Stochastic Processes — Time-Indexed Randomness

Status: placeholder. This page will become the dedicated treatment of stochastic processes, covering everything that random experiments (the subject of part0_what_is_a_distribution.md) deliberately leaves out: time indexing, autocorrelation, stationarity, power spectral density, and ergodicity.

The main probability series (parts 1–6 + applied_sensors) only needs single random variables, so this material is not on the critical path for those parts. Read it after part 0 once the question “what about a whole window of samples, not just one?” starts to matter — typically when you reach Part II (Signals and Measurement) or any time-series work.

Why This Page Exists¶

Part 0 introduces the random process (single trial, no time index) and notes in passing that a stochastic process is the time-indexed version. That distinction is enough to keep parts 1–6 honest, but it leaves real questions on the table:

• What does it mean for two samples in a vibration stream to be “related”?

• Why does a 1 s window of accelerometer data look noticeably different from a 10 s window of the same machine?

• How do you describe a noise signal whose statistics drift over an exposure?

• What’s actually being measured by a power spectral density plot?

• When does averaging over time tell you the same thing as averaging over many trials?

Each of these needs the language of stochastic processes. This page builds that language from first principles.

Intended Outline¶

The sections below are placeholders. Each will follow the same applied-first pattern as the rest of the probability series: one concrete sensor scenario → the math needed to describe it → numbers → cross-domain examples.

1. From Random Variable to Stochastic Process¶

• A stochastic process as a collection of random variables indexed by time: $\{X(t) : t \in T\}$

• The two ways to look at one: a single realisation (one waveform) vs the ensemble (the distribution at each $t$)

• Discrete-time vs continuous-time processes

• Worked example: vibration sensor stream as a discrete-time stochastic process

2. Mean and Variance Over Time¶

• $\mu(t) = \mathbb{E}[X(t)]$ — the mean function

• $\sigma^2(t) = \text{Var}(X(t))$ — the variance function

• Why these can change over time — and why we often want them not to

• Worked example: motor warm-up — the noise floor’s $\sigma$ changes over the first 10 minutes

3. Autocorrelation — Are Successive Samples Independent?¶

• Definition: $R_X(t_1, t_2) = \mathbb{E}[X(t_1) X(t_2)]$

• Intuition: how much knowing $X(t_1)$ tells you about $X(t_2)$

• The autocorrelation function $R_X(\tau)$ for a stationary process

• Worked example: white noise vs band-limited noise vs a periodic signal — three very different autocorrelation shapes

• Why this matters: many statistical tools (CLT included) assume independent samples; autocorrelation tells you when that assumption is broken

4. Stationarity¶

• Strict-sense stationary (SSS): the joint distribution is invariant to time shifts

• Wide-sense stationary (WSS): only $\mu$ and $R_X(\tau)$ are time-invariant

• Why WSS is what engineers actually use

• How to check for stationarity in a real signal (rolling-mean / rolling-variance plots)

• Worked example: a healthy motor as WSS; a degrading one as non-stationary

5. Power Spectral Density (PSD)¶

• The Wiener–Khinchin theorem: $S_X(f) = \mathcal{F}\{R_X(\tau)\}$

• Reading a PSD plot: where is the energy?

• Why PSD is the natural language for vibration, audio, and EEG analysis

• Worked example: bearing fault frequency as a peak in the PSD that wasn’t there on the healthy machine

6. Ergodicity¶

• The question: when does a time average equal an ensemble average?

• $\frac{1}{T}\int_0^T X(t)\,dt \;\overset{?}{=}\; \mathbb{E}[X]$

• Why ergodicity matters: it’s the assumption that lets you estimate population statistics from one long recording

• Worked example: estimating noise-floor $\sigma$ from a 60 s vibration recording instead of repeating the experiment 1,000 times

7. Common Stochastic Processes¶

A short tour of the named processes that show up everywhere in signal processing and ML:

• White noise — the canonical “memoryless” process; flat PSD

• Coloured noise — pink, brown, etc.; PSD shapes you’ll see in real sensors

• Random walk — cumulative sum of white noise; non-stationary, integrated noise

• Wiener process (Brownian motion) — continuous-time random walk; appears in finance, diffusion models, SDE-based generative models

• Markov chains — discrete-state, memoryless transitions; foundational for HMMs and reinforcement learning

• Poisson process — continuous-time generalisation of the Poisson distribution from part3_poisson.md; arrival times of rare events

8. Where Stochastic Processes Show Up Downstream¶

Until This Page Is Filled In¶

The vocabulary table below is the minimum useful summary. It’s the same one in part0_what_is_a_distribution.md §6 — repeated here so this file is self-contained.