Take a continuous function $f:[-1,1]\to\mathbb{R}$ and a sequence of independent random variables $X_1,X_2,\ldots$ uniformly distributed in $[-1,1]$.
Define $Y_n=\max\{f(X_1),f(X_2),\ldots,f(X_n)\}$. Then almost surely (that's a joke, I'm unaware if this is actually true) there is a theorem which would give us almost sure convergence $Y_n\to \max_{x\in[-1,1]}f(x)$. This would model the process of finding the maximum value of $f$ by evaluating it at random points and keeping the largest value found.
- Is there a better sequence of $X_k$ which gives faster convergence for "most functions"? Like say the $X_k$ are supported near Chebyshev nodes somehow (wild guess)? Maybe we drop the independence assumption in some way and adapt our $X_k$ so that we disfavor testing points near points we have already tested.
- Are there known estimates for rates of convergence of something like this? Has someone already looked into this?