2. Weak Convergence in DE [ 0,∞)
Society for Industrial and Applied Mathematics eBooks · 1981
In this chapter we will review the theory of weak convergence for processes whose state space is a metric space (E, r) (r being the metric) and whose sample paths are right continuous and have left-hand limits. Weak convergence refers to convergence (in a sense defined below) of the probability distributions of random variables taking values in a metric space (or more general topological space). Consequently, when we talk about weak convergence for processes we are thinking of the process as a single random variable with values in a space of E-valued functions. In our case, this space is DE [ 0,∞)= {x: [ 0,∞)→E |for all t≧0 lim s→t+ x (s)=x (t)and lim s→t− x (s)exists }, i.e., the space of right continuous, E-valued functions having limits from the left.