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Probability II / Random Walks on the Integers

Simple Random Walks — Symmetric and Asymmetric

A trading desk that models tomorrow's price as "up a tick or down a tick, independently of today" has, without necessarily naming it, written down a random walk. This note makes that object precise: a particle on the integers that steps +1+1 or −1-1 at each tick, independently of its history, and asks the two questions every model built on it eventually needs answered — where is it likely to be, and does it return to where it started? The random walk is also the vault's first genuinely new object since Probability I: it is a process, a whole sequence of dependent random variables evolving in time, rather than a single random variable or a fixed vector of them. Everything in this unit — Markov chains, branching processes, Poisson processes, queues — is built by generalising some feature of the random walk defined here.
Learning objectives. By the end of this note you should be able to:
  • write the random walk SnS_n as a partial sum of i.i.d. ±1\pm 1 steps and state its mean and variance as a function of nn;
  • distinguish the symmetric (p=12p=\tfrac12) and asymmetric (p≠12p \ne \tfrac12) walks and predict their long-run qualitative behaviour;
  • verify the spatial-homogeneity, temporal-homogeneity, and Markov properties of the walk directly from the i.i.d.-steps definition;
  • connect the walk, via the CLT, to its diffusive scaling limit.

In this lesson

  1. The random walk as a sum of i.i.d. steps
  2. Three structural properties
  3. The three regimes of long-run behaviour

The full lesson, with worked examples and exercise solutions, is for members.

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