Independent Statistics

Independent statistics are random variables whose joint distribution is the product measure of their marginal distributions. For two real random variables and , this is equivalent to

for all real numbers and (Durrett 2019, p. 47).

When and have finite second moments, independence implies that their covariance is zero, so they are uncorrelated. The converse fails even when all moments exist (Durrett 2019, pp. 48-49). For example, if has a uniform distribution on and , then , so their covariance is zero. They are dependent, since the events and are identical and have probability , rather than the product .