but not all, of any particular time period under study. They can therefore miss some of the largest deviations from the standard model, such as financial crises, debt crises and systemic risk in the banking system that occur only infrequently. Nassim Nicholas Taleb has argued that a very important part of empirical reality in finance and investment is non-ergodic. An even statistical distribution of probabilities, where the system returns to every possible state an infinite number of times, is simply not the case we observe in situations where “absorbing states" are reached, a state where ruin is seen. The death of an individual, or total loss of everything, or the devolution or dismemberment of a nation state and the legal regime that accompanied it, are all absorbing states. Thus, in finance, path dependence matters. A path where an individual, firm or country hits a "stop"—an absorbing barrier, "anything that prevents people with skin in the game from emerging from it, and to which the system will invariably tend. Let us call these situations ruin, as the entity cannot emerge from the condition. The central problem is that if there is a possibility of ruin, cost benefit analyses are no longer possible."—will be non-ergodic. All traditional models based on standard probabilistic statistics break down in these extreme situations. In Social Sciences In the social sciences the ergodicity concept surfaces as group level data often gives a poor indication of individual level variation, as individual standard deviations (SDs) tend to be almost eight times larger than group level SDs of the same people. Subsequently a third of the individual observations falls outside a 99.9% confidence interval of group level data. Definition for discrete-time systems Formal definition Let be a measurable space. If is a measurable function from to itself and a probability measure on then we say that is -ergodic or is an ergodic measure for if preserves and the following condition holds: For any such that either or . In other words there are no -invariant subsets up to measure 0 (with respect to ). Recall that preserving (or being -invariant) means that for all (see also measure-preserving dynamical system). Note that some authors (eg, "An introduction to infinite ergodic theory" by Anderson, p. 21) relax the requirement that is -invariant to the requirement that pullbacks of measure-zero sets are measure-zero, i.e., the pushforward measure is singular with respect to .