A stochastic process is written {Xt}t∈T\{X_t\}_{t\in T}. Stochastic-process theory is needed because random variables at different indices usually have a specific dependence structure.

A stochastic process describes the joint relationships within a family of random variables.

1. From Marginal to Joint Distributions

1.1 Individual distributions do not determine a process

Consider two processes:

  • In the first, every Xt∼N(0,1)X_t\sim N(0,1) and X1,X2,…X_1,X_2,\dots are mutually independent.
  • In the second, one variable Z∼N(0,1)Z\sim N(0,1) is sampled and every time point is set to Xt=ZX_t=Z.

At any individual time, both processes satisfy Xt∼N(0,1)X_t\sim N(0,1), but their dynamics are entirely different. The first resamples at every step; the second remains fixed after the initial draw. Their distinction lies in how XsX_s and XtX_t vary together.

1.2 Joint distributions

For one random variable XtX_t, its distribution can be studied through P(Xt≤x)P(X_t\le x). Studying two times s,ts,t requires the joint distribution P(Xs≤x,  Xt≤y)P(X_s\le x,\;X_t\le y).

A joint distribution describes how variables occur together. Two variables can both have standard normal marginal distributions while being independent, strongly positively correlated, strongly negatively correlated, or related by a nonlinear dependence. Marginal distributions cannot distinguish these cases.

1.3 Finite-dimensional distributions

A stochastic process may contain infinitely many random variables. Analysis selects a finite set of times t1,t2,…,tnt_1,t_2,\dots,t_n and studies the joint distribution of (Xt1,Xt2,…,Xtn)(X_{t_1},X_{t_2},\dots,X_{t_n}).

These distributions are the process’s finite-dimensional distributions. If the joint distribution is known for every finite selection of time points and the distributions satisfy the required consistency conditions, the probability structure of the process is determined.

2. Statistics That Describe Dependence

2.1 The mean function

For each time point, define m(t)=E[Xt]m(t)=E[X_t].

m(t)m(t) is the process’s mean function. If m(t)=0m(t)=0, the process is centered at zero at every time. If m(t)=tm(t)=t, its mean level rises linearly over time.

The mean function describes the center at each time but does not determine relationships across times.

2.2 Variance and covariance

Var⁡(Xt)\operatorname{Var}(X_t) measures variability at one time. The linear relationship between two times is described by covariance: C(s,t)=Cov⁡(Xs,Xt)C(s,t)=\operatorname{Cov}(X_s,X_t).

When C(s,t)>0C(s,t)>0, a high XsX_s tends to accompany a high XtX_t. When C(s,t)<0C(s,t)<0, the two tend to move in opposite directions.

C(s,t)=0C(s,t)=0 only rules out linear correlation; it does not guarantee independence:

Independence implies zero correlation, but zero correlation does not necessarily imply independence.

2.3 Autocorrelation and lag

Stochastic-process analysis focuses especially on correlations within one process at different indices. When comparing XtX_t with Xt+τX_{t+\tau}, τ\tau is the lag. The standardized correlation coefficient is

ρ(s,t)=Cov⁡(Xs,Xt)Var⁡(Xs)Var⁡(Xt).\rho(s,t)= \frac{\operatorname{Cov}(X_s,X_t)} {\sqrt{\operatorname{Var}(X_s)\operatorname{Var}(X_t)}}.

If ρ(t,t+τ)\rho(t,t+\tau) rapidly approaches zero as τ\tau grows, the process has short memory. Persistent correlation indicates longer temporal dependence. These relationships matter in time-series analysis, signal processing, finance, and physical systems.

3. Special Structures and Process Types

3.1 Independent and identically distributed variables are a special case

X1,X2,…X_1,X_2,\dots may be:

  • independent and identically distributed;
  • independent with different distributions;
  • identically distributed but correlated;
  • differently distributed with complex dependence.

The i.i.d. assumption is one convenient special structure. Many important processes arise precisely from dependence across time.

A random walk, for example, satisfies Xt+1=Xt+ϵt+1X_{t+1}=X_t+\epsilon_{t+1}.

Xt+1X_{t+1} depends on XtX_t, even though the increments ϵt\epsilon_t may be i.i.d.

3.2 Structural conditions define process families

Markov processes, Poisson processes, Gaussian processes, and random walks impose different structural conditions on {Xt}\{X_t\}:

  • a Markov process specifies a conditional-dependence structure;
  • a Poisson process specifies the distribution and independence of counting increments;
  • a Gaussian process requires every finite collection of time points to have a multivariate Gaussian distribution;
  • a random walk accumulates random increments into its state.

Learning a process family largely means learning the special relationships it requires among its random variables.

4. Core Concepts

Elementary probability often studies the distribution of one random variable. Stochastic-process theory asks how a family of random variables changes jointly.

Marginal distributions provide information about individual times. Joint distributions, covariance, autocorrelation, and general dependence describe relationships across times. The next concept is whether these statistical laws remain unchanged under a shift in time: stationarity.