Random walk with drift
Topic of Dependence along a series, part of Time series · Also in Random walks and first passage
Random walk with drift
| Type | Topic |
| Taught in | Dependence along a series · Random walks and first passage |
| Part | Time series · Stochastic processes |
A random walk whose increments have non-zero mean \mu: S_n=S_0+\sum_{i\le n}Y_i with \mathbb{E}Y_i=\mu. Its mean moves like n\mu while its spread grows like \sqrt n, so in the long run the drift dominates.
In Dependence along a series
An autoregression of order 1 writes X_t=\beta X_{t-1}+\varepsilon_t, with the innovations \varepsilon_t independent of each other. A random walk is the case \beta=1, and it wanders off. A spring temperature might have \beta a little above 1, a drift toward summer, plus a daily innovation. A winter temperature might have \beta a little below 1. The best guess of today’s temperature, before the day is observed, is often yesterday’s. A Markov chain is the same skeleton: the next state depends on the current state and a draw.
In Random walks and first passage
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
Drift and scale of a random walk · Gambler’s ruin · Four-state random walk · Wold decomposition · Autoregression and moving averages · Stationarity of a time series · White noise · Variogram