Lumpable

Lumpable refers to a property of a stochastic process, typically a finite Markov chain, that allows aggregation of states without losing the Markov property. A chain is lumpable with respect to a partition of its state space into blocks Bi if the distribution of the next state depends only on the current block, not on the specific state within the block. Equivalently, for all blocks Bi and for all states x, y in Bi and for all blocks Bj, the probability that Xt+1 falls in Bj given Xt = x equals the corresponding probability given Xt = y. When this holds, the process Yt that records the block Bi containing Xt is itself a Markov chain with a transition matrix P* whose entry P*(i,j) equals that common probability.

A simple way to view it is: if all states within a block behave identically with respect to transitions to other blocks, you can treat the entire block as a single aggregated state. This enables a reduced model that preserves the Markov property and often simplifies analysis and computation.

Examples typically involve symmetry or structure in the transition dynamics. For instance, in a chain with two blocks, the conditional probabilities from any state in the first block to the second block are all the same, and similarly for transitions within or to the first block.

Applications include model reduction in stochastic modeling, queuing networks, reliability engineering, and population genetics, where large state spaces are common but grouping is natural. Limitations include the fact that not every partition is lumpable; verifying lumpability requires checking the transition probabilities within each block. If exact lumpability does not hold, approximate lumping or projection methods may be used.