An Invariance Principle for Certain Probability Limit Theorems¶
Donsker, M. D. (1951). An Invariance Principle for Certain Probability Limit Theorems.
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Primes¶
- Random Walk
- The central limit theorem gives the full continuum picture: \(S_n / \sqrt{n}\) converges to a standard normal, and the rescaled path \(S_{\lfloor nt \rfloor} / \sqrt{n}\) converges to Brownian motion (Donsker's theorem) — the diffusion limit and self-similarity invariant, with the probability density of the walk satisfying the discrete heat equation whose continuum limit is the diffusion equation \(\partial_t u = \tfrac12 \partial_{xx} u\).
This sourceProves the functional central limit theorem: the rescaled random-walk path converges to Brownian motion.
- The central limit theorem gives the full continuum picture: \(S_n / \sqrt{n}\) converges to a standard normal, and the rescaled path \(S_{\lfloor nt \rfloor} / \sqrt{n}\) converges to Brownian motion (Donsker's theorem) — the diffusion limit and self-similarity invariant, with the probability density of the walk satisfying the discrete heat equation whose continuum limit is the diffusion equation \(\partial_t u = \tfrac12 \partial_{xx} u\).
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