Define what the system can be.
States may be discrete categories, counts, continuous measurements or high-dimensional configurations.
Side 122
Stochastic Processes
Random variables extended through time: systems whose next state is uncertain but whose patterns can still be modeled through transition structure, rates and distributions.
state→transition→time→distribution→long-run behavior
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A stochastic process is a collection of random variables indexed by time or another dimension.
The model must specify the state space, index set and dependence linking observations across the index.
Define how the process is ordered.
Time can be discrete or continuous, and other ordered dimensions are possible.
One realization traces one possible history.
Observed data are usually one or a few paths drawn from an underlying process.
Successive values are often not independent.
Autocorrelation and transition structure are central objects of study.
Markov models compress history into the current state.
The Markov property is a modeling assumption: conditional on the present state, additional past history does not change the next-step distribution.
Assign probabilities between states.
A transition matrix describes one-step movement in a discrete-state chain.
Repeated transitions generate longer-run behavior.
Matrix powers propagate state distributions through time.
Some chains settle into stable proportions.
Stationarity describes a distribution unchanged by the transition dynamics, not a frozen path.
Initial conditions can lose influence.
The rate of convergence depends on structure such as connectivity and transition probabilities.
Event arrivals can be modeled as random processes.
Counting processes focus on how many events occur by a given time and how gaps between events behave.
Model independent arrivals at a constant average rate.
The simple Poisson model is useful precisely because deviations from it become diagnostically meaningful.
Study the waiting time between events.
In a Poisson process these waiting times follow an exponential distribution.
Allow event intensity to vary.
Nonhomogeneous models capture predictable changes in arrival intensity over time.
Generalize beyond exponential waiting.
Renewal processes allow broader waiting-time distributions while preserving repeated-event structure.
Continuous-state processes model noisy motion and accumulation.
Diffusion models combine systematic drift with random fluctuation.
Accumulate independent or dependent steps.
Random walks form discrete foundations for many diffusion models and hitting-time questions.
Use continuous paths with Gaussian increments.
It is a canonical model for accumulated random fluctuation.
Add systematic directional movement.
Separating drift from noise helps distinguish trend from stochastic variation.
Ask when a process reaches a boundary.
Threshold-crossing times connect stochastic dynamics to reliability, finance and decision models.
Random does not mean structureless. Stochastic-process models describe regularities in how uncertainty evolves, including dependence, persistence, arrival rates and long-run distributions.