12 Stochastic Processes — Time-Indexed Randomness
This page is a placeholder outline. The main probability series only needs single random variables, so this material is not on the critical path. Read it once the question “what about a whole window of samples, not just one?” starts to matter — typically when you move into time-series work or signal processing.
12.1 Why this page exists
The early probability chapters introduce the random variable — single trial, no time index. That distinction is enough to keep those chapters 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.
12.2 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.
12.2.1 1. From random variable to stochastic process
- A stochastic process as a collection of random variables indexed by time: \{X(t) : t \in T\}
- Two ways to look at one: a single realisation (one waveform) vs the ensemble (the distribution at each t)
- Discrete-time vs continuous-time
- Worked example: vibration sensor stream as a discrete-time process
12.2.2 2. Mean and variance over time
- \mu(t) = E[X(t)] — the mean function
- \sigma^2(t) = \operatorname{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
12.2.3 3. Autocorrelation — are successive samples independent?
- Definition: R_X(t_1, t_2) = 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 shapes
- Why this matters: many statistical tools (CLT included) assume independent samples; autocorrelation tells you when that assumption is broken
12.2.4 4. Stationarity
- Strict stationarity: the joint distribution of any subset is invariant to time shifts
- Wide-sense stationarity (WSS): mean is constant in time and the autocorrelation depends only on lag
- Why WSS is enough for most engineering work
- Worked example: a healthy bearing is WSS; a failing bearing isn’t
12.2.5 5. Power spectral density
- Definition: Fourier transform of the autocorrelation function
- Worked example: 1/f noise, white noise, narrowband interference
12.2.6 6. Ergodicity
- When can you replace ensemble averages with time averages?
- The bridge between “many independent experiments” and “one long recording”
If you’d like this filled in earlier, raise an issue against the repo. The material is well-defined; it’s mostly a writing-time question.