On the distribution of recurrence times and the exact asymptotics of Lempel-Ziv coding

Let x=(...,x_-1,x₀,x₁,...) be a realization from the discrete, stationary ergodic source X={X_n;n∈Z}. We investigate the asymptotics of the recurrence time R_n defined as the first time that the n-block x₁ⁿ=(x₁,x₂,..,x_n) recurs in the past of x. We provide a natural framework for deducing the exact asymptotic behaviour of R_n, and we extract from it precise information about the behaviour of the pointwise redundancy of an idealized version of Lempel-Ziv coding. Similar results are obtained for the waiting time W_n defined as the first time that the initial n-block x₁ⁿ from x occurs in an independent realization y