Fundamentals

6-10 Hours

Stochastic Process Fundamentals

Master the core concepts: definitions, classification, sample functions, and numerical characteristics

Definitions & Structure - The Stochastic Process

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $T$ be a parameter set (usually time). A stochastic process is a collection of random variables:

\{X(t, \omega) : t \in T, \omega \in \Omega\}

It can be viewed in two ways:

Fixed $t$ : $X(t, \cdot)$ is a random variable on $\Omega$ .
Fixed $\omega$ : $X(\cdot, \omega)$ is a deterministic function of $t$ , called a sample function or trajectory.

Classification of Processes

Type	Time $T$	State $S$	Example
Discrete Time, Discrete State	Discrete	Discrete	Bernoulli process, Random Walk
Continuous Time, Discrete State	Continuous	Discrete	Poisson process
Continuous Time, Continuous State	Continuous	Continuous	Brownian motion

Numerical Characteristics - Mean Function

\mu_X(t) = E[X(t)]

Represents the average behavior of the process at time $t$ .

Variance Function

\sigma_X^2(t) = \text{Var}(X(t)) = E[(X(t) - \mu_X(t))^2]

Measures the spread around the mean at time $t$ .

Autocorrelation Function

R_X(t_1, t_2) = E[X(t_1)X(t_2)]

Describes linear relationship between values at two different times.

Autocovariance Function

C_X(t_1, t_2) = E[(X(t_1)-\mu(t_1))(X(t_2)-\mu(t_2))]

Relationship: $C_X(t_1, t_2) = R_X(t_1, t_2) - \mu_X(t_1)\mu_X(t_2)$

Stationarity - Strict-Sense Stationarity (SSS)

All statistical properties are invariant to time shifts. For any $\tau, n, t_1, \dots, t_n$ :

F_{X(t_1+\tau), \dots, X(t_n+\tau)} = F_{X(t_1), \dots, X(t_n)}

Wide-Sense Stationarity (WSS)

A process $X(t)$ is WSS if:

$E[X(t)] = \mu$ (constant)
$R_X(t_1, t_2) = R_X(t_1 - t_2)$ (depends only on lag)

Gaussian Processes - Definition & Properties

A process $\{X(t), t \in T\}$ is a Gaussian Process if for any finite set of times $t_1, \dots, t_n$ , the random vector $(X(t_1), \dots, X(t_n))$ has a multivariate normal distribution.

Complete Characterization

A Gaussian process is completely determined by its mean function $\mu_X(t)$ and autocovariance function $C_X(t_1, t_2)$ .

Fundamental Theorems - Kolmogorov Extension Theorem

For any consistent family of finite-dimensional distributions $F_{t_1, \dots, t_n}(x_1, \dots, x_n)$ , there exists a probability space $(\Omega, \mathcal{F}, P)$ and a stochastic process $\{X(t)\}$ such that its finite-dimensional distributions are exactly the given family.

Significance: This guarantees that we can define a process simply by specifying its finite-dimensional distributions.

Worked Examples - Example 1: Random Phase Cosine Wave

Problem:

Let $X(t) = A \cos(\omega t + \Theta)$ , where $A, \omega$ are constants and $\Theta \sim U(0, 2\pi)$ . Find $\mu_X(t)$ and $R_X(t_1, t_2)$ .

Solution:

Mean: $\mu_X(t) = E[A \cos(\omega t + \Theta)] = 0$ (by symmetry).

Autocorrelation: Using trigonometric identities, $R_X(t_1, t_2) = \frac{A^2}{2} \cos(\omega(t_1-t_2))$ .

Key Insight: This process is Wide-Sense Stationary (WSS).

Example 2: Poisson Process Characteristics

Problem:

Let $\{N(t), t \geq 0\}$ be a Poisson process with rate $\lambda$ . Find $\mu_N(t)$ and $C_N(t_1, t_2)$ .

Solution:

Mean: $\mu_N(t) = E[N(t)] = \lambda t$ .

Autocovariance: For $t_1 < t_2$ , using independent increments: $C_N(t_1, t_2) = \lambda \min(t_1, t_2)$ .

Key Insight: The Poisson process is not stationary because its mean depends on time.

Example 3: Simple Random Walk

Problem:

Let $X_n = \sum_{i=1}^n Y_i$ where $Y_i$ are i.i.d. with $P(Y_i=1)=p, P(Y_i=-1)=q=1-p$ . Find $\mu_X(n)$ and $\sigma_X^2(n)$ .

Solution:

Mean: $\mu_X(n) = n(p-q)$ .

Variance: $\sigma_X^2(n) = 4npq$ .

Practice Quiz

Questions

Correct

Accuracy

What is the fundamental definition of a stochastic process?

Not attempted

What is the difference between a random variable and a stochastic process?

Not attempted

What are finite-dimensional distributions?

Not attempted

What is the mean function

\mu_X(t)

of a stochastic process?

Not attempted

What is the autocorrelation function

R_X(t_1, t_2)

Not attempted

What is the difference between strict-sense and wide-sense stationarity?

Not attempted

What is a Gaussian process?

Not attempted

What is a sample function (or trajectory)?

Not attempted

What does the Kolmogorov Extension Theorem guarantee?

Not attempted

What is an independent increment process?

Not attempted

Frequently Asked Questions

What is the difference between a random variable and a stochastic process?

A random variable maps outcomes to numbers (a single value per trial). A stochastic process maps outcomes to functions of time (a whole trajectory per trial). You can think of a stochastic process as a 'dynamic' random variable that evolves over time.

Why are finite-dimensional distributions so important?

According to the Kolmogorov Extension Theorem, the family of all finite-dimensional distributions completely characterizes the stochastic process (under mild consistency conditions). They contain all the statistical information about the process.

What is the difference between strict-sense and wide-sense stationarity?

Strict-sense stationarity requires all statistical properties (distributions) to be invariant under time shifts. Wide-sense (or weak) stationarity only requires the first two moments (mean and autocorrelation) to be invariant. Wide-sense is much easier to verify and is sufficient for many applications like signal processing.

Can a process be wide-sense stationary but not strict-sense?

Yes. For example, a process with a constant mean and autocorrelation but whose higher-order moments change over time would be wide-sense but not strict-sense stationary. However, for Gaussian processes, the two concepts are equivalent.

What is a sample function?

A sample function (or realization/trajectory) is the specific time function x(t) observed for a single outcome ω of the random experiment. Once the experiment is performed, the sample function is deterministic.