Why the law of large numbers works: counting sequences
Flip a fair coin \(N\) times. By symmetry all \(2^N\) sequences are equally likely. Nothing is sampled: every bar and dot is the exact \(\mathrm{Bin}(m) = \binom{N}{m} / 2^N\), read off row \(N\) of Pascal's triangle. Count how many of them have a fraction of heads inside the window, and watch that fraction tend to one as N grows. The same counts, read through three lenses, give the law of large numbers (LLN), the central limit theorem (CLT) and large deviations.
Derivations are in the lecture notes (Sections 1, 5–7 and 10). Each lens rescales the same exact counts, which by Stirling are \[ \mathrm{Bin}(m) \simeq \frac{e^{-N I(\nu)}}{\sqrt{2\pi N \nu(1-\nu)}} , \] and its density factor must also balance the \(1/\sqrt N\) prefactor: \(N/\sqrt N = \sqrt N \to \infty\) in the frequency lens (the diverging peak \(\approx \sqrt{2N/\pi}\) of the delta), \(\sqrt N/\sqrt N = 1\) in the \(\sqrt N\) lens (the fixed peak \(1/\sqrt{\pi/2} \approx 0.80\) of \(\mathcal{N}(\tfrac12, \tfrac14)\)), and \((\ln N)/(2N) \to 0\) in the log lens, where only \(I(\nu)\) survives.