# Yet again on conditions for Dobrushin's Central limit theorem for non-homogeneous Markov chains

Alexander Veretennikov, Aisha Nurieva

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Alexander Veretennikov, Aisha Nurieva. Yet again on conditions for Dobrushin's Central limit theorem for non-homogeneous Markov chains. 2025. hal-05321900

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# Yet again on conditions for Dobrushin's Central limit theorem for non-homogeneous Markov chains

A.Yu. Veretennikov*, A.I. Nurieva†

October 20, 2025

## Abstract

Yet one more version of sufficient conditions is offered in the problem of Central Limit Theorem in the array scheme for non-homogeneous Markov Chains, with a possibility that Markov-Dobrushin's ergodic coefficient may not satisfy the "standard" Dobrushin's condition, even with $\alpha_n \equiv 0$.

**Keywords:** Dobrushin's CLT, Markov chains, ergodic coefficients, array scheme.

**MSC Classification:** 60F05

## 1 Introduction

R.L. Dobrushin established in 1956 an important central limit theorem (CLT in what follows) for non-homogeneous Markov chains in the array scheme [2]. It attracted much attention in the literature. This paper proposes one more extension of this celebrated result different from our recent preprints (arXiv:2406.16156 and arXiv:2506.07287). The first result in this direction by A.A. Markov [10] was followed by the papers [1, 8, 7] and some others, before the seminal publication [2]. In this paper Dobrushin finally established that the sufficient condition very close to necessary (under the additional condition of a non-degeneracy of variances, see the details in what follows) is

$$ \lim_{n \to \infty} n^{1/3} \alpha_n = \infty, $$

<footnote>*Institute for Information Transmission Problems, & RUDN University, Moscow, Russian Federation; alexander.veretennikov2011@yandex.ru</footnote>

<footnote>†National Research University Higher School of Economics, & Institute for Information Transmission Problems, & Institute for Information Transmission Problems, Moscow, Russian Federation; ai_nurieva@mail.ru</footnote>
where $\alpha_n$ is the Dobrushin, or Markov – Dobrushin’s ergodic coefficient (see (1) for the definition in the next section). The optimality of this value 1/3 was confirmed by the Bernstein – Dobrushin example (see [1, ch. 2] and [2]). Among extensions of this result, for Markov chains with memory, see the short communication [12]. In [13] (2005) Dobrushin’s results were re-proved by a simpler martingale method. The present paper uses the same technique with modifications, which result in a conclusion that, actually, $\alpha_n$ may be even identically equal to zero, but the CLT still holds, of course, under certain little additional modifications in the assumptions. We highlight that Dobrushin himself stated this option of a CLT with $\alpha_n = 0$ as a hypothesis in [2, Remark in the section 1.10]. The present paper offers a realisation of this hypothesis.

The motivation for new extensions is due to the basic role of the CLT in mathematical statistics.

The paper consists of 5 sections: Introduction; The setting; Main results, which are theorem 3 and corollary 4; Auxiliary results; Proof of the theorem.

## 2 The setting

Let us consider a non-homogeneous Markov chain on a probability space $(\Omega, \mathcal{F}, \mathcal{P})$. Let $\pi = \pi_{i,i+1}(x, dy)$ be the Markovian transition kernel on $(\mathbf{X}, \mathcal{B}(\mathbf{X}))$ at time $i$, where $(\mathbf{X}, \mathcal{B}(\mathbf{X}))$ is a measurable space with standard requirements on the Markov process (see [3]). The transition probabilities are given by the formula

$$P[X_{i+1} \in A | X_i = x] \stackrel{\text{a.s.}}{=} \pi_{i,i+1}(x, A).$$

In particular, if the initial distribution<fnref n="1" /> is $X_1 \sim \mu$ then the distribution at time $k \ge 2$ is given by the measure $\mu\pi_{1,2}\pi_{2,3} \dots \pi_{k-1,k}$.

For $i < j$ let us denote by $\pi_{i,i+k}$ the k-step transition kernel at time $i$, which has a representation due to the Chapman-Kolmogorov equation,

$$\pi_{i,j} = \pi_{i,i+1}\pi_{i+1,i+2}\dots\pi_{j-1,j}.$$

**Definition 1.** The MD (for Markov and Dobrushin) coefficient<fnref n="2" /> $\delta(\pi)$ is

<footnote n="1">¹Notice that here the initial time is 1, not 0.</footnote>

<footnote n="2">²In [13] it is called just the contraction coefficient.</footnote>
defined by any of the three equivalent expressions,

$$
\begin{align}
\delta(\pi) &= \sup_{\substack{x_1, x_2 \in \mathbf{X}, \\ A \in \mathcal{B}(\mathbf{X})}} |\pi(x_1, A) - \pi(x_2, A)| = \frac{1}{2} \sup_{\substack{x_1, x_2 \in \mathbf{X}, \\ \|f\|_B \le 1}} \left| \int f(y) [\pi(x_1, dy) - \pi(x_2, dy)] \right| \nonumber \\
&= \sup_{\substack{x_1, x_2 \in \mathbf{X}, \\ u \in \mathcal{U}}} |(\pi u)(x_1) - (\pi u)(x_2)|, \quad \text{where } \mathcal{U} = \{u(\cdot) : \sup_{y_1, y_2} |u(y_1) - u(y_2)| \le 1\}. \tag{1}
\end{align}
$$

Here $\|f\|_B = \sup_{x \in \mathbf{X}} |f(x)|$. Clearly, $0 \le \delta(\pi) \le 1$, and $\delta(\pi) = 0$ if and
only if the kernel $\pi(x, dy)$ does not depend on $x$. The kernel $\pi$ is called
non-degenerate if $\delta(\pi) < 1$. To denote integrals, standard notations will be
used,

$$
(\mu\pi)(A) = \int \mu(dx)\pi(x, A), \quad (\pi u)(x) = \int u(y)\pi(x, dy).
$$

Let for any function on $\mathbf{X}$,

$$
\mathrm{Osc}(u) := \sup_{x_1, x_2} |u(x_1) - u(x_2)|,
$$

and for any r.v. $X$ on $(\Omega, \mathcal{F}, \mathbf{P})$,

$$
\mathrm{Osc}(X) := \underset{\mathbb{P}}{\mathrm{ess\,sup}} \, X(\omega) - \underset{\omega \in \Omega}{\mathrm{ess\,inf}} \, X(\omega). \quad (2)
$$

Notice the following inequality for any pair of transition kernels $\pi_1, \pi_2$,

$$
\delta(\pi_1 \pi_2) \le \delta(\pi_1) \delta(\pi_2),
$$

where $\pi_1\pi_2$ is the two-step transition kernel where the first transition is ac-
cording to the matrix $\pi_1$, while the second one is according to $\pi_2$. Similarly,

$$
\delta(\pi_{i,j}) \le \delta(\pi_{i,i+1})\delta(\pi_{i+1,i+2}) \dots \delta(\pi_{j-1,j}).
$$

Let us define the complementary to $\delta$ coefficient by

$$
\alpha(\pi_{i,j}) := 1 - \delta(\pi_{i,j}).
$$

In what follows, EY and D[Y] stand for the expectation and for the variance
of the random variable Y, respectively.

In the array scheme, for any $n \ge 1$ by $\{X_i^{(n)} : 1 \le i \le n\}$ we denote
$n$ values of the Markov chain on $\mathbf{X}$ with the transition kernels $\{\pi_{i,i+1}^{(n)} =$
$\pi_{i,i+1}^{(n)}(x, dy) : 1 \le i \le n-1\}$ and with the initial distribution $\mu^{(n)}$. For
simplicity, it is assumed that all processes $(X^{(n)})$ are defined on the same probability space $(\Omega, \mathcal{F}, \mathcal{P})$. Let

$$ \alpha_n := \min_{1 \le i \le n-1} \alpha(\pi_{i,i+1}^{(n)}). $$

This coefficient plays the main role in Dobrushin's paper [2]. Further, let $\{f_i^{(n)} : \mathbf{X} \mapsto \mathbb{R}, 1 \le i \le n\}$ be real-valued functions on $\mathbf{X}$, and

$$ S_n := \sum_{i=1}^{n} f_i^{(n)}(X_i^{(n)}), \quad n \ge 1. $$

The main result of Dobrushin's paper along with the main corollary is stated in [13]<fnref n="3" /> in the following form; we call it a proposition so as to distinguish it from our result, which is called a theorem. The notation “$\implies$” stands for weak convergence.

**Proposition 1.** Let $C_n := \sup_{1 \le i \le n} \sup_{x \in \mathbf{X}} |f_i^{(n)}(x)| < \infty$. In this case, if

$$ \lim_{n \to \infty} C_n^2 \alpha_n^{-3} \left[ \sum_{i=1}^{n} \mathbf{D}(f_i^{(n)}(X_i^{(n)})) \right]^{-1} = 0, \quad (3) $$

then the CLT holds true in the array scheme:

$$ \frac{S_n - \mathbf{E}(S_n)}{\sqrt{\mathbf{D}(S_n)}} \implies \mathcal{N}(0, 1). \quad (4) $$

**Corollary 2.** Under the assumptions of the theorem, if $\sup_n C_n = C < \infty$, and $\inf_{i,n} \mathbf{D}(f_i^{(n)}(X_i^{(n)})) \ge c > 0$, and if

$$ \lim_{n \to \infty} n^{1/3} \alpha_n = \infty, \quad (5) $$

then weak convergence (4) holds.

# 3 Main results

Notice the following elementary inequalities:

$$ \delta(\pi_{i,j}) \le (1 - \alpha_n)^{j-i}. \quad (6) $$

Let for any $n$ exist two non-random sequences of ones and zeros $(\beta_k^{(n)}, 1 \le k \le n-1)$ and $(\gamma_k^{(n)}, 2 \le k \le n-1)$, both like $(110010...01)$, which will play a major role in what follows; the link between them is that

$$ \gamma_k^{(n)} = 1 \iff \beta_k^{(n)} \beta_{k-1}^{(n)} = 1. \quad (7) $$

<footnote n="3">³Dobrushin's original theorem is stated and proved for $S_n = \sum_{i=1}^{n} X_i^{(n)}$.</footnote>
Let for $1 \le i \le j \le n$,

$$ \kappa_n^{\beta} = \sum_{k=1}^{n} \beta_k^{(n)}, \quad \kappa_{n,j}^{\beta} = \sum_{k=1}^{j} \beta_k^{(n)}, \quad \kappa_{n,i,j}^{\beta} = \sum_{k=i}^{j} \beta_k^{(n)}, \quad \Gamma_n = \sum_{i=2}^{n-1} \gamma_i^{(n)}. $$

Denote

$$ \alpha_n^{\beta} := \min_{1 \le i \le n-1: \beta_i^{(n)} = 1} \alpha(\pi_{i,i+1}^{(n)}), \quad \alpha_n^{\gamma} := \min_{2 \le i \le n-1: \gamma_i^{(n)} = 1} (\alpha(\pi_{i,i+1}^{(n)}) \wedge \alpha(\pi_{i-1,i}^{(n)})). $$

Then<fnref n="4" /> (6) may be relaxed to

$$ \delta(\pi_{i,j}) \le (1 - \alpha_n^{\beta})^{\kappa_{n,i,j}^{\beta}}. \qquad (8) $$

**Definition 2.** We say that the condition $(H_\beta)$ is met if there exist $m_0 > 0$ and $c > 0$ such that for any $n$ the sequence $(\beta_k^{(n)}, 1 \le k \le n-1)$ satisfies the following property: for any pair $(i < j)$ with $j - i \ge m_0$, the inequality

$$ k_j^{\beta} - k_i^{\beta} \ge c(j - i) \qquad (9) $$

is satisfied.

In what follows such a sequence $(\beta_k^{(n)}, 1 \le k \le n-1)$ is fixed for each $n$. The next theorem is the main result of the paper. Both the theorem and corollary allow the case $\alpha_n = 0$ for every $n \ge 1$.

**Theorem 3.** Let the assumption $(H_\beta)$ be satisfied, and let for any $n$

$$ \sup_{1 \le i \le n} \sup_{x \in \mathbf{X}} |f_i^{(n)}(x)| \le C_n < \infty. $$

If conditions

$$ \lim_{n \to \infty} \Gamma_n = \infty, \qquad (10) $$

and

$$ \lim_{n \to \infty} C_n^2 (\alpha_n^\gamma)^{-1} (\alpha_n^\beta)^{-2} \left[ \sum_{i=2}^{n-1} \gamma_i^{(n)} \mathbf{D}(f_i^{(n)}(X_i^{(n)})) \right]^{-1} = 0, \qquad (11) $$

are met, then the CLT holds true,

$$ \frac{S_n - \mathbf{E}(S_n)}{\sqrt{\mathbf{D}(S_n)}} \implies \mathcal{N}(0, 1). \qquad (12) $$

<footnote n="4">⁴Roughly, $\beta_i^{(n)} = 1$ means $\alpha(\pi_{i,i+1}) > 0$, while $\gamma_i^{(n)} = 1$ indicates $\alpha(\pi_{i,i+1}) \wedge \alpha(\pi_{i-1,i}) > 0$.</footnote>
**Corollary 4.** If condition (H_β) is satisfied, sup_n C_n = C < ∞, and if there exists c > 0 such that

$$
\inf_n \inf_{i: \gamma_i^{(n)} = 1} (\mathbf{D}(f_i^{(n)}(X_i^{(n)})) \wedge \mathbf{D}(f_{i+1}^{(n)}(X_{i+1}^{(n)}))) \geq c > 0, \quad (13)
$$

and if the following condition holds

$$
\lim_{n \to \infty} \alpha_n^{\gamma}(\alpha_n^{\beta})^2 \Gamma_n = \infty. \quad (14)
$$

Then the CLT (12) is valid.

NB: No assumption on α_n is imposed; it may even vanish.

4 Auxiliary results

In this section, auxiliary results are presented. Except for the lemmata 6,
10, and 12, they are all taken without changes from the papers [5] and [13];
they are cited without proofs. For the simple, yet essential for the proof of
the main result, the proofs of the lemma 10 and proposition 7 are presented
because they contain certain news in the calculi. We accept the notation

$$
\|Z\|_{L^{\infty}} := \operatorname*{ess\,sup}_{P} |Z(\omega)|. \quad (15)
$$

The proofs of the lemmata 6 and 12 may be found in the preprint [11].

**Proposition 5.** Let for any $n \ge 1$ the process $\{(W_i^{(n)} : 0 \le i \le n)\}$ be a martingale with respect to a filtration $(G_i^{(n)})$, $W_0^{(n)} = 0$, and let $\xi_i^{(n)} := W_i^{(n)} - W_{i-1}^{(n)}$. If

$$
\max_{1 \le i \le n} \|\xi_i^{(n)}\|_{L^\infty} \to 0, \quad \sum_{i=1}^{n} \mathrm{E}[(\xi_i^{(n)})^2 | G_i^{(n)}] \to 1 \quad \text{in } L^2 \quad (16)
$$

(i.e., $\max_{1 \le i \le n} \operatorname*{ess\,sup}_{P} |\xi_i^{(n)}(\omega)| \to 0$, and $\mathrm{E} \left| \sum_{i=1}^n \mathrm{E}[(\xi_i^{(n)})^2 | G_i^{(n)}] - 1 \right|^2 \to 0$),
then weak convergence takes place,

$$
W_n^{(n)} \implies \mathcal{N}(0, 1).
$$

See [5, Theorem 3.2], or [9, Theorem 5.5.8], and more general martingale
CLT may be found in [6, Ch. VII.5].

Everywhere in what follows, as in [13] and in order not to care about
centering, we will assume that the functions {$f_i^{(n)}$} are such that

$$
\mathrm{E}[f_i^{(n)}(X_i^{(n)})] = 0, \quad 1 \le i \le n, n \ge 1.
$$
Otherwise, the expectations may be just subtracted. Let

$$Z_k^{(n)} := \sum_{i=k}^{n} \mathbb{E}[f_i^{(n)}(X_i^{(n)})|X_k^{(n)}], \quad (17)$$

so that

$$Z_k^{(n)} = \begin{cases} f_k^{(n)}(X_k^{(n)}) + \sum_{i=k+1}^{n} \mathbb{E}[f_i^{(n)}(X_i^{(n)})|X_k^{(n)}], & 1 \le k \le n-1, \\ f_n^{(n)}(X_n^{(n)}), & k=n. \end{cases}$$

Notice that $\mathbb{E}Z_k^{(n)} = 0$. For any $1 \le k \le n-1$, the representation holds,

$$f_k^{(n)}(X_k^{(n)}) = Z_k^{(n)} - \mathbb{E}[Z_{k+1}^{(n)}|X_k^{(n)}]. \quad (18)$$

More than that, for any $2 \le k \le n-1$ the same expression may be equivalently represented as $(Z_k^{(n)} - \mathbb{E}[Z_k^{(n)}|X_{k-1}^{(n)}]) + (\mathbb{E}[Z_k^{(n)}|X_{k-1}^{(n)}] - \mathbb{E}[Z_{k+1}^{(n)}|X_k^{(n)}])$. It follows that the sum $S_n$ admits the following martingale representation,

$$S_n = \sum_{i=1}^{n} f_i^{(n)}(X_i^{(n)}) = Z_1^{(n)} + \sum_{k=2}^{n} [Z_k^{(n)} - \mathbb{E}[Z_k^{(n)}|X_{k-1}^{(n)}]].$$

As it is highlighted in [13], this crucial transformation was proposed in the important paper [4] devoted to a “simple” CLT for Markov chains<fnref n="5" />. Since all summands in the right-hand side of the latter formula are evidently non-correlated, we have,

$$\mathbb{D}(S_n) = \sum_{k=2}^{n} \mathbb{D}(Z_k^{(n)} - \mathbb{E}[Z_k^{(n)}|X_{k-1}^{(n)}]) + \mathbb{D}(Z_1^{(n)}), \quad (19)$$

where $\mathbb{D}(Z_1^{(n)}) = \mathbb{E}(Z_1^{(n)})^2$. Let

$$\xi_k^{(n)} = \frac{1}{\sqrt{\mathbb{D}(S_n)}} [Z_k^{(n)} - \mathbb{E}[Z_k^{(n)}|X_{k-1}^{(n)}]]. \quad (20)$$

Then the process $M_k^{(n)} = \sum_{\ell=1}^k \xi_\ell^{(n)}$ is a martingale with respect to the filtration $\mathcal{F}_k^{(n)} = \sigma\{X_\ell^{(n)} : 1 \le \ell \le k\}$ for $1 \le k \le n$. The proof of the CLT will be based on the approximation of the expression $\frac{S_n}{\sqrt{\mathbb{D}(S_n)}}$ by the random variable $M_n^{(n)}$ and due to a martingale CLT, as in [13], see the proposition 5 in what follows. Therefore, the task is to verify both conditions in (16). By $f_i^{(n)}\pi_{i,j}f_j^{(n)}$ we denote the product of the functions $f_i^{(n)}$ and $\pi_{i,j}f_j^{(n)}$.

Notice that for any $i < j$, $n$, and function $f_j^{(n)}$,

$$\mathrm{Osc}(\pi_{i,j}(f_j^{(n)})) \le \delta(\pi_{i,j}) \mathrm{Osc}(f_j^{(n)}). \quad (21)$$

<footnote n="5">⁵Notice, however, that [2] was published much earlier than [4].</footnote>
**Lemma 6.** Under the conditions of the theorem, for any $1 \le i \le j \le n$,

$$ \|\pi_{i,j} f_j^{(n)}\|_B \le 2C_n(1 - \alpha_n^\beta)^{\kappa_j^\beta - \kappa_i^\beta}, \quad \text{Osc}(\pi_{i,j}(f_j^{(n)})^2) \le 2C_n^2(1 - \alpha_n^\beta)^{\kappa_j^\beta - \kappa_i^\beta}; \qquad (22) $$

also, for $1 \le l < i \le j \le n$,

$$ \mathrm{Osc}(\pi_{l,i}(f_i^{(n)}\pi_{i,j}f_j^{(n)})) \le 6C_n^2(1-\alpha_n^\beta)^{\kappa_i^\beta - \kappa_l^\beta}(1-\alpha_n^\beta)^{\kappa_j^\beta - \kappa_i^\beta} = 6C_n^2(1-\alpha_n^\beta)^{\kappa_j^\beta - \kappa_l^\beta}. \qquad (23) $$

*Proof.* See [11, Lemma 8]. □

<figure><img src="image_3.png" /></figure>

The next result provides a crucial lower bound for the variance of the sum, which is quite important for the proof of the theorem. For a version with $\alpha_n^\gamma$ replaced by a smaller constant $\alpha_n$ (without $\gamma_i^{(n)}$ under the summation) see [13, proposition 3.2]; for the original Dobrushin's lower bound with $\alpha_n$ and with another multiplier instead of 1/4 see [2, §4].

**Proposition 7.** Under the assumptions of the theorem 3, for any $n \ge 1$,

$$ D(S_n) \ge \frac{\alpha_n^\gamma}{4} \sum_{i=2}^{n-1} \gamma_i^{(n)} D(f_i^{(n)}(X_i^{(n)})). $$

For the proof, the following auxiliary statements are necessary.

**Lemma 8 (Lemma 4.1, [13]).** Let $\lambda$ be a probability measure on $X \times X$ with marginals $\alpha$ and $\beta$ respectively. Let $\pi(x_1, dx_2)$ and $\hat{\pi}(x_2, dx_1)$ be the corresponding transition probabilities in the two directions so that $\alpha\pi = \beta$ and $\beta\hat{\pi} = \alpha$. Let $f(x_1)$ and $g(x_2)$ be square integrable functions with respect to $\alpha$ and $\beta$, respectively. If

$$ \int f(x_1) \alpha(dx_1) = \int g(x_2) \beta(dx_2) = 0 $$

then,

$$ \left| \int f(x_1) g(x_2) \lambda(dx_1, dx_2) \right| \le \sqrt{\delta(\pi)} \|f\|_{L_2(\alpha)} \|g\|_{L_2(\beta)}. $$

**Lemma 9 (Lemma 4.2, [13]).** Let functions $f(x_1)$ and $g(x_2)$ be square integrable with respect to $\alpha$ and $\beta$ respectively. Then,

$$ E\left[ (f(X_1) - g(X_2))^2 \right] \ge \alpha(\pi)D(f(X_1)), $$

and

$$ E\left[ (f(X_1) - g(X_2))^2 \right] \ge \alpha(\pi)D(g(X_2)). $$
Proof of Proposition 7. Applying Lemma 9 to the Markov pairs $\{(X_k^{(n)}, X_{k+1}^{(n)}) : \gamma_k^{(n)} = 1\}$ with $f(X_k^{(n)}) = E[Z_{k+1}^{(n)} | X_k^{(n)}]$ and $g(X_{k+1}^{(n)}) = Z_{k+1}^{(n)}$, we get$^6$

$$ \begin{aligned} & \gamma_k^{(n)} \mathrm{E} \left[ \left( Z_{k+1}^{(n)} - \mathrm{E} \left[ Z_{k+1}^{(n)} \mid X_k^{(n)} \right] \right)^2 \right] \ge \gamma_k^{(n)} \alpha_{k,k+1} \mathrm{E} \left[ \left( Z_{k+1}^{(n)} \right)^2 \right] \tag{24} \\ & \ge \alpha_n^{\gamma} \gamma_k^{(n)} \mathrm{E} \left[ \left( Z_{k+1}^{(n)} \right)^2 \right], \end{aligned} $$

or, equivalently,

$$ \alpha_n^{\gamma} \gamma_k^{(n)} \mathrm{E} \left[ \left( Z_{k+1}^{(n)} \right)^2 \right] \le \gamma_k^{(n)} \mathrm{E} \left[ \left( Z_{k+1}^{(n)} - \mathrm{E} \left[ Z_{k+1}^{(n)} \mid X_k^{(n)} \right] \right)^2 \right]. $$

Notice that also

$$ \begin{aligned} & \gamma_k^{(n)} \mathrm{E} \left[ \left( Z_k^{(n)} - \mathrm{E} \left[ Z_k^{(n)} \mid X_{k-1}^{(n)} \right] \right)^2 \right] \ge \gamma_k^{(n)} \alpha_{k-1,k} \mathrm{E} \left[ \left( Z_k^{(n)} \right)^2 \right] \tag{25} \\ & \ge \alpha_n^{\gamma} \gamma_k^{(n)} \mathrm{E} \left[ \left( Z_k^{(n)} \right)^2 \right], \end{aligned} $$

or, equivalently,

$$ \alpha_n^{\gamma} \gamma_k^{(n)} \mathrm{E} \left[ \left( Z_k^{(n)} \right)^2 \right] \le \gamma_k^{(n)} \mathrm{E} \left[ \left( Z_k^{(n)} - \mathrm{E} \left[ Z_k^{(n)} \mid X_{k-1}^{(n)} \right] \right)^2 \right] $$

These two inequalities (24) and (25) are both valid with the same multiplier $\gamma_k^{(n)}$ because of the assumption that $\gamma_k^{(n)} = 1$ is equivalent to the fact that

$$ \alpha_{k,k+1}^{(n)} \wedge \alpha_{k-1,k}^{(n)} \ge \alpha_n^{\gamma}. $$

We highlight that because of this multiplier $\gamma_k^{(n)}$, these two inequalities are not identical (as it may seem from the first sight). On the other hand, we have from (18), for $1 \le k \le n - 1$ that

$$ \begin{aligned} \mathrm{D} \left( f_k^{(n)} \left( X_k^{(n)} \right) \right) &= \mathrm{E} \left[ \left( f_k^{(n)} \left( X_k^{(n)} \right) \right)^2 \right] = \mathrm{E} \left( Z_k^{(n)} - \mathrm{E} \left[ Z_{k+1}^{(n)} \mid X_k^{(n)} \right] \right)^2 \\ &\le 2 \mathrm{E} \left[ \left( Z_k^{(n)} \right)^2 \right] + 2 \mathrm{E} \left[ \left( \mathrm{E} \left[ Z_{k+1}^{(n)} \mid X_k^{(n)} \right] \right)^2 \right] \\ &\le 2 \mathrm{E} \left[ \left( Z_k^{(n)} \right)^2 \right] + 2 \mathrm{E} \left[ \left( Z_{k+1}^{(n)} \right)^2 \right]. \end{aligned} $$

<footnote>$^6$Recall that the r.v. $Z_{k+1}^{(n)}$ is an a.s. version of a function of the r.v. $X_{k+1}^{(n)}$, see (17).</footnote>
Hence, summing over $k$, and noting that $f_n^{(n)}(X_n^{(n)}) = Z_n^{(n)}$, we get,

$$
\begin{align*}
& \alpha_n^{\gamma} \sum_{2 \le k \le n-1: \gamma_k^{(n)} = 1} \mathbf{D} \left( f_k^{(n)} \left( X_k^{(n)} \right) \right) \\
& \le 2 \alpha_n^{\gamma} \sum_{k=2}^{n-1} \gamma_k^{(n)} \mathbf{E} \left[ \left( Z_k^{(n)} \right)^2 \right] + 2 \alpha_n^{\gamma} \sum_{k=2}^{n-1} \gamma_k^{(n)} \mathbf{E} \left[ \left( Z_{k+1}^{(n)} \right)^2 \right] \\
& \le (24)-(25) \left( 2 \sum_{k=2}^{n-1} \gamma_k^{(n)} \mathbf{E} \left[ \left( Z_k^{(n)} - \mathbf{E} \left[ Z_k^{(n)} \mid X_{k-1}^{(n)} \right] \right)^2 \right] \right. \\
& \qquad \left. + 2 \sum_{k=2}^{n-1} \gamma_k^{(n)} \mathbf{E} \left[ \left( Z_{k+1}^{(n)} - \mathbf{E} \left[ Z_{k+1}^{(n)} \mid X_k^{(n)} \right] \right)^2 \right] \right) \\
& \le 4 \left[ \sum_{k=1}^{n-1} \mathbf{E} \left[ \left( Z_{k+1}^{(n)} - \mathbf{E} \left[ Z_{k+1}^{(n)} \mid X_k^{(n)} \right] \right)^2 \right] + \mathbf{E} \left[ \left( Z_1^{(n)} \right)^2 \right] \right] = 4 \mathbf{D} (S_n),
\end{align*}
$$

the latter equality due to the decomposition (19). So,

$$
\mathbf{D}(S_n) \geq \frac{\alpha_n^\gamma}{4} \sum_{k=2}^{n-1} \gamma_k^{(n)} \mathbf{D}(f_k^{(n)}(X_k^{(n)})).
$$

For the ratio $||Z_k^{(n)}||_{L^\infty} / \sqrt{\mathbf{D}(S_n)}$, we have an estimate in the following lemma; recall the definition of $Z_k^{(n)}$ in (17).

**Lemma 10.** Under the assumptions of the theorem 3, the equality holds,

$$
\lim_{n \to \infty} \sup_{1 \le k \le n} \frac{\|Z_k^{(n)}\|_{L^\infty}}{\sqrt{\mathbf{D}(S_n)}} = 0. \qquad (26)
$$

Recall that the value $\|Z_k^{(n)}\|_{L^\infty}$ is nonrandom, see the definition (15).

*Proof.* As in [11, Lemma 12], we obtain the bound

$$
\|Z_k^{(n)}\|_{L^\infty} \le 2C_n \frac{(cm_0 + 1)/c}{\alpha_n^\beta}.
$$

Here $c$ and $m_0$ are the constants from the condition $(H_\beta)$, see (9).
So, applying the inequality from the proposition 7, we get,

$$ \sup_{1 \le k \le n} \frac{\|Z_k^{(n)}\|_{L^\infty}}{\sqrt{D(S_n)}} \le 4 C_n \left( \alpha_n^\gamma (\alpha_n^\beta)^2 \sum_{i=1}^n \gamma_i^{(n)} D(f_i^{(n)}(X_i^{(n)})) \right)^{-1/2} . $$

Due to the condition (11) this implies the inequality (26), as required. □

Recall the definition (2) of the semi-norm “Osc” for a random variable.

**Lemma 11** ([13], lemma 3.3). Let $\{Y_l^{(n)} : 1 \le l \le n\}$ and $\{G_l^{(n)} : 1 \le l \le n\}$, for $n \ge 1$, are, respectively, the sequence of non-negative random variables and sigma-fields such that $\sigma\{Y_1^{(n)}, \dots, Y_l^{(n)}\} \subset G_l^{(n)}$. Suppose that

$$ \lim_{n \to \infty} \mathrm{E} \left[ \sum_{l=1}^{n} Y_l^{(n)} \right] = 1 \quad \text{and} \quad \sup_{1 \le i \le n} \left\| Y_i^{(n)} \right\|_{L^\infty} \le \epsilon_n, $$

where $\lim_{n \to \infty} \epsilon_n = 0$. Suppose in addition that

$$ \lim_{n \to \infty} \sup_{1 \le l \le n-1} \mathrm{Osc} \left( \mathrm{E} \left[ \sum_{j=l+1}^{n} Y_j^{(n)} \mid \mathcal{G}_l^{(n)} \right] \right) = 0. $$

Then

$$ \lim_{n \to \infty} \sum_{l=1}^{n} Y_l^{(n)} = 1 \quad \text{in } L^2. $$

Further, denote

$$ v_j^{(n)} := \mathrm{E}[(\xi_j^{(n)})^2 \mid \mathcal{F}_{j-1}^{(n)}], $$

where $\xi_j^{(n)}$ are defined$^7$ in (20). These random variables are measurable with respect to the sigma-fields $\mathcal{G}_j^{(n)} = \mathcal{F}_{j-1}^{(n)}$ for $2 \le j \le n$, respectively.

**Lemma 12** ([11], Lemma 14). Under the assumptions of the theorem 3 the convergence

$$ \sup_{2 \le l \le n-1} \mathrm{Osc} \left( \mathrm{E} \left[ \sum_{j=l+1}^{n} v_j^{(n)} \mid \mathcal{F}_{l-1}^{(n)} \right] (\omega) \right) = o(1), \quad n \to \infty, $$

holds true.

NB: Notice that here and in what follows, expressions $o(1)$ are understood in the non-random way, due to the convention about “Osc” for random variables, see (2).

$$ {}_7\xi_j^{(n)} = \frac{1}{D(S_n)} [Z_k^{(n)} - \mathrm{E}[Z_k^{(n)} | X_{k-1}^{(n)}]] $$
# 5 Proof of the theorem 3

*Proof.* It follows from the lemma 10 that it suffices to show convergence

$$M_n^{(n)}/\sqrt{D(S_n)} \implies \mathcal{N}(0, 1).$$

This property follows from the proposition 5, because it was already verified that (perepisat', my nemnogo na druguyu formulirovku CLT soshliemsia)

$$(a) \sup_{2 \le k \le n} \left\| \xi_k^{(n)} \right\|_{L^\infty} \to 0, \quad n \to \infty;$$

$$(b) E \left| \sum_{k=2}^{n} E \left[ \left( \xi_k^{(n)} \right)^2 \mid \mathcal{F}_{k-1}^{(n)} \right] \right|^2 \to 1, \quad n \to \infty.$$

Here, part (a) follows from the lemma 10, while part (b) follows from the lemma 11 and from part (a). The theorem is proved. □

# References

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[11] Aisha Nurieva, On Dobrushin's Central Limit Theorem for Non-homogeneous Markov Chains in the Array Scheme, II, https://www.preprints.org/manuscript/202509.2020; doi: 10.20944/preprints202509.2020 (2025)

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