# The Central Limit Theorem for Random Weights

• Jean Jacod
• Philip Protter
Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 67)

## Abstract

This chapter presents the Central Limit Theorems associated with the Laws of Large Numbers of Chap. : 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 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 $$\sqrt {\varDelta _{n}}$$.

## Keywords

Central Limit Theorem Polynomial Growth Left Endpoint Random Weight Bibliographical Note
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

1. 1.Institut de MathématiquesUniversité Paris VI – Pierre et Marie CurieParisFrance
2. 2.Department of StatisticsColumbia UniversityNew YorkUSA