Redundancy rates for renewal and other processes

Upper and lower bounds, both of order √n, are obtained on minimax redundancy of universal lossless codes for the class of renewal processes. This is the first example of an interesting model class with strong redundancy rate o(n) but not O(log n). For the same class, the nonexistence of weak-rate bounds of smaller order than √n is also shown. The methods extend to provide upper and lower redundancy rate bounds of order n^(k+1)/(k+2) for the class of processes that are Markov renewal of order k. The weak-rate methods also extend to show the nonexistence of o(n) weak-rate bounds for the class of regenerative processes