Abstract
The simple random walk is the generic example of a discrete time temporal evolution on an integer state space. In this chapter it is defined and simple combinatorics are provided in the computation of its distribution. Two possible characteristic long-time properties, (point) recurrence and transience, are identified in the course of the analysis. Recurrence is a form of “stochastic periodicity” in which the process revisits a state (or arbitrarily small neighborhood) infinitely often, while transience refers to the phenomena in which there are at most finitely many returns.
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Notes
- 1.
See BCPT p. 124, Cor. 6.15.
- 2.
See BCPT p. 125, Lemma 3.
- 3.
See BCPT p. 129.
- 4.
This exercise treats a very special case of a more elaborate contemporary theory of random walk on graphs initiated by Dvoretsky and Erdos (1951).
References
Dvoretsky A, Erdos P (1951) Some problems on random walk in space. In: Proceedings of the second Berkeley symposium on mathematical statistics and probability, pp 353–367.
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Bhattacharya, R., Waymire, E.C. (2021). The Simple Random Walk I: Associated Boundary Value Distributions, Transience, and Recurrence. In: Random Walk, Brownian Motion, and Martingales. Graduate Texts in Mathematics, vol 292. Springer, Cham. https://doi.org/10.1007/978-3-030-78939-8_2
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