A key ingredient for the asymptotic normality proof, as outlined in Chapter 8, is that the normalized score vector can be expressed as a linear function of random variables ζ n which converge in distribution, cf. Assumption 8.1(g). In this chapter we present central limit theorems (CLTs) which can be used to imply this distributional convergence of ζ n in the important case where ζ n , can be expressed as a normalized sum of random variables. We give two alternative CLTs.