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On the central limit question under \(\rho\)-mixing and information regularity. (English) Zbl 1024.60010

Berkes, I. (ed.) et al., Limit theorems in probability and statistics. Fourth Hungarian colloquium on limit theorems in probability and statistics, Balatonlelle, Hungary, June 28-July 2, 1999. Vol I. Budapest: János Bolyai Mathematical Society. 235-248 (2002).
Let \((\Omega,{\mathcal F},P)\) be a probability space. For any two \(\sigma\)-fields \({\mathcal A}\) and \({\mathcal B}\subset {\mathcal F}\), define the “maximal correlation” \(\rho({\mathcal A},{\mathcal B})= \sup|\text{corr}(f,g)|\), where the supremum is taken over all pairs of square-integrable random variables \(f\) and \(g\) which are \({\mathcal A}\)-measurable and \({\mathcal B}\)-measurable, respectively. Further, define the “coefficient of information” \[ I({\mathcal A},{\mathcal B})=\sup \sum^I_{i=1} \sum^J_{j=1}P (A_i\cap B_j)\log {P(A_i\cap B_j)\over P(A_i)P (B_j)}, \] where the supremum is taken over all pairs of partitions \(\{A_1, \dots, A_I\}\) and \(\{B_1, \dots, B_J\}\) of \(\Omega\) such that \(A_i\in{\mathcal A}\) for each \(i\) and \(B_j\in{\mathcal B}\) for each \(j\). For a strictly stationary sequence \(\{X_k, k\in Z\}\) define the dependence coefficients \[ \rho(n)= \rho\bigl( \sigma (X_k,k \leq 0),\sigma (X_k,k\geq n)\bigr),\quad I(n)=I\bigl( \sigma(X_k,k \leq 0),\sigma(X_k,k \geq n)\bigr). \] We say that the sequence is “\(\rho\)-mixing” if \(\rho(n) \to 0\) and “information regular” if \(I(n)\to 0\) as \(n\to \infty\). The author constructs an example of a strictly stationary sequence of real-valued random variables with \(EX_0=0\) and \(EX_0^2 <\infty\) which is both \(\rho\)-mixing and information regular, but fails to satisfy a central limit theorem.
For the entire collection see [Zbl 1009.00022].

MSC:

60F05 Central limit and other weak theorems
60G10 Stationary stochastic processes
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