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,

(1)

Their alphabet or domain will be denoted by $X$. The energy of state $ν∈Xn$ is

(2)

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:

(3)
(4)
(5)

where β = 1/kT, k is the Boltzmann constant, and T is the absolute temperature; $N=Xn$ is the number of configurations, which is the result of combining n events and $|X|$ possibilities for each event.

The two-sequence energy is

(6)

and the sequence-dependent partition function is given by

(7)
(8)

Theorem

(Independence limit). Let ν be a stochastic, linear chain with memory [Eq. (1)] sequentially constructed i: 1 → n. Let Z and Zνbe the normal (equilibrium) and the sequence-dependent partition functions [Eqs. (3) and (5) and Eqs. (7) and (8), respectively], and let$Ei=Exi′;xi−1′,…,x1′$and$Ei′=Exi′;xi−1,…,x1$be the energies of object$xi′$relative to two different partial sequences [Eq. (2)]. If the normalized energy difference |Ei$Ei′$|/kT → 0, ∀i, then Z/Zν → 1.

We want to show that
(9)
for mild memory effects. We next provide two alternative demonstrations.

Proof 1.
For mild memory effects, we can express the partial energies as follows:
where $ϵxi′;xi−1′,xi−1,…,x1′,x1$ is the difference between energies associated with different, partial sequences ($xi−1′$, …, $x1′$) and (xi−1, …, x1).
For all i, $ϵxi′;xi−1′,xi−1,…,x1′,x1$ fulfills
(10)
(11)
Then,
where $ϵν′ν≡∑i=1nϵ(xi′;xi−1′,xi−1,…,x1′,x1)$ is a small two-sequence energy. Consequently,
(12)
where
(13)
can be interpreted as a probability in ν′ with free parameter ν. Certainly, 0 ≤ pν(ν) ≤ 1, ∀ν′, ν = 1, …, N, and $∑ν′=1Npν′(ν)=1$, ∀ν = 1, …, N. We now expand $exp−βϵν′ν$ in the limit βϵ ($xi′$; $xi−1′$, xi−1, …, $x1′$, x1) → 0 [see Eqs. (10) and (11)],
(14)
Then,
(15)
where we have introduced $εν≡−∑ν′=1Npν′(ν)βϵν′ν$, a small number. Indeed, $εν≤∑ν′=1Nβϵν′ν→0$ for βϵνν → 0; we have used the following inequalities:
with aj, bj, and pj being real numbers and 0 ≤ pj ≤ 1, ∀j = 1, …, m. Therefore, εν → 0, which concludes the proof of Eq. (9) and the theorem.

Proof 2.
For mild memory effects, we express $xi′$, i = 1, …, n,
(16)
Then, $βEν′−Eν′ν$
(17)
The difference between partition functions is as follows:
(18)
To continue, we prove the following inequality:
(19)
which arises from the remainder in Taylor’s theorem for f(z) = ez. Indeed, since f(z) = f(0) + f′(c)z, for |c| ≤ |z|, it follows that ez = 1 + ecz, thus making |ez − 1| = |ecz| = ec|z| ≤ e|z||z|.
We now use this inequality in Eq. (18) followed by Eq. (17). We obtain
Therefore,
which concludes the proof.

1.
J. R.
Arias-Gonzalez
,
J. Chem. Phys.
145
,
185103
(
2016
).