Not All Universal Source Codes Are Pointwise Universal

Let Cn be a lossless code for n-tuples over a finite alphabet and ln denote its length function. The code sequence {Cn} is said to be universal if for every stationary source X = (X1, X2, . . .) 1 n Eln(X ) −→ H(X), where X = (X1, . . . , Xn) and H(X) denotes the entropy rate of X. It is said to be pointwise universal if for every stationary and ergodic source, with probability one, 1 n ln(X ) −→ H(X). Pointwise universality implies universality. We show that the converse is not true by constructing a universal sequence of schemes that is not pointwise universal, hence establishing the latter as a stronger notion of universality.