10 Definition (Uniform Convergence in Probability):Q(0;Xn) converges uniformly in probability to Qo(0)if sup(;)-Qo()P() 0∈Θ More precisely,we have that for all e>0, Poo[sup Q(0;Xn)-Qo(0)I>e]0 0∈⊙ Why isn't pointwise convergence enough?Uniform convergence guarantees that for almost all realizations,the paths in 0 are in the e-sleeve.This ensures that the maximum is close to 00.For pointwise convergence,we know that at each 0,most of the realizations are in the e-sleeve,but there is no guarantee that for another value of 0 the same set of realizations are in the sleeve. Thus,the maximum need not be near 00.10 Definition (Uniform Convergence in Probability): Q(θ; Xn) converges uniformly in probability to Q0(θ) if sup θ∈Θ |Q(θ; Xn) − Q0(θ)| P (θ0) → 0 More precisely, we have that for all > 0, Pθ0 [sup θ∈Θ |Q(θ; Xn) − Q0(θ)| > ] → 0 Why isn’t pointwise convergence enough? Uniform convergence guarantees that for almost all realizations, the paths in θ are in the -sleeve. This ensures that the maximum is close to θ0. For pointwise convergence, we know that at each θ, most of the realizations are in the -sleeve, but there is no guarantee that for another value of θ the same set of realizations are in the sleeve. Thus, the maximum need not be near θ0.