A subsequence principle in probability theory
A survey paper by an author who contributed a large part of the theory of the so-called subsequence principle. Let (\(\Omega\),\(\Sigma\),\(\mu)\) be a measure space and \(g_ n\), \(n\in {\mathbb{N}}\), a sequence of measurable functions \(\Omega\) \(\to {\mathbb{R}}\). Under certain conditions there is a subsequence of \(\{g_ n\}\) behaving as a sequence of exchangeable or even i.i.d. random variables. E.g. if \(\int | g_ n| dP\leq M\) there is a subsequence \(\{f_ n\}\) of \(\{g_ n\}\) such that any subsequence of \(\{f_ n\}\) satisfies the strong law of large numbers. If \(\int | g_ n|^ 2dP\leq M\), a similar result holds with central limit theorem and log log law. The development of the theory is described, especially the fundamental results of \textit{D. J. Aldous}, see the foregoing review, Zbl 0571.60027. The connection with gap theorems and results on functions \(g_ n(x)=f(\lambda_ nx)\), \(\Omega ={\mathbb{R}}\), \(\lambda_{n+1}/\lambda_ n\geq q>1\), is described. The state of the problem for strongly measurable \(g_ n:\Omega \to E\) a Banach space, is reviewed.
scientific article; zbMATH DE number 3911383
scientific article