= Markovization
<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 $p(c\mid a,b)$, define $Y_n=(X_{n-1},X_n)$. The transition from $(a,b)$ goes to $(b,c)$ with <probability> $p(c\mid a,b)$.
A stationary sequence cycling through $0,0,1$ 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 $00\to01\to10\to00$. 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.
Back to article page
