Stochastic Processes
A stochastic process is a collection of random variables indexed by time, describing how randomness evolves.
Randomness through time
A stochastic process is a family of random variables {X_t} indexed by time t. It models systems whose state is uncertain and changes: queue lengths, particle positions, signal noise, sensor readings. A single run of the process is called a sample path or realization.
Discrete versus continuous time
Time can advance in steps (discrete-time processes like Markov chains) or flow continuously (continuous-time processes like the Poisson process and Brownian motion). The state space can likewise be discrete or continuous, giving four broad combinations.
Key examples
- Poisson process: events arriving at a constant rate, with exponential gaps.
- Random walk: a running sum of independent steps.
- Brownian motion: the continuous limit of a random walk, with normal increments.
- Markov chains: memoryless jumps between states.
Stationarity and ergodicity
A process is stationary if its statistical properties do not change over time, and ergodic if time averages along one long path equal averages across many paths. Ergodicity is what lets a single long simulation stand in for an ensemble — a practical necessity in many computations.
Where they appear
Stochastic processes describe counting statistics in detectors (Poisson), thermal noise in electronics, and the diffusion of particles. In simulation, discrete-event models advance a system through random events drawn from these processes, and time-series methods fit process models to observed data to forecast and quantify uncertainty.