# Difference between uniform laws of large numbers and law of large numbers

How to understand the difference between uniform laws of large numbers and law of large numbers? In particular, what does the word "uniform" mean?

• Uniform LLN is a statement about sequences of random functions. LLN's are about sequences of random variables. One could view LLN as a (very) special case of ULLN, where the functions are constant. Oct 1, 2020 at 12:44

...what does the word "uniform" mean?

Uniform refers to the topology (metric, if you'd like) of uniform convergence. The uniform metric $$\|\cdot\|_{\infty}$$ between two functions $$f, g :X \rightarrow \mathbb{R}$$ is $$\| f-g \|_{\infty} = \sup_{x \in X} |f(x) - g(x)|,$$ i.e. $$f$$ and $$g$$ are close when, well, their values are uniformly close across $$x$$.

For a sequence of random functions $$f_n$$ and a random function $$f$$, a uniform LLN is a statement of the form "$$\| f_n-f \|_{\infty}$$ converges to zero (say) in probability". That is, for $$n$$ sufficiently large, $$f_n$$ is uniformly close to $$f$$ with high probability.

One example of the Uniform LLN is the Glivenko–Cantelli Theorem, i.e. convergence of empirical CDF's to the population CDF, uniformly in probability. In this case, $$X$$ is the real line---possible outcomes of a random variable. This in turn leads to the Kolmogorov-Smirnov test. (You can find some related simulation in this question).

Another context in which one encounters Uniform LLN is MLE. In this case, $$X$$ is the parameter space and likelihood functions are random functions defined on $$X$$. MLE is consistent because the likelihood functions converge uniformly in probability to a non-random function that is maximized at the true parameter, with curvature at the maximum being Fisher information.

...the difference between uniform laws of large numbers and law of large numbers?

One can view LLN as a (very) special case of ULLN, where the functions are constant, but this is perhaps not the best perspective.