A subsequence principle in probability theory.

*(English)*Zbl 0571.60028A 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 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.

The development of the theory is described, especially the fundamental results of 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.

Reviewer: A.J.Stam