Step 03 · Stochastic Series
Stationary Processes
A record can be read as a single long experiment only if its rules hold steady while time passes. That steadiness is called stationarity, and it is the quiet assumption beneath almost every later result.
The idea in one sentence
A process is stationary when a shift of the clock changes nothing about the joint behaviour of its values. Take a block of moments, slide it forward by any amount, and the picture should look the same. The values themselves may wander, sometimes far, but the rules that govern the wandering must not drift.
Weak and strict forms
The strict form demands that the entire joint density of every block be unaffected by a shift in time. The weak form asks for much less: that the mean be constant, that the variance be constant, and that the covariance between two moments depend only on the distance between them and not on where they sit. The weak form is the one used most often, because it is what the covariance instruments of the previous step can actually test and estimate.
| Feature | Strict | Weak |
|---|---|---|
| whole joint density | shift-invariant | not required |
| mean | constant | constant |
| variance | constant | constant |
| covariance | shift-invariant | depends on lag only |
A normal stationary process is a special convenience. When the joint densities are normal, the weak conditions are enough to guarantee the strict ones, because a normal density is fully described by its means and its covariances. The two forms then coincide, and the analyst may work with the lighter conditions without any loss.
What breaks without it
When stationarity fails, averages behave badly. A record that climbs steadily through time has no fixed mean to estimate, and the ordinary average of its values drifts as the record lengthens. A record whose spread widens over time has no fixed variance to quote. These failures are not curiosities; they are the common case in nature, and the practical response is to transform the record, for instance by differencing, so that the transformed version regains the steadiness the instruments require.
Differencing as a repair
Take the difference between each value and its predecessor and you obtain a new record of changes rather than levels. A record that wanders without a settled level often becomes stationary once differenced, because the wandering level is removed and only the change is kept. This single idea, stated plainly, underpins a great deal of practical series work.
Reading a record for steadiness
Before trusting any summary, split the record in half and compare the two halves. If the means and the spreads sit close together, steadiness is plausible. If they diverge sharply, the record is telling you that a single summary would be a half-truth. This simple check asks little and prevents the most common error in the whole subject, which is to summarise a moving thing with a fixed number.
A test by halves
The simplest check for steadiness is to cut a record in half and summarise each half on its own. If the two halves tell nearly the same story, steadiness is plausible. If they tell different stories, the whole record should not be summarised by a single figure. The check is crude and it catches the most common failure, which is a slow drift that quietly changes the level while the reader's attention is on something else.
It is worth repeating the cut at the quarter points, not only at the middle, so that a record which is steady at the ends and restless in the middle does not slip through. Each extra cut asks almost nothing, and together the cuts build a picture of how the record's behaviour moves through time. Where the picture is flat, the steady instruments may be trusted; where it rises or falls, the record must be reshaped before any steady instrument is used.