State space
Topic of Stochastic processes, part of Stochastic processes · Also in Discrete-time Markov chains
State space
| Type | Topic |
| Taught in | Stochastic processes · Discrete-time Markov chains |
| Part | Stochastic processes |
The set S of values a stochastic process can take, such as \{0,1,2,\ldots\} for a count or a finite set of weather types. Together with the index set T (discrete or continuous time) it fixes what kind of process a model is, and a Markov model is only as good as the choice of state.
In Stochastic processes
From Stochastic processes
A stochastic process on a probability space (\Omega,\mathcal{F},\mathrm{P}), with state space S and index set T, is a family \{X_t:t\in T\} of S-valued random variables. If T=\{0,1,2,\ldots\}, time is discrete. If T=[0,\infty), time is continuous.
In Discrete-time Markov chains
Suppose tomorrow’s weather depends on today’s weather and also on whether it has already rained for three days. Two states cannot store a run of length three. A state such as (\text{today's weather},\text{current run length}) can. The state space gets larger, and the transition rule becomes local in time again. Compressing the past into a state makes the one-step calculation possible. Compressing too far drops information the next step uses.
Time homogeneity is a separate assumption. A chain can be Markov at every time with a matrix P_n that changes with n, for example a morning matrix and an evening matrix. The law is then
\mu_n=\mu_0 P_0 P_1\cdots P_{n-1},
not a power of one matrix. Solving \pi P=\pi for a single matrix does not describe that process.
Used in
See also
Stochastic process · Discrete-time Markov chain · Markov chain · Stationary and independent increments · What a stochastic process models · State, clock, and assumptions · Toolkit for stochastic processes · Modelling checklist for a stochastic process