Principles of Random Walk¶
Spitzer, F. (1964). Principles of Random Walk. Graduate Texts in Mathematics.
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Primes¶
- Random Walk
- From this single law follow the walk's other structural properties: the path is self-similar (statistically the same when rescaled by \(\sqrt{n}\) in space against \(n\) in time), it is the discrete substrate of diffusion (its scaling limit is Brownian motion and its density obeys the diffusion equation), and in low dimensions it is recurrent (certain to return to the origin) while in high dimensions it is transient (it wanders off and never comes back).
This sourceAuthoritative monograph on random walks, including self-similarity, the diffusion limit, and the recurrence/transience dichotomy across dimensions.
- From this single law follow the walk's other structural properties: the path is self-similar (statistically the same when rescaled by \(\sqrt{n}\) in space against \(n\) in time), it is the discrete substrate of diffusion (its scaling limit is Brownian motion and its density obeys the diffusion equation), and in low dimensions it is recurrent (certain to return to the origin) while in high dimensions it is transient (it wanders off and never comes back).
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