The Analysis of Data, volume 1


8.12. Notes

More information on the topics in this chapter is available in (Feller, 1971), (Breiman, 1992), (Billingsley, 1995), (Ash, 1999), (Resnick, 1999), (Kallenberg, 2002). Our description follows most closely (Billingsley, 1995} and the first chapter of (Ferguson1996}. Applications of limit theorems in statistics are described in (Serfling, 1980), (Ferguson, 1996), (Var der Vaart, 1998), (DasGupta, 2008) and applications in information theory are described in (Shannon, 1948), (Cover, 2005}.

The theory described in this chapter has been complemented by additional important results in limit theory. The uniform strong law of large numbers (Ferguson, 1996) extends the strong law of large numbers to apply uniformly over a set of parameters. Berry-Essen theory explore the accuracy of the Gaussian approximation in the central limit theorem (Berry, 1941). Multiple extensions of the iid central limit theorem are available, for example in the research papers (Hoeffding, 1948), (Berk, 1973) (McLeish, 1974), (Romano, 2000) or the monograph (Davidson, 1994). A converse to Shceffe's theorem is described in (Boos, 1985), (Sweeting1986). The law of the iterated logarithm complements the law of large numbers and the central limit theorem in describing the behavior of a sum of iid RVs (Billingsley, 1995).