Circular representation of infinitely extended sequences

All data in a space-ordered or time-ordered series are always observed for a finite distance or time. A sample variance of the data represents a formalized means of capturing the extent of observed variations over a finite interval. Different types of variances abound and for the existence of a “true” particular variance, assumptions are made regarding ergodicity, stationarity and statistical independence of the random variables. This writing is not about variance per se but rather about the fact that proper estimations of variance come from understanding the implications of these basic assumptions. In particular, there is usually an underlying assumption of ergodicity. Ergodicity means that we treat one statistical average as an ensemble of smaller statistical averages. If we use the ergodic assumption within a data set, then we acknowledge its generalization as an included assumption for data outside the set, namely before and after, in the case of a time series. Ergodicity implies that any series will have a likelihood of recurrence which inversely depends on the number of independent observations. If only one series is ever observable, absolute recurrence of that series is the consequence under the implied assumption of ergodicity and an assumption that a “true” variance indeed exists. This yields the model that time-ordered data may be treated as circular or wrapped