Drift and scale of a random walk
Section of Random walks and first passage, part of Stochastic processes
Drift and scale of a random walk
| Type | Section |
| Taught in | Random walks and first passage |
| Part | Stochastic processes |
Let Y_1,Y_2,\ldots be independent and identically distributed increments, and let S_n=x+\sum_{i=1}^n Y_i. If \mu=\mathbb{E} Y_1 and \sigma^2=\operatorname{Var}(Y_1) are finite, then
\mathbb{E} S_n=x+n\mu, \qquad \operatorname{Var}(S_n)=n\sigma^2.
The strong law gives (S_n-x)/n\to\mu with probability 1 when the mean exists. The central limit theorem gives, when 0<\sigma^2<\infty,
\frac{S_n-x-n\mu}{\sigma\sqrt{n}}\;\xrightarrow{d}\; \mathrm{N}(0,1).
Drift accumulates like n. Typical fluctuations are of order \sqrt{n}. Neither statement is a first-passage probability. A first-passage event depends on the whole path, not on the single variable S_n. The four modes of the arrow \xrightarrow{d}, and the conditions under which a limit of distributions controls an expectation, are on Expectation and convergence.
Used in
See also
Random walk with drift · Gambler’s ruin · Four-state random walk · Expected duration of a random walk · Solving the recurrences from scratch · Why the boundary problem is well posed · First-passage time