Invariance under dependence by mixing
D. R. Jensen · Lecture notes-monograph series · 1990
This paper is concerned with arrays of conditionally independent random elements that become dependent by mixing.The principal focus is the preservation of properties known to hold under independence.Findings are reported in the context of limit theory, including laws of large numbers and central limit theory, and topics in statistical inference.Several standard results, ranging from Berry-Esseen bounds in central limit theory to the use of Friedman's (1937) test in the analysis of two-way data, are seen to remain valid under certain models for dependence.The class of limit laws for standardized sums is expanded to include dependent cases, as are bounds on rates of convergence to these limits. Introduction.Independence and conditional independence are central to probability theory and its applications, supporting the theory of Markov chains, Bayesian analysis, limit theory, and the foundations of statistical inference.See Dawid (1979), for example.A systematic study of conditionally independent events dates back at least to de Finetti (1937), who postulated these as models for conditional independence in a random environment.More recently, conditionally independent events have been studied as models for weak dependence (cf.Dykstra et al. (1973), Shaked (1977), and Tong (1980), for example), but there the primary focus centers on inequalities relating unconditional joint probabilities to products of marginal probabilities.The assumption of independence pervades much of mathematical statistics, including essential portions of parametric and nonparametric statistical inference.That independence is a fundamental mathematical concept is not in doubt.Much less clear is the extent to which it mimics reality.Indeed, apart from highly specialized models, there appear to be no omnibus empirical tests for genuine independence.On physical grounds it even may be argued that observable phenomena at best can be only conditionally independent, owing to the common background energy attributed to the big bang.