Prerequisites & Notation

Before You Begin

This chapter introduces stochastic processes, building on the probability and random variable theory from Parts I--III. You should be comfortable with joint distributions, expectation, covariance, and the Gaussian distribution.

  • Joint distributions fX,Y(x,y)f_{X,Y}(x,y) and marginals (Ch 9)(Review ch09)

    Self-check: Can you compute the marginal PDF from a joint PDF by integration?

  • Expectation, variance Var(X)\text{Var}(X), and covariance Cov(X,Y)\text{Cov}(X,Y) (Ch 7--8)(Review ch08)

    Self-check: Can you compute Var(aX+bY)\text{Var}(aX + bY) for correlated X,YX, Y?

  • The Gaussian distribution N(ΞΌ,Οƒ2)\mathcal{N}(\mu, \sigma^2) and its properties (Ch 8)(Review ch08)

    Self-check: Can you state the PDF of a Gaussian RV and compute its MGF?

  • Covariance matrices Ξ£\boldsymbol{\Sigma} and positive semi-definiteness (Ch 10)(Review ch10)

    Self-check: Can you verify that a matrix is positive semi-definite using eigenvalues?

  • Jointly Gaussian random vectors (Ch 10)(Review ch10)

    Self-check: Do you know that uncorrelated jointly Gaussian RVs are independent?

Notation for This Chapter

Symbols introduced in this chapter for describing stochastic processes and their second-order statistics.

SymbolMeaningIntroduced
{X(t),t∈T}\{X(t), t \in \mathcal{T}\} or {Xn,n∈I}\{X_n, n \in \mathcal{I}\}Stochastic process (continuous-time or discrete-time)s01
X(t,Ο‰)X(t, \omega) or Xn(Ο‰)X_n(\omega)Sample path for a fixed outcome Ο‰βˆˆΞ©\omega \in \Omegas01
Ft1,ldots,tn(x1,ldots,xn)F_{t_1, \\ldots, t_n}(x_1, \\ldots, x_n)Finite-dimensional joint CDF (fdds)s01
muX(t)\\mu_X(t) or mu[n]\\mu[n]Mean function E[X(t)]\mathbb{E}[X(t)]s01
rXX(t1,t2)r_{XX}(t_1, t_2) or rxx[n,m]r_{xx}[n,m]Autocorrelation function E[X(t1)Xβˆ—(t2)]\mathbb{E}[X(t_1)X^*(t_2)]s03
cXX(t1,t2)c_{XX}(t_1, t_2) or cxx[n,m]c_{xx}[n,m]Autocovariance function Cov(X(t1),X(t2))\text{Cov}(X(t_1), X(t_2))s03
rXY(tau)r_{XY}(\\tau)Cross-correlation of jointly WSS processess04
cXY(tau)c_{XY}(\\tau)Cross-covariance of jointly WSS processess04
tau=t1βˆ’t2\\tau = t_1 - t_2Time lag (for WSS processes)s02
langleXrangleT\\langle X \\rangle_TTime average 12Tβˆ«βˆ’TTX(t) dt\frac{1}{2T}\int_{-T}^{T} X(t)\,dts05
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