random walk: a standalone construct

The term random walk was coined by Karl Pearson in a brief 1905 letter to Nature, where he posed the problem of a walker who takes a series of steps, each of fixed length but in a randomly chosen direction, and asked where the walker is likely to be found afterwards. In probability theory a random walk is a mathematical path built from a succession of random steps, the simplest case being a point on the integers that moves one unit right or left at each tick with equal probability, so that its position is the running total of independent random increments. Lord Rayleigh had earlier solved a mathematically equivalent problem for the combination of sound vibrations, and George Pólya's 1921 study of walks on lattices proved the now-classic recurrence result that a walk on the one- or two-dimensional integer grid returns to its starting point with probability one, while a walk in three dimensions may wander off and never come back. The construct is a landmark because it is the discrete building block from which much of modern probability is assembled: taken to a continuous limit it yields Brownian motion and diffusion, and it supplies working models for phenomena as varied as molecular motion, population genetics, and fluctuating prices in financial markets.[1][2][3]

Unit

A stochastic process: a sequence of positions given by the running total of independent random steps

Recovery means

Not applicable

Measured by

Specified by its step distribution and analysed through quantities such as expected displacement, mean squared displacement (which grows in proportion to the number of steps), and return or recurrence probabilities