Discretization of Processes pp 273-296 | Cite as

# The Central Limit Theorem for Random Weights

## Abstract

*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 \(\sqrt {\varDelta _{n}}\).

## Keywords

Central Limit Theorem Polynomial Growth Left Endpoint Random Weight Bibliographical Note## Preview

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