A universal compression code can act as an estimator for the distribution of a finite alphabet finite-order Markov source. This property of universal codes was exploited by Jacob and Bansal in [5] to propose a modification of the CUSUM test in order to solve the change detection problem when the post-change distribution is not known. The performance of this test was proven to be asymptotically optimal for a memoryless sources and class of sources with memory under Lorden's minimax formulation. This study was further extended in [9] where performance of the modified CUSUM for an i.i.d. setting under Lai's criterion involving a constraint on the probability of false alarm within a window (PFAW) was analyzed. In this paper, we introduce a modified version of the window limited CUSUM (WL-CUSUM) test by incorporating strongly universal code. We closely follow the work of Lai [8] in order to prove the asymptotic optimality for the test under the PFAW criterion. We further prove the asymptotic optimality of the modified WL-CUSUM test in the Bayesian setting.

We consider the problem of designing weakly universal codes for stationary and ergodic processes with countable alphabet and present a set of algorithms. First two algorithms use a combination of an integer coding algorithm and Lempel-Ziv algorithms (incremental parsing based algorithm and one based on recurrence times). Third algorithm converts the source into a finite alphabet process in step one through an integer coding algorithm and then uses LZ-78 in second step. Asymptotic optimality of all three is proved in full generality. We make use of Shannon-McMillan-Breiman theorem for countable alphabet and its extension for asymptotically mean stationary processes

Under weak restriction on the distortion measure the continuity of the. single letter distortion measure is established with respect to a suitable distance measure on the sequence space of elements drawn from the representation alphabet. This in turn translates into properties of the distortion rate function. Using subtleties about the behavior of the sequence distortion measure we develop a proof of convexity of the distortion rate function in full generality. A generalization of the optimality of Lempel Ziv algorithm with regard to finite state codes and stationary and ergodic sources with finite alphabet is presented for a more general class of universal codes applied to individual sequences.