We previously introduced the independence limit theorem, within the context of directional, stochastic chains with memory.1 For the early proof, we considered that partition functions for two different stochastic chains with memory do not strongly depend on the present. In the following, we prove this theorem for any configuration, i.e., for configurations that include any present and past events.
To herein discuss on a self-contained material, we bring in the necessary equations and include the theorem statement before its demonstration.
Let ν be a sequence of objects, x1, …, xn, that stem from a multivariate random variable X,
Their alphabet or domain will be denoted by . The energy of state is
where the partial energy E(xi; xi−1, …, x1) is the energy of object xi, provided that the previous objects (which constitute a partial sequence) are (x1, …, xi−1).
The equilibrium partition function reads as follows:
where β = 1/kT, k is the Boltzmann constant, and T is the absolute temperature; is the number of configurations, which is the result of combining n events and possibilities for each event.
The two-sequence energy is
and the sequence-dependent partition function is given by
(Independence limit). Let ν be a stochastic, linear chain with memory [Eq. (1)] sequentially constructed i: 1 → n. Let Z and Zν be the normal (equilibrium) and the sequence-dependent partition functions [Eqs. (3) and (5) and Eqs. (7) and (8), respectively], and let and be the energies of object relative to two different partial sequences [Eq. (2)]. If the normalized energy difference |Ei − |/kT → 0, ∀i, then Z/Zν → 1.