Previous $i\;$ stages

Under this model, the history I-state is truncated. Any actions or observations received earlier than $i$ stages ago are dropped from memory. An I-map, ${\kappa}_i$, is defined as

$${\kappa}_i({\eta}_k) = (u_{k-i},\ldots,u_{k-1},y_{k-i+1},\ldots,y_k) ,$$ (11.41)

for any integer $i > 0$ and $k > i$. If $i \leq k$, then the derived I-state is the full history I-state, (11.14). The advantage of this approach, if it leads to a solution, is that the length of the I-state no longer grows with the number of stages. If $X$ and $U$ are finite, then the derived I-space is also finite. Note that ${\kappa}_i$ is sufficient in the sense defined in Section 11.2.1 because enough history is passed from stage to stage to determine the derived I-states.