Fat tails preclude ergodicity
Fat tails preclude ergodicity because estimation of the various moments of the distribution will depend upon the specific time period and will tend not to converge to stable values even with increasingly large time periods.
For example, a sliding window view of the realizations of a random variable over time will observe very different means across windows. Without a large realization, the mean will tend to decay, then jump up when a large value is realized. Sample means change with sample size in fat-tailed distributions
This jumpiness precludes ergodicity — the mean will be unstable through time, which eliminates the timelessness of ergodic processes.
Time is what prevents everything from happening at once. To simply assume that economic processes are ergodic and concentrate on ensemble averages – and a fortiori in any relevant sense timeless – is not a sensible way for dealing with the kind of genuine uncertainty that permeates open systems such as economies. – Randomness, Fat Tails and Ergodicity – A Keynesian Perspective on Knightian Uncertainty
Fat tails should be expected because Interaction generates non-normality and interactions are quite common in the real world.
If Fat Tails should be expected and fat tails preclude ergodicity, then it follows that ergodicity should not be expected. In the real world, ergodicity is the exception, not the rule.
Tails should not be unexpected, for they are the rule. As the world becomes increasingly integrated – financially, economically, socially – interactions among the moving parts may make for potentially fatter tails. Catastrophe risk may be on the rise. – Randomness, Fat Tails and Ergodicity – A Keynesian Perspective on Knightian Uncertainty
Despite the empirical rarity of true Ergodicity, much of financial and economic theory implicitly imply Ergodicity in its assumptions and conclusions.