Reduced-memory likelihood processing of point processes

The problems of reduced-memory modeling and processing of regular point processes are studied. The
m
-memory processes and processors are defined as those whose present (incremental) behavior depends only on the present observation of counts and the stored values of the preceding
m
instants of occurrence. Characterization theorems for
m
-memory point processes and homogeneous reduced-memory point processes are obtained. Under proper optimization criteria, optimal reduced-memory "moving-window" information processors for point processes are derived. The results are applied to study reduced-memory processors for doubly stochastic Poisson processes (DSPP's) and to characterize
m
-memory DSPP's. Finally, a practically implementable scheme of a distribution-free l-memory processor is presented.