On some topological properties of stable measures (Q2564224)

The object of study in a class of probability measures, called stable measures, on the measurable space \((\times^\infty_{t=1} \mathbb{R}^K,{\mathcal B}(\times^\infty_{t=1} \mathbb{R}^K))\), where \({\mathcal B}(\times^\infty_{t=1} \mathbb{R}^K)\) are the Borel sets. These measures, introduced by \textit{M. Kurz} [Econ. Theory 4, No. 6, 859-876 (1994; Zbl 0811.90021)], have as defining property that they fulfill the pointwise ergodic theorem, not for all \(L^1\) function but for indicator functions of cylinder sets only. The set of ergodic stable measures can be partitioned into sets \(A(\overline{\mu})\) where \(\overline{\mu}\) on \((\times^\infty_{t=1} \mathbb{R}^K,{\mathcal B}(\times^\infty_{t=1} \mathbb{R}^K))\), is the (stationary) empirical distribution: \(\mu\) and \(\nu\) belong to \(A(\overline{\mu})\) if for all cylinder sets \(C\), we have \(\frac{1}{N} \sum^{N-1}_{j=0} 1_c(T^jx)\to \overline{\mu}(C)\) \(\mu\)-a.s. and \(\nu\)-a.s.\ . The topological properties of the set of stable measures and the sets \(A(\overline{\mu})\) are studied. These sets are closed in the topology of the sup-norm but not in the topology of weak convergence. Next, the paper introduces sets of uniformly stable measures for which the rate of convergence of the empirical distribution is uniform (a weaker requirement is used when the support of the empirical distribution is not countable). These sets are shown to be closed also in the topology of weak convergence. The paper then turns to study a class of stable probability measures called, SIDS measures, introduced in a previous paper of \textit{C. K. Nielsen} [Econ. Theory 8, No. 3, 399-422 (1996; Zbl 0859.90031)]. They are the countably infinite product of a realization of a stationary stochastic process on a countable set of probability measures on the measurable space \((\mathbb{R}^K,{\mathcal B}(\mathbb{R}^K))\). It is shown using results from \textit{F. Topsøe} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 14, 239-250 (1970; Zbl 0185.46701)] that the set of all SIDS measures in \(A(\overline{\mu})\) is the union of an increasing sequence of sets of uniformly stable SIDS measures when the support of the empirical distribution is countable. In the uncountable case a similar result is obtained when the individual probability measures of the infinite product have bounded support and are absolutely continuous with respect to Lebesgue measure.