Chapter 02

Stochastic Processes

The language of uncertain signals: stochastic processes, their mean and autocovariance, weak vs strong stationarity, the Toeplitz condition that makes a covariance valid, white noise, and Wold's decomposition — everything the exam's first problem is built on.

Reading: ~32 min Interactive: 1 widgets Source: Polimi MIDA1 2025/26 — Lecture 3 (Stochastic processes) · Polimi MIDA1 2025/26 — Appendix A (Probability reminders)

01 · Definitions

Processes and their first two moments

A stochastic process is an infinite sequence of random variables — one signal per outcome of a single random experiment. We never know it exactly, only its statistics, and two of them carry almost everything we need.

Formally a discrete-time process is $v(t) = \phi(s, t)$ , where $s$ is the outcome of a random experiment and $t$ the time index. Fix the outcome $s$ and you get one ordinary signal — a realization. Fix the time $t$ and you get a random variable. A full probabilistic description would assign a joint distribution to every finite collection of samples — infinitely many functions. We almost always work instead with the first- and second-order characteristics:

Mean & covariance

$\mu(t) = \mathbb{E}[v(t)], \qquad \gamma(t_1,t_2) = \mathbb{E}\big[(v(t_1)-\mu(t_1))(v(t_2)-\mu(t_2))\big].$

The mean is the value the signal fluctuates around; the autocovariance measures how the process at two different times moves together. Setting $t_1=t_2=t$ recovers the variance $\gamma(t,t)=\mathrm{Var}[v(t)]\ge 0$ . Two normalised cousins appear constantly: the autocorrelation $\tilde\gamma(t_1,t_2)=\mathbb{E}[v(t_1)v(t_2)]=\gamma(t_1,t_2)+\mu(t_1)\mu(t_2)$ (equal to $\gamma$ when the mean is zero), and the Pearson correlation coefficient $\rho = \gamma(t_1,t_2)/\sqrt{\gamma(t_1,t_1)\gamma(t_2,t_2)}$ , which lives in $[-1,1]$ by Cauchy–Schwarz.

This is the exam's first problem

On every recent structured MIDA1 paper, Problem 1 is “Mean, covariance and spectrum”: you are handed a system that filters white noise and asked — is the output stationary? compute $\mu$ , $\gamma(0)$ , a lag-covariance $\gamma(\tau)$ , and the spectrum at one frequency. It is pure, boxed-answer computation. Graders want the number and a one-line justification of stationarity. Master the mechanics in this chapter and the spectrum piece in Chapter 4, and you own eight points.

02 · Core

Stationarity: weak vs strong

Almost everything simplifies if the process’s statistics do not drift over time. A process is strongly stationary if all its joint distributions are invariant to a time shift — a complete but unusable description. The workable version keeps only the first two moments:

✓

Weak (wide-sense) stationarity

A process is weakly stationary if (i) the mean is constant, $\mu(t)=\mu$ ; (ii) the variance is constant; and (iii) the autocovariance depends only on the lag $\tau = t_2 - t_1$ , so we write $\gamma(\tau)$ . A Gaussian weakly-stationary process is automatically strongly stationary.

A weakly stationary process is completely characterised by a number (its mean) and a function of one variable (its covariance $\gamma(\tau)$ ) — two processes with the same pair are “equivalent”. And if the process is ergodic, its statistics can be recovered from a single realization: the sample average converges to the expected value, $\frac1N\sum_{t=1}^N v(t) \to \mathbb{E}[v]$ . That is what lets us estimate anything from one recorded signal.

Exam-format worked example Is y(t) stationary, and what are its statistics?

The setup

Let $x(t) = a\,x(t-1) + e(t)$ with $e(\cdot)\sim\mathrm{WN}(m_e,\lambda_e^2)$ and $|a|<1$ , and let $y(t) = x(t) + \eta(t)$ with $\eta(\cdot)\sim\mathrm{WN}(0,\lambda_\eta^2)$ independent of $e$ . This is the shape of every Problem-1 process. Take $a=\tfrac12$ , $m_e=0$ , $\lambda_e^2=1$ , $\lambda_\eta^2=3$ .

Stationarity

$x$ is an AR(1) with pole $a=\tfrac12$ inside the unit circle, hence a stable filter driven by white noise — stationary. $y$ is the sum of two stationary, independent processes, so $y$ is stationary. Always argue it this way: stable-filter-of-white-noise + independent-stationary = stationary.

Mean

Take expectations of the state equation: $\mu_x = a\,\mu_x + m_e \Rightarrow \mu_x = m_e/(1-a) = 0$ . Then $\mu_y = \mu_x + \mathbb{E}[\eta] = 0$ .

Variance of x

Square the AR(1) equation and use $\mathbb{E}[x(t-1)e(t)]=0$ : $\gamma_x(0)=a^2\gamma_x(0)+\lambda_e^2$ , so $\gamma_x(0)=\dfrac{\lambda_e^2}{1-a^2}=\dfrac{1}{1-1/4}=\dfrac43$ . (This is the Yule–Walker variance of Chapter 3.)

Lag-1 covariance of y

For $\tau\ne 0$ the independent white $\eta$ drops out, so $\gamma_y(1)=\gamma_x(1)=a\,\gamma_x(0) =\tfrac12\cdot\tfrac43=\dfrac23$ . The measurement noise $\eta$ only ever touches $\gamma_y(0)$ .

Now do one yourself — the July 2026 Problem 1, with an AR(1) that has a non-zero-mean noise (watch the mean feed $\mu$ while the variance feeds $\gamma_x(0)$ ):

2026-07-q12026Q01Mean, covariance & spectrummedium8 pts

Consider the stochastic process generated by the system $S$: $$x(t)=\tfrac13 x(t-1)+e(t),\qquad y(t)=x(t)+x(t-1)+\eta(t)$$ with $e(\cdot)\sim\mathrm{WN}(2,2)$, $\eta(\cdot)\sim\mathrm{WN}(-1,1)$, and $e\perp\eta$. (a) Is $y(t)$ stationary? (b) Compute $E[y(t)]$. (c) Compute $\gamma_x(0)$. (d) Compute $\gamma_y(1)$. (e) Compute the power spectral density $\Gamma_y(\omega)$ at $\omega=\pi$.

03 · Property

What makes a valid covariance

The autocovariance of a stationary process is not an arbitrary function. Three properties hold, and the third is deeper than it looks:

Non-negative variance

$\gamma(0)=\mathrm{Var}[v(t)]\ge 0$ — the covariance at lag zero is the variance.

Bounded & even

$\lvert\gamma(\tau)\rvert \le \gamma(0)$ (Cauchy–Schwarz — a signal is most correlated with itself) and $\gamma(\tau)=\gamma(-\tau)$ (an even function).

Toeplitz PSD

The Toeplitz matrix built from $\gamma(0),\gamma(1),\dots$ must be positive semi-definite for every size $N$ .

A plausible-looking γ(τ) can still be illegal

The first two properties are necessary but not sufficient. An even, bounded sequence is a genuine covariance function iff its Toeplitz matrix is positive semi-definite for all $N$ . Exam traps ask “can this be the covariance of a stationary process?” — a sequence can pass the eyeball test and still fail because a $3\times3$ (or larger) Toeplitz minor goes negative.

Proof Why the Toeplitz matrix must be PSD

Stack $N$ centred samples into a vector $V = [\,v(0)-\mu,\ \dots,\ v(N-1)-\mu\,]^\top$ . Its covariance matrix is $\mathbb{E}[VV^\top]$ , whose $(i,j)$ entry is $\gamma(i-j)$ — exactly the Toeplitz matrix of the covariance sequence. Any covariance matrix is positive semi-definite (for any vector $c$ , $c^\top \mathbb{E}[VV^\top] c = \mathbb{E}[(c^\top V)^2]\ge 0$ — it is the variance of a scalar random variable). Since this must hold for every window length $N$ , the whole family of Toeplitz matrices is PSD. That is why “is this a valid $\gamma$ ?” reduces to a definiteness check.

04 · Building block

White noise: the atom

White noise is the simplest stationary process and the building block of every model to come. It is a sequence of independent, identically distributed variables:

White noise

$\mathbb{E}[v(t)] = 0, \qquad \gamma(\tau) = \begin{cases}\lambda^2 & \tau = 0\\ 0 & \tau \ne 0\end{cases}, \qquad v(\cdot)\sim\mathrm{WN}(0,\lambda^2).$

Because $\gamma(\tau)=0$ for every $\tau\ne0$ , knowing the past tells you nothing about the future: white noise is completely unpredictable. If the distribution is Gaussian we write $\mathrm{WGN}(0,\lambda^2)$ . This is the same white noise as Chapter 1’s optimal residual — the two ideas are the same object seen from two directions.

05 · Structure

Wold’s decomposition

How can an arbitrary stationary process be built from white noise? Wold’s theorem says any stationary process splits into two uncorrelated pieces:

Wold

$v(t) = \tilde v(t) + \hat v(t),$

where $\tilde v$ is purely deterministic — knowing its past predicts its entire future exactly (e.g. a constant, an alternating $\pm$ sequence, or a sinusoid $\bar v\cos(\omega_0 t + \theta)$ ) — and $\hat v$ is purely non-deterministic: everything you cannot pin down from the past. The crucial fact is that the non-deterministic part is a convolution of white noise,

$\hat v(t) = \sum_{k\ge 0} w(k)\,\eta(t-k) = W(z)\,\eta(t),\qquad W(z) = \sum_{k\ge 0} w(k)z^{-k},$

i.e. the output of a dynamical filter driven by white noise. That is precisely the object the next chapter parameterises as MA, AR and ARMA models. Because a non-zero mean is just a purely deterministic constant component (predicted exactly), we lose nothing by developing the theory for zero-mean processes, where autocovariance and autocorrelation coincide.

key

Why Wold matters

Wold turns “model a stationary signal” into “find the filter $W(z)$ that colours white noise into it”. Prediction (Chapter 5) and identification (Chapters 6–7) are both about recovering that filter — and the residual left after the best filter is, once more, white noise.

A zero-mean stationary process has γ(0)=2, γ(1)=3. Can this be a valid autocovariance function?

Load-bearing ideas

A stochastic process is summarised by its mean $\mu$ and autocovariance $\gamma(\tau)$ ; the Pearson coefficient normalises correlation into $[-1,1]$ .
Weak stationarity = constant mean and variance + covariance depending only on the lag $\tau$ . Ergodicity lets you estimate the statistics from one realization.
$\gamma$ is even, bounded by $\gamma(0)$ , and — the deep condition — has a positive semi-definite Toeplitz matrix; that is what makes a candidate $\gamma$ a legal covariance.
White noise $\mathrm{WN}(0,\lambda^2)$ has $\gamma(\tau)=0$ for $\tau\ne0$ and is completely unpredictable.
Wold: every stationary process = a purely deterministic part + white noise pushed through a filter $W(z)$ . Modelling and prediction are about recovering that filter.

Exam radar. Re-derive the StepThrough cold before the exam: given $x=ax(t-1)+e$ , $y=x+\eta$ , produce stationarity + $\mu_y$ + $\gamma_x(0)$ + $\gamma_y(1)$ . Keep the $\lvert\gamma(\tau)\rvert\le\gamma(0)$ validity trap in mind, and remember independent measurement noise touches only $\gamma_y(0)$ .