Ergodicity, explained.
Ergodicity concerns when a time average along a process agrees with an average across its statistical ensemble. It depends on the process and the quantity being averaged.
Why it happens
A snapshot of many parallel outcomes and one person's long sequence answer different questions. Multiplicative changes can create a striking gap between expected wealth and typical compound growth without that gap alone proving an ergodic theorem.
Ergodicity concerns conditions under which averages over time agree with averages across a statistical ensemble. The property depends on the process and on the observable being averaged.
Look for this pattern
Specify whether you are averaging wealth levels, percentage changes, or logarithmic growth. Then distinguish a finite simulation from a statement about limiting averages over time.
A worked example
Two multiplicative changes
Start at 100 and apply a 50% gain followed by a 40% loss.
The arithmetic mean of the two percentage changes is +5%, but the balance becomes 100 × 1.5 × 0.6 = 90.
Compounding depends on products. An average percentage change alone does not describe the wealth path.
A common misconception
“Any difference between the mean and median proves non-ergodicity.”
Different summary statistics can disagree in many distributions. Ergodicity requires a specified process, observable, and averaging limit.
What this explanation leaves out
- A difference between a mean and a median alone does not establish non-ergodicity. Finite-horizon simulations illustrate differences between paths and populations, not a proof of an ergodic theorem.
Why distinguish time and ensemble averages?
A policy chosen from the average across possible worlds may not match the experience of one repeated path. The distinction helps identify which outcome the decision-maker actually faces.
Does the average you are using describe many people at once or one person's experience through time?