MDM Reference • Stochastic processes
Stochastic Processes: Core Notions
A stochastic process is a family of random variables indexed by a running quantity, most often time. It is the formal home of every irregular record, and its notions allow a single realised record to be reasoned about.
The index set and the state
The index set names the instants at which the process is observed; it may be discrete, as in readings taken at fixed intervals, or continuous, as in an unbroken record. The state space names the values the process may take. Fixing both is the first act of any modelling, and a change in either changes the object under study.
Mean and autocovariance
The mean of a process is a function of the index, giving the expected value at each instant. The autocovariance is a function of two instants, giving the covariance of the readings there. For a process with constant mean and an autocovariance that depends only on the separation of the two instants, the label stationary is used. Stationarity is a strong simplifying assumption, and processes that violate it require their own treatment.
Paths and ensembles
A single record is one realised path of the process. Statements about the process are statements about the whole collection of paths, of which only one is seen. The distance between the path and the process is the same distance met earlier between a sample and a population, and the same discipline is needed to cross it.
Ergodicity
A process is ergodic when an average taken along one long path settles to the same value as an average taken across the whole collection. Ergodicity is the licence that lets a single record speak for the process, and without it the record describes only itself. The condition is not automatic, and a process may be stationary yet not ergodic.
Special families
- The white process has uncorrelated readings at distinct instants.
- The random walk accumulates such readings and drifts without a fixed level.
- The autoregressive process feeds each reading back into the next, producing persistence.
- The Markov chain moves among a finite set of states with fixed transition weights.
These families are the workhorses of the subject, and each illustrates a different kind of memory.
Why the subject matters here
The wave forms studied in this compendium are paths of stochastic processes, and the spectral and correlation instruments of the neighbouring cells are properties of those processes. A reader who holds the process notion firmly will find that the earlier statistical material and the later wave material are two readings of one subject rather than two separate ones.
Why memory matters
Two processes with the same mean and the same dispersion may behave quite differently because they remember their past in different measures. A process that forgets swiftly looks ragged from instant to instant, while one that remembers slowly wanders in long sweeps. The autocovariance is the quantity that records this difference, and it is the reason two records of equal spread may carry quite different information.
A caution on stationarity
Stationarity is often assumed for convenience, and the assumption is sometimes wrong. A record that drifts in level, or whose spread grows with time, is not stationary, and the standard summaries will mislead if they are used without change. The remedy is to look at the record before assuming, and to allow a moving level or a changing spread where the record shows one.