First Extension: Random Weights

Abstract

In this chapter a Law of Large Numbers is proved for an extension of the functionals V ⁿ(f,X) and V′ⁿ(f,X). For the first one, the summands $f(X_{i \varDelta _{n}}-X_{(i-1) \varDelta _{n}})$ are replaced by $F(\omega ,(i-1) \varDelta _{n},X_{i \varDelta _{n}}-X_{(i-1) \varDelta _{n}})$ for a function F on Ω×ℝ₊×ℝ^d, where d is the dimension of X, and likewise for the functional V′ⁿ(f,X). The results are perhaps obvious generalizations of those of Chap. 3, the main difficulty being to establish the assumptions on F ensuring the convergence.

The motivation for this is to solve parametric statistical problems for discretely observed processes: in the last section one considers the solution of a (continuous) stochastic differential equation whose diffusion coefficient depends smoothly on a parameter θ. Then, on the basis of discrete observation along a regular grid, one shows how the previous Laws of Large Numbers can be put to use for constructing estimators of θ which are consistent, as the discretization mesh goes to 0.