Some Probability Theory on C[0,1] and D[0,1]

We have seen in Chapter II that to prove asymptotic normality by von Mises’ method it is necessary to show that a statistical functional is differentiable and that the remainder term of its von Mises expansion satisfies the convergence condition (2.7). In this chapter we show that statistical functionals induce functionals on the space D[0,1] of functions on [0,1] with at most discontinuities of the first kind, and that problems of differentiability and convergence can be considered in this setting. Both the differentiability of the functional and the convergence of the remainder depend on the choice of topology on the domain of the functional. A stronger topology will allow more functionals to be differentiable, but will interfere with the convergence of the remainder. We shall use the uniform topology on D[0,1] and we shall show that with this topology the remainder term satisfies the convergence condition (2.7). This result will first be proved on C[0,1], the space of continuous functions on [0,1] with the uniform topology, and then be extended to D[0,1]. In the following chapters we shall show that wide classes of statistical functionals induce Hadamard differentiable functionals on D[0,1] with the uniform topology, and therefore with this choice of topology we are able to construct a broadly applicable von Mises calculus.