TY - JOUR
T1 - Entropy and recurrence rates for stationary random fields
AU - Ornstein, Donald
AU - Weiss, Benjamin
PY - 2002/6
Y1 - 2002/6
N2 - For a stationary random field {x(u): u ∈ ℤd}, the recurrence time Rn(x) may be defined as the smallest positive k, such that the pattern {x(u): 0 ≤ ui < n} is seen again, in a new position in the cube {0 ≤ |ui| < k}. In analogy with the case of d = 1, where the pioneering work was done by Wyner and Ziv, we prove here that the asymptotic growth of Rn(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.
AB - For a stationary random field {x(u): u ∈ ℤd}, the recurrence time Rn(x) may be defined as the smallest positive k, such that the pattern {x(u): 0 ≤ ui < n} is seen again, in a new position in the cube {0 ≤ |ui| < k}. In analogy with the case of d = 1, where the pioneering work was done by Wyner and Ziv, we prove here that the asymptotic growth of Rn(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.
KW - Countable alphabet
KW - Entropy
KW - Random fields
KW - Recurrence times
KW - Shannon-McMillan-Breiman (SMB) theorem
UR - http://www.scopus.com/inward/record.url?scp=0036612006&partnerID=8YFLogxK
U2 - 10.1109/TIT.2002.1003848
DO - 10.1109/TIT.2002.1003848
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AN - SCOPUS:0036612006
SN - 0018-9448
VL - 48
SP - 1694
EP - 1697
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 6
ER -