= 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 $X_1,X_2,\ldots$ be independent identically distributed random variables with mean zero and finite variance $\sigma^2>0$. Put $S_k=\sum_{j=1}^kX_j$, 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>.<Probability: Theory and Examples (fifth edition) by Rick Durrett>[Theorem 8.1.4, Example 8.1.6, printed page 392; linearly interpolated random walk and its maximum]{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 $a>0$,
$$
\lim_{n\to\infty}\Pr\left(\max_{0\le k\le n}S_k/\sqrt n\ge a\right)=2(1-\Phi(a)),
$$
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.
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