 Probabilistic invariance principle

A probabilistic invariance principle is a limit theorem saying that a suitably rescaled random process converges to a universal limiting process. A basic instance is the functional central limit theorem: many different microscopic increment laws produce the same limiting Brownian motion after centering and variance normalization.

Let $X_1,X_2,\dots$ be independent identically distributed random variables with mean zero and finite variance ${\sigma }^2>0$. Put $S_k={\sum }_{j=1}^k{X}_j$, and interpolate the values $S_k/(\sigma \sqrt{n})$ linearly at times $k/n$. The resulting random continuous function $W_n$ converges in distribution in $C([0,1])$, with its uniform norm, to standard Brownian motion.^[ref]

This is stronger than convergence at each fixed time: it controls continuous functionals of the whole path. For a simple symmetric random walk, the maximum satisfies, for $a>0$,using the Brownian reflection principle and the standard normal distribution function $\Phi$. For $a=1$ this is about $0.3173$. The maximum here is one-sided, rather than the maximum absolute displacement. Convergence of finite-dimensional distributions alone would not establish a path-space theorem; control of path fluctuations, expressed through tightness of probability measures, is also needed.