Editing Wiener process (section)

==Wiener process as a limit of random walk==
Let <math>\xi_1, \xi_2, \ldots</math> be [[Independent and identically distributed random variables|i.i.d.]] random variables with mean 0 and variance 1. For each ''n'', define a continuous time stochastic process
<math display="block">W_n(t)=\frac{1}{\sqrt{n}}\sum\limits_{1\leq k\leq\lfloor nt\rfloor}\xi_k, \qquad t \in [0,1].</math>
This is a random step function. Increments of <math>W_n</math> are independent because the <math>\xi_k</math> are independent. For large ''n'', <math>W_n(t)-W_n(s)</math> is close to <math>N(0,t-s)</math> by the central limit theorem. [[Donsker's theorem]] asserts that as <math>n \to \infty</math>, <math>W_n</math> approaches a Wiener process, which explains the ubiquity of Brownian motion.<ref>Steven Lalley, Mathematical Finance 345 Lecture 5: Brownian Motion (2001)</ref>