Stochastic Processes 1: What Is a Stochastic Process?

The shortest useful definition is:

A stochastic process is a family of random variables organized by an index set.

Understanding the definition requires two further questions: why organize many random variables, and which relationships among them matter? The answer begins with basic probability.

1. Starting with Probability

1.1 Random experiments, events, and probabilities

When a six-sided die is rolled, the outcome can be $1,2,3,4,5,$ or $6$. The result is unknown before the roll, so rolling the die is a random experiment.

The set of all possible outcomes is the sample space, conventionally denoted by $\Omega$. For the die, $\Omega=\{1,2,3,4,5,6\}$. If the question is whether the roll exceeds 3, the corresponding set $A=\{4,5,6\}$ is an event.

The probability $P(A)$ measures how likely the event is. A fair die gives $P(A)=P(X>3)=3/6=1/2$.

• A random experiment is an operation with an uncertain result.
• An event is a set of outcomes of interest.
• A probability measures the likelihood of an event.

1.2 Random variables

Let $X$ denote the number shown by the die, so $X\in\{1,2,3,4,5,6\}$. Then $X$ is a random variable.

More precisely, a random variable is a function that maps outcomes in the sample space to numbers: $X:\Omega\rightarrow\mathbb{R}$, $\omega\mapsto X(\omega)$.

If one trial produces a 4, the realization can be written as $X(\omega)=4$. The random variable $X$, its realized value 4, and the event $X>3$ are three distinct objects.

1.3 Probability distributions and expectations

A probability distribution specifies the probability of each possible value. A fair die satisfies $P(X=1)=P(X=2)=\cdots=P(X=6)=\frac16$.

For a discrete random variable, the expected value is $E[X]=\sum_x xP(X=x)$.

The expected value of a fair die is 3.5. No roll can produce 3.5; the expectation is the value approached by the average of many repeated trials. Stochastic-process theory frequently studies $E[X_t]$, the mean behavior of the random variable at index $t$.

1.4 Probability and statistics

Probability theory starts from a known random model and derives the outcomes it can produce. Given a fair die, for example, probability theory computes the chance of rolling more than 3.

Statistics starts from observations and infers the underlying model. If a die produces a 6 in 400 of 1,000 rolls, statistical methods can test whether the die is fair.

Stochastic processes are built on probability theory. Applications usually provide only one or a few observed trajectories, so statistical inference is also essential.

2. From Random Variables to Stochastic Processes

2.1 From a random variable to a random vector

One die roll can be described by a random variable $X$. For 1,000 consecutive rolls, define $X_1,X_2,\dots,X_{1000}$, where $X_i$ is the result of roll $i$.

The tuple $(X_1,X_2,\dots,X_{1000})$ is a 1,000-dimensional random vector. A complete experiment might produce $(3,6,1,4,2,5,\dots,1)$, one realization of that vector.

Repeating the full experiment gives a table such as:

Experiment	$X_1$	$X_2$	$X_3$	$\cdots$	$X_{1000}$
1	3	6	1	$\cdots$	1
2	5	2	4	$\cdots$	2
3	1	3	6	$\cdots$	3

Reading down one column shows the realizations of a fixed random variable across experiments. Reading across one row shows one realization of the entire random vector. Even if the experiment is performed only once, the probability model represents every possible result.

2.2 From a random vector to a stochastic process

A finite tuple such as $(X_1,\dots,X_{1000})$ is a random vector. If the index extends indefinitely to $X_1,X_2,X_3,\dots$, or varies continuously as $X_t$, a finite-dimensional vector is no longer enough.

A stochastic process is written $\{X_t\}_{t\in T}$, where $X_t$ is a random variable, $t$ is an index, and $T$ is the index set.

• $T=\{0,1,2,\dots\}$ gives a discrete-time stochastic process.
• $T=[0,\infty)$ gives a continuous-time stochastic process.

The progression can be viewed as

random variable → random vector → stochastic process

The crucial feature is the joint structure among indexed random variables, not merely their number.

3. Indices, Experiments, and Sample Paths

3.1 An index need not be time

$X_t$ may represent the temperature on day $t$, a stock price at time $t$, a random walk after $t$ steps, or a simulation state at iteration $t$.

An index may also represent a generation, sampling order, or another ordered structure. A stochastic process needs a meaningful index structure, but the index need not be physical time.

3.2 A stochastic process is not one curve

A random walk can be written $X_{t+1}=X_t+\epsilon_{t+1}$, where $\epsilon_t$ equals $+1$ or $-1$ with equal probability. Starting from $X_0=0$, two experiments might produce:

0 → 1 → 2 → 1 → 0 → -1 → 0 → ...
0 → -1 → 0 → 1 → 2 → 1 → 2 → ...

Each complete sequence is a realization, also called a sample path. The stochastic process describes all possible sample paths and their probabilities; one observed curve is only one realization.

3.3 Fixing an index or an experiment

The full notation for a process is $X(t,\omega)$, where $t$ is the index and $\omega\in\Omega$ is an outcome in the sample space.

• Fixing $t=t_0$ and varying $\omega$ gives the random variable $X(t_0,\omega)$.
• Fixing $\omega=\omega_0$ and varying $t$ gives the sample path $X(t,\omega_0)$.

Mean functions, autocorrelation functions, Gaussian processes, and time-series analysis all use these two perspectives.

4. From Scalar Processes to Random Fields

4.1 A state can be a vector

$X_t$ need not be a scalar. A two-dimensional random walk can use $X_t=(x_t,y_t)$. A vehicle model might use $X_t=(x_t,v_t,a_t)$ for position, velocity, and acceleration.

The index $t$ defines how states are organized, while $X_t$ defines the state at each index. A process can therefore have a one-dimensional time index and a high-dimensional state.

4.2 The index can also be multidimensional

If $X_{i,j}$ is the pixel value at image position $(i,j)$, the index is two-dimensional. Random objects with multidimensional index sets are usually called random fields.

A captured image has fixed pixels, but a probabilistic image model represents all possible images before observation. Sensor noise can also produce different pixel values when the same scene is photographed repeatedly.

An RGB image can be written $X_{i,j,c}$, and a video can be written $X_{t,i,j,c}$. These objects share one structure:

Object	Notation	Index structure
Random variable	$X$	No additional index
Stochastic process	$X_t$	One-dimensional
Two-dimensional random field	$X_{i,j}$	Two-dimensional
Higher-dimensional random field	$X_{t,i,j,c}$	Multidimensional

An index set organizes a family of random variables that share a joint probability structure.

5. What Stochastic Processes Study

Stochastic processes focus on relationships across indices, not just the distribution of each individual $X_t$. A temperature model, for example, may ask:

• whether a hot day makes another hot day more likely;
• how strongly $X_t$ and $X_{t+1}$ are correlated;
• whether $X_{t-1}$ adds information once $X_t$ is known;
• whether the distribution of $X_{t+h}-X_t$ depends only on $h$;
• whether correlation weakens as observations become farther apart;
• whether the mean $E[X_t]$ changes over time.

A stochastic process does not require $X_1,X_2,\dots$ to be independent or identically distributed. Die rolls may be independent, while stock prices and temperatures usually have strong temporal dependence.

The central objects of study are the joint distributions and dependence structure of the indexed random variables.

6. Core Concepts

The basic form of a stochastic process is $\{X_t\}_{t\in T}$. Each $X_t$ is a random variable, and $T$ is the index set. An observed sequence $x_0,x_1,x_2,\dots$ is one sample path; the full process also describes every other possible path and its probability.

The progression continues as

random variable → random vector → stochastic process → random field

Knowing each marginal distribution does not determine how different indices vary together. Joint distributions, expectations, variances, covariances, autocorrelations, and more general dependencies provide the missing structure.