Markovization enlarges a state description until the conditional law of the next state depends only on the present enlarged state. For a second-order process with transition probabilities , define . The transition from goes to with probability .
A stationary sequence cycling through with uniformly random initial phase illustrates the lost information: after the current value zero, the next value depends on whether the preceding value was zero or one. The pairs follow the deterministic cycle . This finite-memory construction is useful, but a general process need not admit an economical finite-dimensional Markovization; retaining the entire history can be formally possible and computationally unhelpful.
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