Entropy and recurrence rates for stationary random fields

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For a stationary random field {x(u): u /spl isin/ Z/sup d/}, the recurrence time R/sub n/(x) may be defined as the smallest positive k, such that the pattern {x(u): 0 /spl les/ u/sub i/ < n} is seen again, in a new position in the cube {0 /spl les/ |u/sub i/| < k}. In analogy with the case of d = 1, where the pioneering work was done by Wyner and Ziv (1989), we prove here that the asymptotic growth of R/sub n/(x) for ergodic fields is given by the entropy of the random field. The nonergodic case is also treated, as well as the recurrence times of central patterns in centered cubes. Both finite and countable state spaces are treated.