Nonuniform Distribution of Entropy of Processes with a Countable Set of States

In an ergodic stationary (in the narrow sense) process with a finite alphabet, the modulus of the logarithm of the probability of a word divided by its length tends with probability unity to the same constant (specifically, to the entropy) as the length increases. This fact is valid if the alphabet is infinite but has finite entropy. The article constructs a stationary (in the narrow sense) ergodic process with an alphabet whose entropy is infinite and for which the sequence of random quantities made up of the moduli of the logarithms of the probabilities of words of fixed length divided by this length does not have a limiting distribution as the word length tends to infinity.