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Paulo
Paulo
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In the case of (weak) Gibbs measures you can find exponential large-deviation bounds as well, as almost sure estimates for the error term in the convergence to entropy given by the Shannon–McMillan–Breiman formula for both uniformly and non-uniformly expanding shift spaces [here][1][ETDS, 37:7, (2017), 2313-2336]. For broader classes of invariant measures, in opposition to the case of Birkhoff averages, I belive it is not known reasonably mild assumptions under which e.g. some L^1 convergence holds.

In the case of (weak) Gibbs measures you can find exponential large-deviation bounds as well, as almost sure estimates for the error term in the convergence to entropy given by the Shannon–McMillan–Breiman formula for both uniformly and non-uniformly expanding shift spaces [here][1]. For broader classes of invariant measures, in opposition to the case of Birkhoff averages, I belive it is not known reasonably mild assumptions under which e.g. some L^1 convergence holds.

In the case of (weak) Gibbs measures you can find exponential large-deviation bounds as well, as almost sure estimates for the error term in the convergence to entropy given by the Shannon–McMillan–Breiman formula for both uniformly and non-uniformly expanding shift spaces [ETDS, 37:7, (2017), 2313-2336]. For broader classes of invariant measures, in opposition to the case of Birkhoff averages, I belive it is not known reasonably mild assumptions under which e.g. some L^1 convergence holds.

Source Link
Paulo
Paulo
  • 11
  • 2

In the case of (weak) Gibbs measures you can find exponential large-deviation bounds as well, as almost sure estimates for the error term in the convergence to entropy given by the Shannon–McMillan–Breiman formula for both uniformly and non-uniformly expanding shift spaces [here][1]. For broader classes of invariant measures, in opposition to the case of Birkhoff averages, I belive it is not known reasonably mild assumptions under which e.g. some L^1 convergence holds.