three different concepts of convergence for sequences of random variables – almost sure convergence, convergence in \(L_p\) and convergence in probability – and studied their properties. These culminated in many important results that include the Laws of Large Numbers and the Three Series Theorem. In this chapter, we are going to introduce and study another important and natural notion of convergence for sequences of random variables. In the sequel, we are going to use the standard notation \(F_Y\) to denote the distribution function of a real random variable Y. The notion of convergence that we are going to introduce, is essentially based on pointwise convergence of the corresponding sequence of distribution functions, except that it is not demanded at all points. The next section starts with the precise definition.