The Central Limit Theorem for Random Weights

Abstract

This chapter presents the Central Limit Theorems associated with the Laws of Large Numbers of Chap. 7: the summands in the unnormalized functionals are now

$$F\bigl(\omega ,(i-1) \varDelta _n,X_{i \varDelta _n}-X_{(i-1) \varDelta _n}\bigr)$$

for a function F on Ω×ℝ₊×ℝ^d, where d is the dimension of X, and it is the same for the normalized functional upon dividing the increment by $\sqrt {\varDelta _{n}}$.

Sections 10.1 and 10.2 are devoted to unnormalized functionals, in two situations: first we treat the case for a “general” test function F, satisfying rather strong regularity assumptions as a function of time; and second, we treat the case for F of the form F(ω,t,x)=G(X _t (ω),x), where G is a (smooth enough) function on (ℝ^d)². The same task is performed for normalized functionals in Sect. 10.3, again in the two cases mentioned before.

Finally, in Sect. 10.4 we present an application to the estimation of a parameter θ for the solution of a (continuous) stochastic differential equation whose diffusion coefficient depends smoothly on θ, and which is observed at the discrete times iΔ _n over a finite time interval [0,T]. In particular, we show how to construct estimators which are asymptotically (mixed) normal with the optimal rate of convergence $\sqrt {\varDelta _{n}}$.