the second hundred · metaphor 118
The cause went quiet long ago — the loss is years back, the argument settled, the reputation earned or lost in another season entirely. So why does it still weigh on today? How far back does the past actually reach into the present, and why do some pasts fade in a week while others press for a lifetime?
We picture memory as a fading echo: loud at first, then quieter, then gone. That's one kind of past — a shock whose influence dies away fast, so that after a little while today owes almost nothing to what came before. But there is another kind, where the influence doesn't die away so much as thin: still there at ten years, at twenty, weaker but never quite zero, a long tail of the far-off cause reaching all the way to now.
Mathematics separates these cleanly. In a short-memory process the correlation between then and now drops off geometrically — halve, halve, halve, gone. In a long-memory process it decays as a slow power law, so gently that the total weight of the past never really closes out. Below is a dial from one to the other, running live.
Short memory vs. long memory
To measure how much a process remembers, ask a simple question at every distance in time: how strongly does today resemble the value k steps ago? Averaged over the whole record, that is the autocorrelation at lag k — one number for each look-back distance, and together they are the process's memory made visible.
For most familiar processes that number falls off geometrically: each extra step back multiplies the resemblance by a fixed fraction, so it halves, halves again, and within a handful of steps is indistinguishable from zero. This is short memory — a past that leaves like an echo, and after a brief while has genuinely left. But tune the process and the decay can slacken into a power law, ρ(k) ∼ k^{2H−1} — a curve that drops steeply at first and then refuses to finish, staying faintly positive out to enormous lags. Now the sum of all those echoes no longer converges: the total weight of the past is, in a real sense, unbounded. That is long memory, and the single knob that governs it is the Hurst exponent H — one half for the forgetful process, climbing toward one for the haunted one.
What to try
The upper panel is a live process — each new value is built by weighting the entire history of random shocks, the far ones included, exactly as a long-memory process demands. The lower panel is its measured autocorrelation, recomputed from the running series, with the power-law theory traced faintly over it. Start at H = 0.5: the series is a jittery white noise, and the autocorrelation is a single spike at lag zero that crashes to the baseline and stays there. The past is gone in one step.
Now drag H rightward. The series stops jittering and begins to wander — long excursions above and below its level, because each value is now partly inherited from a slowly-fading past. In the lower panel the autocorrelation stops crashing and grows a tail, sinking gently across dozens of lags without reaching zero. The lag-1 echo climbs, the reach of the past stretches from one step to many, and the integrated memory — the summed weight of all those echoes — swells without any sign of a ceiling. That runaway sum is the mathematical signature of a past that never quite closes.
The mapping
Some things we live through are short-memory: a bad commute, a snub, a rough night. Their influence decays like an echo — a day or two and today owes them nothing. Others are long-memory. Grief is the clearest: the loss recedes, the acute cause has gone quiet, and still the far past presses on the present — thinner each year, but never fully zero, a power-law tail reaching across a life. Moods and temperament can carry the same signature, a run of heavy weeks correlating with weeks long past. Reputations and institutions too: a single episode, decades gone, still faintly tilts how a name is read today.
Naming the shape changes what you expect. If a hurt is short-memory, waiting works and the arithmetic of "give it time" is exactly right. If it is long-memory, the same waiting brings diminishing but unending returns — the influence keeps thinning, yet you should not be surprised to feel it, faintly, far downstream. The question "when will this stop mattering?" has a different answer for the two, and knowing which one you're in is most of the wisdom.
Read as life lessons
A power-law past never reaches zero. Expecting a long-memory hurt to fully close out sets you up to feel broken when it recurs faintly years on. It was never going to hit zero — only keep thinning.
Two lives can share the same mean and be utterly different underneath — one forgetful, one haunted. Memory lives not in the level but in how the moments correlate across time. Read the autocorrelation, not the average.
Long memory makes runs — good stretches and bad stretches that persist far longer than chance would allow. A long low spell needn't mean a new cause; it can be the same far past, still pressing.
The mapping, exactly
| Mathematics | Life |
|---|---|
| white noise (H = ½) | A shock with no aftermath — today is uncorrelated with yesterday; the past leaves no trace at all. |
| geometric decay | Short memory — a hurt that fades like an echo and, after a brief while, has genuinely gone. |
| power-law tail ρ(k) ∼ k^(2H−1) | The far past still pressing on today — thinning across the years, never quite reaching zero. |
| lag-1 autocorrelation | How much today directly echoes yesterday — the near memory you can feel. |
| divergent Σ ρ(k) | A past whose total weight never closes out — the sense that it all still counts, however faintly. |
| Hurst exponent H | The single dial from forgetful to haunted — how persistent a life, mood, or name turns out to be. |
The honest model
The process is fractionally-integrated noise (a FARIMA(0, d, 0) with d = H − ½). Each value is a weighted sum of all past random shocks, x_t = Σ_k ψ_k · ε_{t−k}, with weights built by the recursion ψ_0 = 1, ψ_k = ψ_{k−1}·(k−1+d)/k, which decay as ψ_k ∼ k^{d−1} — a slow, heavy-tailed kernel that keeps a sliver of the far past in every new value. At d = 0 the kernel collapses to a single term and the process is exactly white noise. The panel really does compute each point this way, in real time, weighting a long buffer of live shocks.
The lower panel's bars are the sample autocorrelation, ρ̂(k) = Σ(x_t−x̄)(x_{t+k}−x̄) ⁄ Σ(x_t−x̄)², measured from the running series — not a formula drawn to look right. The faint overlaid curve is the true FARIMA autocorrelation, ρ(k) = ρ(k−1)·(k−1+d)/(k−d), labelled as theory. They agree in shape; at high H the sample sits a little below theory, which is honest — sample autocorrelation is known to underestimate strong long memory in a finite record, and a single finite life is exactly such a record.
Where the metaphor tears
The line "the cause had gone quiet" is the metaphor's own overreach. In the model nothing persists but correlation — the slow statistical shadow of old shocks around a fixed mean. Real grief often has live causes: anniversaries, reminders, fresh reopenings. Long memory describes a past that presses through structure, not one kept alive by renewed injury; conflating the two flatters a tidy math onto a messier wound.
Our series is stationary — its statistics are the same today, next year, forever; the autocorrelation is fixed. Grief usually is not stationary: it has a trend, it fades, the whole distribution shifts. A power-law tail models persistence around a stable level, not a wound that (mostly) closes. Long memory and non-healing are different claims, and only one of them is kind.
Estimating a Hurst exponent is treacherous even with clean data: a slow trend, a single level-shift, or a regime break can each masquerade as long memory, and the sample autocorrelation is biased in short records. One person's history is one short, non-repeatable series. The metaphor sharpens the question — how does your past decay? — long before it could ever hand you a number.