Generic Points for Stationary Measures via Large Deviation Theory
1999, v.5, Issue 3, 235-267
The construction of generic points by concatenation is discussed in the natural setting of Large Deviation Theory. Our main result is the construction of generic points for any stationary $k$-Markov measure $\alpha$, using only the $(k+1)$-marginals of $\alpha$ (Section 4). This construction is based on the notion of LD-regular sequences and an improvement of results in [J.T. Lewis, C.-E. Pfister and W.G. Sullivan, Entropy, concentration of probability and conditional limit theorems. Markov Processes Relat. Fields, 1995, 1, 319-386] about large deviations of conditioned measures (Section 6). The first part of the paper provides motivation for the concatenation method coming from our recent study of Asymptotic Equipartition Property [J.T. Lewis, C.-E. Pfister, R. Russel, W.G. Sullivan, Reconstruction sequences and equipartition measures: an examination of the asymptotic equipartition property. IEEE Inform. Theory, 1997, 43, 1935-1947].
Keywords: large deviations,ergodic theory,entropy,generic points,normal numbers