Independence in Probability

Probability theory is measure theory specialized by assumptions having to do with stochastic independence. Delete from probability and statistics those theorems that explicitly or implicitly (e.g., by postulating a random sample) invoke independence, and relatively little remains. Or at tempt to estimate probabilities from data without assuming that at least certain observations are independent, and little results. Everyone who has worked with or applied probability is keenly aware of the importance of stochastic independence; experimenters go to some effort to ensure, and to check, that repeated observations are independent. Kolmogorov (1933, 1950) wrote: