MathIsimple

Stochastic Process Fundamentals

Essential Formulas & Mathematical References

Complete collection of formulas for stochastic process definitions, classification, numerical characteristics, and special processes

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1. Stochastic Process Definition
Fundamental mathematical definition and structure

Basic Definition

A stochastic process with parameter set TT and sample space SS:

{X(t,ω);t∈T,ω∈S} or {X(t);t∈T}\{X(t,\omega); t \in T, \omega \in S\} \text{ or } \{X(t); t \in T\}
State Space

Set of all possible values:

I={x:X(t)=x possible}I = \{x : X(t) = x \text{ possible}\}
Sample Function

Fixed outcome realization:

x(t)=X(t,ω0)x(t) = X(t,\omega_0)
2. Process Classification Criteria
Classification based on parameter set and state space properties
Parameter Set Classification
Discrete Time: T={t1,t2,…}T = \{t_1, t_2, \ldots\}
Continuous Time: T=[a,b]T = [a,b] or T=RT = \mathbb{R}
State Space Classification
Discrete State: II at most countable
Continuous State: II is an interval
Four Standard Categories
1. Discrete T, Discrete I
2. Discrete T, Continuous I
3. Continuous T, Discrete I
4. Continuous T, Continuous I
3. Finite-Dimensional Distribution Functions
Complete statistical characterization through distribution families
One-Dimensional Distribution
FX(x;t)=P{X(t)≤x},x∈R,t∈TF_X(x;t) = P\{X(t) \leq x\}, \quad x \in \mathbb{R}, t \in T

Family: {FX(x;t);t∈T}\{F_X(x;t); t \in T\}

n-Dimensional Distribution
FX(x1,…,xn;t1,…,tn)=P{X(t1)≤x1,…,X(tn)≤xn}F_X(x_1,\ldots,x_n;t_1,\ldots,t_n) = P\{X(t_1) \leq x_1, \ldots, X(t_n) \leq x_n\}

where n≥2n \geq 2, t1,…,tn∈Tt_1, \ldots, t_n \in T distinct, x1,…,xn∈Rx_1, \ldots, x_n \in \mathbb{R}

Compatibility Conditions
Marginal Consistency:
lim⁡xn→+∞FX(x1,…,xn;t1,…,tn)=FX(x1,…,xn−1;t1,…,tn−1)\lim_{x_n \to +\infty} F_X(x_1,\ldots,x_n;t_1,\ldots,t_n) = F_X(x_1,\ldots,x_{n-1};t_1,\ldots,t_{n-1})
Permutation Invariance: Distribution unchanged under reordering of time indices
4. Numerical Characteristics
Essential statistical functions for process description
Mean Function
μX(t)=E[X(t)]\mu_X(t) = E[X(t)]

Average level at time tt

Mean Square Function
ψX(t)=E[X2(t)]\psi_X(t) = E[X^2(t)]

Second moment at time tt

Variance Function
σX2(t)=Var[X(t)]=E[(X(t)−μX(t))2]=ψX(t)−[μX(t)]2\sigma_X^2(t) = \text{Var}[X(t)] = E[(X(t) - \mu_X(t))^2] = \psi_X(t) - [\mu_X(t)]^2

Fluctuation measure around mean at time tt

Autocorrelation Function
RX(t1,t2)=E[X(t1)X(t2)],t1,t2∈TR_X(t_1,t_2) = E[X(t_1)X(t_2)], \quad t_1,t_2 \in T

Linear correlation between values at different times

Special case: RX(t,t)=ψX(t)R_X(t,t) = \psi_X(t)
Autocovariance Function
CX(t1,t2)=Cov(X(t1),X(t2))=RX(t1,t2)−μX(t1)μX(t2)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)

Correlation after removing mean effects

Special case: CX(t,t)=σX2(t)C_X(t,t) = \sigma_X^2(t)
5. Second-Order Moment Processes
Processes with finite second moments and their properties
Definition

{X(t);t∈T}\{X(t); t \in T\} is a second-order moment process if:

E[X2(t)]<∞,∀t∈TE[X^2(t)] < \infty, \quad \forall t \in T
Cauchy-Schwarz Guarantee

For second-order processes, autocorrelation function exists:

[E∣X(t1)X(t2)∣]2≤E[X2(t1)]E[X2(t2)]<∞[E|X(t_1)X(t_2)|]^2 \leq E[X^2(t_1)]E[X^2(t_2)] < \infty
6. Gaussian (Normal) Processes
Special properties and complete characterization
Definition

{X(t);t∈T}\{X(t); t \in T\} is Gaussian if for any n≥1n \geq 1 and times t1,…,tn∈Tt_1, \ldots, t_n \in T:

(X(t1),X(t2),…,X(tn))∼Nn(μ,Σ)(X(t_1), X(t_2), \ldots, X(t_n)) \sim N_n(\boldsymbol{\mu}, \boldsymbol{\Sigma})

where μ=(μX(t1),…,μX(tn))\boldsymbol{\mu} = (\mu_X(t_1), \ldots, \mu_X(t_n)) and Σij=CX(ti,tj)\boldsymbol{\Sigma}_{ij} = C_X(t_i, t_j)

Complete Characterization

All statistical properties determined by:

• Mean function: μX(t)\mu_X(t)
• Autocovariance: CX(t1,t2)C_X(t_1,t_2)
Key Properties
1. Linear Transformation: If Y(t)=∑aiX(ti)Y(t) = \sum a_i X(t_i), then {Y(t)}\{Y(t)\} is Gaussian
2. Independence Equivalence: CX(t1,t2)=0⇔X(t1)⊥X(t2)C_X(t_1,t_2) = 0 \Leftrightarrow X(t_1) \perp X(t_2)
7. Two-Dimensional Processes & Cross-Relationships
Joint processes and inter-process relationships
Cross-Correlation Function
RXY(t1,t2)=E[X(t1)Y(t2)],RYX(t1,t2)=E[Y(t1)X(t2)]R_{XY}(t_1,t_2) = E[X(t_1)Y(t_2)], \quad R_{YX}(t_1,t_2) = E[Y(t_1)X(t_2)]

Linear correlation between different processes

Cross-Covariance Function
CXY(t1,t2)=Cov(X(t1),Y(t2))=RXY(t1,t2)−μX(t1)μY(t2)C_{XY}(t_1,t_2) = \text{Cov}(X(t_1),Y(t_2)) = R_{XY}(t_1,t_2) - \mu_X(t_1)\mu_Y(t_2)

Correlation after removing mean effects

Uncorrelatedness Condition

Processes X(t)X(t) and Y(t)Y(t) are uncorrelated if:

CXY(t1,t2)=0,∀t1,t2∈TC_{XY}(t_1,t_2) = 0, \quad \forall t_1,t_2 \in T
Sum Process Characteristics

For Z(t)=X(t)+Y(t)Z(t) = X(t) + Y(t) with uncorrelated X(t)X(t) and Y(t)Y(t):

μZ(t)=μX(t)+μY(t)\mu_Z(t) = \mu_X(t) + \mu_Y(t)
CZ(t1,t2)=CX(t1,t2)+CY(t1,t2)C_Z(t_1,t_2) = C_X(t_1,t_2) + C_Y(t_1,t_2)
Quick Reference Summary
Key Relationships
  • • σX2(t)=RX(t,t)−[μX(t)]2\sigma_X^2(t) = R_X(t,t) - [\mu_X(t)]^2
  • • CX(t,t)=σX2(t)C_X(t,t) = \sigma_X^2(t)
  • • RX(t,t)=ψX(t)=E[X2(t)]R_X(t,t) = \psi_X(t) = E[X^2(t)]
Process Types
  • • Second-order: E[X2(t)]<∞E[X^2(t)] < \infty
  • • Gaussian: All finite-dim. dists. are normal
  • • Uncorrelated: CXY(t1,t2)=0C_{XY}(t_1,t_2) = 0
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