State, clock, and assumptions
Section of Stochastic processes, part of Stochastic processes
State, clock, and assumptions
| Type | Section |
| Taught in | Stochastic processes |
| Part | Stochastic processes |
Name the state X_t before any calculation: what it measures, its units, and the set S it lives in. Then name the clock. A generation in a branching process is not elapsed time, and the number of arrivals at a desk is not the number of calls still waiting. Then say which increments, individuals, or waiting times are independent. Those three choices decide which theorems apply.
Two properties are about shifts of the clock.
Definition 1 (Stationary and independent increments) The process has stationary increments when the difference X_{t+h}-X_t has a distribution that depends on the length h and not on the time t. It has independent increments when increments over disjoint intervals are independent.
Neither property says that the levels X_t are identically distributed. Stationary increments fix the law of a change over an interval of length h. The level X_t can still spread out as t grows, which is what a Poisson count and a random walk both do.
Used in
See also
Stochastic process · Stationary and independent increments · What a stochastic process models · Toolkit for stochastic processes · Modelling checklist for a stochastic process · What a model can identify · Markov’s inequality · Chebyshev’s inequality