quality:correction, structural:expansion
Uniform laws of large numbers

There are extensions of the law of large numbers to collections of estimators, where the convergence is uniform over the collection; thus the name uniform law of large numbers.

Suppose f (x, θ) is some function defined for θ ∈ Θ, and continuous in θ. Then for any fixed θ, the sequence { f (X, θ), f (X, θ),...} will be a sequence of independent and identically distributed random variables, such that the sample mean of this sequence converges in probability to E. This is the pointwise (in θ) convergence.

A particular example of a uniform law of large numbers states the conditions under which the convergence happens uniformly in θ. If

Then E is continuous in θ, and

sup θ ∈ Θ ‖ 1 n ∑ i = 1 n f (X i, θ) − E ‖ → P 0. {\displaystyle \sup _{\theta \in \Theta }\left\|{\frac {1}{n}}\sum _{i=1}^{n}f(X_{i},\theta)-\operatorname {E} \right\|{\overset {\mathrm {P} }{\rightarrow }}\ 0.}

This result is useful to derive consistency of a large class of estimators (see Extremum estimator).