Random f-expansions
A stationary process \(\{\phi_ n\}^{\infty}_{n=0}\) defined on a probability space (\(\Omega\),A,P) is said to be continued fraction mixing if there exists a sequence \(\{\epsilon_ n\}\) of nonnegative real numbers with \(\lim_{n\to \infty}\epsilon_ n=0\) such that, for all \(m,n\in N\), \(A\in \sigma (\{\phi_ k:k\leq m\})\) and \(B\in \sigma (\{\phi_ k:k\geq m+n\}),\) \[ | P(A\cap B)-P(A)\cdot P(B)| \leq \epsilon_ n\cdot P(A)\cdot P(B). \] The following generalization of a theorem of P. Lévy on continued fraction expansions is proved. Suppose that \(\{\phi_ n\}^{\infty}_{n=0}\) is a nonnegative, continued fraction mixing stationary process defined on (\(\Omega\),A,P) such that \(E(\phi_ 0)=\infty\), and that \(L(X)=E(\phi_ 0\wedge x)\) is regularly varying with index 1-\(\gamma\) as \(x\uparrow \infty\) \((0<\gamma \leq 1)\). Then \[ dist (b(n))^{-1}\sum^{n}_{k=1}\phi_ k\to dist Y_{\gamma}\quad as\quad n\to \infty, \] where b(n)\(\sim n\cdot L(b(n))\) and \(Y_{\gamma}\) is a nonnegative random variable with Laplace transform \(E(e^{-pY_{\gamma}})=e^{-\Gamma (2-\gamma)p^{\gamma}}\) \((p>0)\). The theorem is proved by showing that the base of the tower over the associated shift transformation with height function \([\phi_ 0]\) is a Darling-Kac set, defined as follows. Let (X,\({\mathcal B},m,T)\) be a conservative ergodic measure-preserving transformation on a \(\sigma\)-finite nonatomic measure space and let \(\hat T\) be the operator on \(L^ 1(X,B,m)\) dual to T. Then a set \(A\in B\), \(0<m(A)<\infty\), is called a Darling-Kac (DK) set for T if there are constants \(a_ n>0\) such that \[ \| a_ n^{- 1}\sum^{n}_{k=1}\hat T^ k1_ A-1\|_{L^{\infty}(A)}\to 0\quad as\quad n\to \infty. \] For DK sets an asymptotic renewal equation and a distributional limit theorem for the number of visits (due to D. A. Darling and M. Kac) hold. These are the basic properties of DK sets used in the proof of the theorem. The main lemma of the paper gives sufficient conditions for a set to be a DK set in terms of mixing properties of the appertaining induced transformation. The theorem is first applied to f-expansions with finite invariant measures, where the crucial mixing property follows from a theorem of R. L. Adler. By developing results of the reviewer on interval maps with indifferent fixed points the author also obtains distributional limit theorems for f-expansions with infinite invariant measures. To mention a special case, let \(T(x)=1-((1/x))\), \(\phi (x)=[1/x]+1\), \(\phi_ k(x)=\phi (T^ kx)\) (x\(\in (0,1)\), \(k\geq 0)\). Then (1/n)\(\sum^{n-1}_{k=0}\phi_ k\to 3\) in measure.
scientific article; zbMATH DE number 4076192