68 Modeling and In verse Problems in the Presence of Uncertainty
θ. In [7], the almost sure convergence of J
N
OLS
(θ; Y ) to J
0
(θ) is demonstrated
constructively, that is, by building a set A ∈ F (which does not depend on
θ) with Prob{A} = 1 such that J
N
OLS
(θ; Y ) → J
0
(θ) for each ω ∈ A and for
each θ ∈ Ω
θ
. T his construction relie s upon the separability of the parameter
space Ω
θ
(assumption (A2)) as well as the compactness of the space [t
0
, t
f
]
(assumption (A3)). The alternative a pproach of Gallant uses a consequence
of the Glivenko–Cantelli theorem [25, p. 158] to demonstrate a uniform (with
respect to θ) strong law of large numbers. The proof relies upon the dominated
conve rgence theorem [32, p. 246], and hence the dominating function b. As
a res ...