Stochastic Process vs Time Series
Stochastic Process
A stochastic process $\{X_t\}_{t \in T}$ is a collection of random variables indexed by $T$ (the time index). In this course, $T = \{0, 1, 2, \ldots\}$ (discrete time).
Notation
- $t$ — time index
- $X_t$ — random variable at time $t$ (the state of the process)
Time Series
A time series is a sequence of observations $x_1, x_2, \ldots, x_n$ — one realization of an underlying stochastic process.
| Stochastic Process | Time Series | |
|---|---|---|
| Nature | Theoretical model (random variables) | Observed data (numbers) |
| Notation | $\{X_t\}$ (capital) | $\{x_t\}$ (lowercase) |
| Role | What we model | What we measure |
We only ever observe one realization. To estimate parameters, we need stationarity — treating observations at different times as if from the same distribution.