§ Vibes of Weiner Processes
- Caveat Emptor: This is totally non-rigorous, and it taken from physics / computer graphics.
- The actual formalism requires quite a lot of machinery to setup the right measure space and topology to talk about convergence of processes to produce brownian motion.
§ Information definition of weiner process / brownian motion
- (Continuity) For each time $t$, associate a random variable $W_t$ that is almost surely continuous in $t$.
- (Independent Increments) For any two times $s, t$ ( $s \leq t$, then "random increment" $W_t - W_s$ is independent of any past state $W_p$ (for all $0 \leq p \leq s$)
- (Gaussian Incremenets) Each increment $W_t - W_s ~ N(0, t - s)$. That is, it is a normal distribution with mean 0, variance $(t - s)$.
§ Simulating Weiner process: Donsker's theorem
- Consider IID sequences $X_1, \dots X_n$.
- Define a continuous function $W[n](t) \equiv 1/\sqrt{n} \sum_{i=1}^{\texttt{floor}(tn)} X_i$ for $t \in [0, 1]$.
- Donsker's theorem : As $n \to \infty$, $W_n$ converges ()