Generic Points for Stationary Measures via Large Deviation Theory

#### J.T. Lewis, C.-E. Pfister, W.G. Sullivan

1999, v.5, Issue 3, 235-267

ABSTRACT

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