Donsker's Invariance Principle
For independent identically distributed increments with mean zero and variance one, rescaled interpolated partial-sum paths converge in distribution to Brownian motion. The path-space statement also controls continuous functionals such as maxima.
Before opening the references
Central Limit Theorem · Brownian Motion · Convergence in Distribution · Uniform Convergence
Where to read
Start here: Probability: Theory and Examples (Version 5)
Rick Durrett · Book · English · Free full text
§8.1 Donsker's Theorem, Theorems 8.1.2 and 8.1.4 (hypotheses and path-space limit), Example 8.1.6 (maximum), and the proof of Theorem 8.1.4
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