Stochastic Process Fundamentals

Master the core concepts of stochastic processes: definitions, classification, sample functions, finite-dimensional distributions, and numerical characteristics

6-10 hoursIntermediate to Advanced8 lessons

Learning Objectives

Understand the rigorous definition of stochastic processes and their mathematical structure
Master classification of processes by parameter sets and state spaces
Analyze sample functions and their properties in stochastic modeling
Apply finite-dimensional distribution theory for process characterization
Calculate and interpret numerical characteristics of stochastic processes
Understand Gaussian processes and their special properties

1. Stochastic Process Definition & Structure

Core Definition

Let $S$ be the sample space of a random experiment (containing all possible outcomes $\omega$ ), and $T$ be a parameter set (usually representing time, where $T \subset \mathbb{R}$ , such as $T = \{1,2,\ldots\}$ or $T = [0,+\infty)$ ).

Definition:

If for any $t \in T$ , there exists a random variable $X(t,\omega)$ defined on $S$ , then the family of two-parameter functions

\{X(t,\omega); t \in T, \omega \in S\}

is called a stochastic process, denoted as $\{X(t); t \in T\}$ .

State & State Space

For fixed $t$ , the value $X(t)$ is called thestate at time $t$ . The set of all possible states forms the state space $I$ .

Examples:

• Target hits: $I = \{0,1,2,\ldots,n\}$
• Voltage signal: $I = [-a,a]$

Sample Functions

For fixed $\omega$ (one trial outcome), $X(t,\omega)$ is a deterministic function of $t$ , called a sample functionor realization, denoted as $x(t) = X(t,\omega_0)$ .

Physical Meaning:

One complete observation of the random process over time

2. Classification of Stochastic Processes

Four-Category Classification

Based on parameter set

T

(discrete/continuous) and state space

I

(discrete/continuous)

Category	Parameter Set T	State Space I	Example
1. Discrete Time, Discrete State	At most countable (e.g., $\{1,2,\ldots\}$ )	At most countable	Binomial process $S_n$ (hits in first $n$ trials)
2. Discrete Time, Continuous State	At most countable	Interval (e.g., $\mathbb{R}$ )	Temperature sequence $X_n$ (daily temperature, $I \in [-30,40]$ )
3. Continuous Time, Discrete State	Real interval (e.g., $T = [0,+\infty)$ )	At most countable	Claim count process $N(t)$ (claims in $(0,t]$ , $I = \{0,1,2,\ldots\}$ )
4. Continuous Time, Continuous State	Real interval	Interval	Random phase cosine $X(t) = a\cos(\omega t + \Theta)$ , voltage fluctuation

Classic Examples with Analysis

Example 1: Binomial Process

Independent target shooting with hit probability $p$ . Let $S_n$ represent total hits in first $n$ trials.

Parameter set: $T = \{1,2,\ldots\}$ (discrete time)

State space: $I = \{0,1,\ldots,n\}$ (discrete state)

Sample functions: Non-decreasing step functions

Property: $S_n - S_{n-1} \in \{0,1\}$ (at most one hit per trial)

Example 2: Random Phase Cosine Wave

$X(t) = a\cos(\omega t + \Theta)$ where $t \in \mathbb{R}$ , $a, \omega > 0$ are constants, $\Theta \sim U(0, 2\pi)$

Parameter set: $T = \mathbb{R}$ (continuous time)

State space: $I = [-a,a]$ (continuous state)

Sample functions: $x(t) = a\cos(\omega t + \theta_0)$ (cosine curves with different phases)

Property: All realizations have same frequency but random phase

3. Finite-Dimensional Distribution Family

Mathematical Foundation

Complete characterization of stochastic processes through finite-dimensional distributions

One-Dimensional Distribution

For fixed $t \in T$ , the distribution function of random variable $X(t)$ :

F_X(x;t) = P\{X(t) \leq x\}, \quad x \in \mathbb{R}

The collection $\{F_X(x;t); t \in T\}$ forms the one-dimensional distribution family.

n-Dimensional Distribution

For any $n \geq 2$ and distinct times $t_1, t_2, \ldots, t_n \in T$ :

F_X(x_1,x_2,\ldots,x_n;t_1,t_2,\ldots,t_n) = P\{X(t_1) \leq x_1, X(t_2) \leq x_2, \ldots, X(t_n) \leq x_n\}

where $x_1,\ldots,x_n \in \mathbb{R}$ .

Compatibility Conditions

The finite-dimensional distribution family must satisfy:

• Consistency: Lower-dimensional distributions are marginals of higher-dimensional ones
• Permutation invariance: Distribution unchanged under time reordering
• Example: $\lim_{x_2 \to +\infty} F_X(x_1,x_2;t_1,t_2) = F_X(x_1;t_1)$

4. Numerical Characteristics of Stochastic Processes

Essential Functions

Simplified statistical description when finite-dimensional distributions are complex

Mean Function

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

Average level of the process at time $t$ , representing the "central tendency" of sample functions.

Variance Function

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

Fluctuation degree around the mean at time $t$ .

Autocorrelation Function

R_X(t_1,t_2) = E[X(t_1)X(t_2)], \quad t_1,t_2 \in T

Describes linear correlation between process values at times $t_1$ and $t_2$ .

Special cases: When $t_1 = t_2 = t$ , $R_X(t,t) = E[X^2(t)]$ (mean square value)

Autocovariance Function

C_X(t_1,t_2) = \text{Cov}(X(t_1),X(t_2)) = R_X(t_1,t_2) - \mu_X(t_1)\mu_X(t_2)

Linear correlation after removing mean effects.

Key relationships:

• $\sigma_X^2(t) = C_X(t,t)$ (variance as special case)
• $\sigma_X^2(t) = R_X(t,t) - [\mu_X(t)]^2$

Second-Order Processes

Definition

If for any $t \in T$ , the mean square value exists:

E[X^2(t)] < \infty

then $\{X(t); t \in T\}$ is called a second-order moment process.

Properties

• Mean function, autocorrelation, and autocovariance functions all exist
• Guaranteed by Cauchy-Schwarz inequality: $[E|X(t_1)X(t_2)|]^2 \leq E[X^2(t_1)]E[X^2(t_2)] < \infty$
• Example: Random phase cosine wave $X(t) = a\cos(\omega t + \Theta)$ is second-order since $E[X^2(t)] = a^2/2 < \infty$

5. Gaussian (Normal) Processes

Definition and Superior Properties

Definition

If for any $n \geq 1$ and any times $t_1, t_2, \ldots, t_n \in T$ , the n-dimensional random vector $(X(t_1), X(t_2), \ldots, X(t_n))$ follows an n-dimensional normal distribution, then $\{X(t); t \in T\}$ is called a Gaussian process or normal process.

Key Properties

1. Complete characterization by second-order characteristics:
All statistical properties determined by mean function $\mu_X(t)$ and autocovariance function $C_X(t_1,t_2)$
2. Linear transformation invariance:
If $\{X(t)\}$ is Gaussian, then $Y(t) = \sum_{i=1}^k a_i X(t_i)$ is also Gaussian for constants $a_1, \ldots, a_k$
3. Uncorrelation equivalence to independence:
For Gaussian processes, $C_X(t_1,t_2) = 0 \Leftrightarrow X(t_1) \perp X(t_2)$ (uncorrelated ⟺ independent)

Example: Distribution Calculation

Let $\{X(t); t \geq 0\}$ be a Gaussian process with $\mu_X(t) = t$ and $C_X(t_1,t_2) = t_1 t_2 + 1$ . Find the distribution of $X(2)$ .

Solution:

• Mean: $\mu_X(2) = 2$
• Variance: $\sigma_X^2(2) = C_X(2,2) = 2 \times 2 + 1 = 5$
• Therefore: $X(2) \sim N(2, 5)$

Next Steps & Applications

Practice Problems

Reinforce understanding with classification exercises, numerical characteristics calculations, and Gaussian process problems.

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Formula Reference

Quick access to key formulas for stochastic process fundamentals and numerical characteristics.

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