Two-state trajectory

A two-state trajectory (also termed two-state time trajectory or a trajectory with two states) is a dynamical signal that fluctuates between two distinct values: ON and OFF, open and closed, ${\displaystyle +/-}$, etc. Mathematically, the signal ${\displaystyle X(t)}$ has, for every ${\displaystyle t,}$ either the value ${\displaystyle X(t)=c_{\mathrm {off} }}$ or ${\displaystyle X(t)=c_{\mathrm {on} }}$.

In most applications, the signal is stochastic; nevertheless, it can have deterministic ON-OFF components. A completely deterministic two-state trajectory is a square wave. There are many ways one can create a two-state signal, e.g. flipping a coin repeatedly.

A stochastic two-state trajectory is among the simplest stochastic processes. Extensions include: three-state trajectories, higher discrete state trajectories, and continuous trajectories in any dimension.