HYPOTHESIS TESTING FOR DEPENDENT VARIABLES WITH UNBALANCED DATA

José Mauricio Gómez Julián

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José Mauricio Gómez Julián. HYPOTHESIS TESTING FOR DEPENDENT VARIABLES WITH UNBALANCED DATA. 2025. hal-04959818v2

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Distributed under a Creative Commons Attribution 4.0 International License
HYPOTHESIS TESTING FOR
DEPENDENT VARIABLES WITH
UNBALANCED DATA

José Mauricio Gómez Julián

josemgomez@protonmail.ch, jose.gomezjulian@ucr.ac.cr

Table of Contents

<table><tr><td>INTRODUCTION</td><td>4</td></tr><tr><td>1. Basic Formal Definitions</td><td>6</td></tr><tr><td>1.1. Basic Terms</td><td>6</td></tr><tr><td>1.1.1. Triangular Arrays (TAC)</td><td>6</td></tr><tr><td>1.1.2. Correlation-Weighted Sums (WSC)</td><td>7</td></tr><tr><td>1.1.3. Mixingale Processes (MPC)</td><td>7</td></tr><tr><td>1.2. Hypotheses and Errors</td><td>8</td></tr><tr><td>2. Unified Theoretical Framework</td><td>8</td></tr><tr><td>2.1. Unified Convergence Metric</td><td>8</td></tr><tr><td>2.1.1. Tightness Conditions</td><td>9</td></tr><tr><td>2.1.1.1. Uniform Tightness Theorem for Convergence of Triangular Arrays</td><td>9</td></tr><tr><td>2.1.1.2. Uniform Tightness Theorem for Convergence of Correlation-Weighted Sums</td><td>10</td></tr><tr><td>2.1.1.3. Uniform Tightness Theorem for Convergence of Mixingale Processes</td><td>10</td></tr><tr><td>2.1.2. Strict Metric Equivalence Theorem</td><td>11</td></tr><tr><td>2.1.2.1. Necessary Conditions (⇒)</td><td>11</td></tr><tr><td>2.1.2.1.1. Convergence in Triangular Arrays</td><td>11</td></tr><tr><td>2.1.2.1.2. Convergence in Correlation-Weighted Sums</td><td>12</td></tr><tr><td>2.1.2.1.3. Convergence in Mixingale Processes</td><td>12</td></tr><tr><td>2.1.2.2. Sufficient Conditions (<=)</td><td>12</td></tr><tr><td>2.1.2.2.1. For Triangular Arrays</td><td>13</td></tr><tr><td>2.1.2.2.2. For Correlation-Weighted Sums</td><td>13</td></tr><tr><td>2.1.2.2.3. For Mixingale Processes</td><td>13</td></tr><tr><td>2.1.2.3. Exponential Dependence Corollary</td><td>13</td></tr><tr><td>2.1.3. For Convergence Via Triangular Arrays</td><td>14</td></tr><tr><td>2.1.4. For Convergence Via Correlation-Weighted Sums</td><td>15</td></tr><tr><td>2.1.5. For Convergence Via Mixingale Processes</td><td>15</td></tr><tr><td>2.2. Fundamental Theorem</td><td>16</td></tr><tr><td>2.2.1. Asymptotic Equivalence Theorem</td><td>16</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>2.2.1.1. Proof for Triangular Arrays</td>
      <td>16</td>
    </tr>
    <tr>
      <td>2.2.1.2. Proof for Correlation-Weighted Sums</td>
      <td>16</td>
    </tr>
    <tr>
      <td>2.2.1.3. Proof for Mixingale Processes</td>
      <td>16</td>
    </tr>
    <tr>
      <td>2.2.1.4. Corollary of the Asymptotic Equivalence Theorem (Strong Consistency Corollary)</td>
      <td>17</td>
    </tr>
    <tr>
      <td>2.3. Critical Regions</td>
      <td>17</td>
    </tr>
    <tr>
      <td>2.3.1. Formal Definition of the Rejection Region</td>
      <td>17</td>
    </tr>
    <tr>
      <td>2.3.1.1. Critical Values Lemma</td>
      <td>17</td>
    </tr>
    <tr>
      <td>2.3.2. Optimality of the Neyman-Pearson Lemma for Dependent Variables</td>
      <td>18</td>
    </tr>
    <tr>
      <td>2.4. Comparison of Experiments</td>
      <td>19</td>
    </tr>
    <tr>
      <td>2.5. Confidence Limits</td>
      <td>20</td>
    </tr>
    <tr>
      <td>2.5.1. Uniform Intervals</td>
      <td>20</td>
    </tr>
    <tr>
      <td>2.5.1. Proofs of Confidence Limits</td>
      <td>20</td>
    </tr>
    <tr>
      <td>2.5.1.1. Proof for Triangular Arrays</td>
      <td>20</td>
    </tr>
    <tr>
      <td>2.5.1.2. Proof for Weighted Sums</td>
      <td>21</td>
    </tr>
    <tr>
      <td>2.5.1.3. Proof for Mixingale Processes</td>
      <td>22</td>
    </tr>
    <tr>
      <td>2.5.2. Comparison of Approaches</td>
      <td>23</td>
    </tr>
    <tr>
      <td>2.6. Measurable Structure and Filtrations</td>
      <td>23</td>
    </tr>
    <tr>
      <td>2.6.1. Definition of Adapted Filtration</td>
      <td>23</td>
    </tr>
    <tr>
      <td>2.6.2. Lemma of σ-Algebra Consistency</td>
      <td>23</td>
    </tr>
    <tr>
      <td>3. Error Control and Bayesian Extensions</td>
      <td>24</td>
    </tr>
    <tr>
      <td>3.1. Type I Error Control with Dependence</td>
      <td>24</td>
    </tr>
    <tr>
      <td>3.2. Extension of the Neyman-Pearson Lemma</td>
      <td>24</td>
    </tr>
    <tr>
      <td>3.3. Bayes' Theorem for Type I and II Errors</td>
      <td>25</td>
    </tr>
    <tr>
      <td>3.4. Connection with Martingale Theorems</td>
      <td>26</td>
    </tr>
    <tr>
      <td>3.4.1. Posterior Convergence Theorem as Martingale</td>
      <td>26</td>
    </tr>
    <tr>
      <td>3.5. Comparison of Mixing</td>
      <td>26</td>
    </tr>
    <tr>
      <td>4. Convergence, Errors, and Bayesian Probabilities</td>
      <td>27</td>
    </tr>
    <tr>
      <td>4.1 Fundamental Relationship: Γ<sub>j</sub>, α<sup>(j,i)</sup> and β<sup>(j,i)</sup></td>
      <td>27</td>
    </tr>
    <tr>
      <td>4.5. Triangular Arrays</td>
      <td>27</td>
    </tr>
    <tr>
      <td>4.3. Interpretation of Bayesian Integrals</td>
      <td>28</td>
    </tr>
    <tr>
      <td>4.3.1. For α<sup>(j,i)</sup></td>
      <td>28</td>
    </tr>
    <tr>
      <td>4.3.2. For β<sup>(j,i)</sup></td>
      <td>28</td>
    </tr>
    <tr>
      <td>4.4. Bayes' Theorem for Error Control</td>
      <td>28</td>
    </tr>
    <tr>
      <td>4.4.1. Theorem on the Posterior Rate of Error Convergence under Dependence</td>
      <td>28</td>
    </tr>
    <tr>
      <td>4.4.1.1. Proof</td>
      <td>29</td>
    </tr>
    <tr>
      <td>4.4.1.2. Corollary</td>
      <td>29</td>
    </tr>
    <tr>
      <td>4.4.2. Variables Involved</td>
      <td>30</td>
    </tr>
    <tr>
      <td>4.4.3. Conceptual Derivation</td>
      <td>30</td>
    </tr>
  </tbody>
</table>
<table><tr><td>4.4.4. Implications</td><td>31</td></tr><tr><td>4.5. Relationship between Convergence, Errors, and Bayesian Probabilities (Explicit Inequalities)</td><td>31</td></tr><tr><td>4.5.1. Non-Asymptotic Probabilistic Bounds</td><td>31</td></tr><tr><td>4.5.1.1. Bernstein's Inequality for Mixingales</td><td>31</td></tr><tr><td>4.5.1.1.1. Specification of Triangular Arrays</td><td>32</td></tr><tr><td>4.5.1.1.2. Correlation-Weighted Sums</td><td>32</td></tr><tr><td>4.5.1.1.3. Mixingale Processes</td><td>33</td></tr><tr><td>4.6. Summary Diagram of the Relationships Between Convergence Methods, Errors, and Bayesian Framework</td><td>35</td></tr><tr><td>5. REFERENCES</td><td>36</td></tr></table>
Abstract

This research develops a unified theoretical framework to evaluate the asymptotic convergence of multiple dependent random variables with unbalanced data toward a common population mean (μ) under weak dependence conditions (α-mixing). Three approaches based on the law of large numbers are employed: triangular arrays (TAC) and weighted correlation sums (WSC) for the strong version, and mixingale processes (MPC) for the weak version. This aims to verify whether deviations from μ are statistically insignificant in the limit. A theorem of strict metric equivalence is proved under specific necessary and sufficient conditions, including an important corollary on exponential dependence. The proposed framework extends the Neyman-Pearson lemma for dependent variables and establishes uniform confidence bounds, incorporating careful consideration of the measurable structure and associated filtrations. Rigorous Type I error control with dependence is developed, and Bayesian extensions are established to incorporate prior information about the dependence structure. The fundamental relationships between convergence, errors, and Bayesian probabilities are made explicit through non-asymptotic inequalities. The work demonstrates that the three approaches are asymptotically equivalent under α-mixing dependence with polynomial decay rate, providing a unified theoretical basis for convergence analysis in weak dependence scenarios. The proposed architecture has been designed with sufficient flexibility to incorporate additional methodologies in future developments.

INTRODUCTION

The relationship between the tendency toward uniformity in sectoral surplus value rates and non-
local varieties of the law of large numbers was established by Gómez Julián (2022), who
demonstrated their theoretical and logical equivalence for both strong and weak versions. This
theoretical foundation was later complemented with a computational implementation (Gómez
Julián, 2025a) that numerically verified the convergence to the global average in U.S. economic
sectors classified as productive, according to the criteria developed in Gómez Julián (2025b) and
using the data compiled in Gómez Julián (2023).

For this numerical verification, three fundamental methodological approaches were selected
among the various analytical frameworks studied in the literature: triangular arrays, which
leverage the strong law of large numbers through stepped matrix structures; correlation-weighted
sums, which are also based on the strong version but explicitly incorporate the dependence
structure; and mixingale processes, which use the weak version to capture temporal
dependencies with decreasing memory. The choice of these three approaches among those
studied by Gómez Julián (2022) was due to their adequate combination of objective economic
dynamics and input-output matrix structure.

Triangular arrays require that the weights form a matrix structure with rows of increasing size (1, 2, 3, ..., *n*). In economics, input-output matrices organize sectors and subsectors in an interdependent network, where each level can be represented as a row in a triangular array. For example, a primary sector can be broken down into subsectors (agriculture, mining), and these, in turn, into specific activities. This hierarchy allows adjusting dimensions to meet mathematical requirements (such as powers of natural numbers), facilitating the application of the LLN. Thus, the asymptotic convergence of sectoral surplus value rates toward a global average reflects how the structural interdependencies of input-output matrices can be modeled under conditions of statistical stability.

For their part, correlation-weighted sums of random variables allow reflecting, through weights (i.e., correlations), structural changes (e.g., the evolution of intersectoral dependencies resulting
from economic shocks, technological changes, variations in demand, etc.), as well as the evolution of the productive structure of the studied economy toward convergence/divergence to certain equilibria. Input-output matrices capture dependency relationships between sectors (e.g., the automotive sector depends on the steel sector). By assigning weights that reflect these correlations, the LLN ensures that, even with interdependencies, sector averages converge to a global theoretical value. This is essential for validating assumptions such as the uniformity of surplus value rates, since adjusted weights (based on occurrence probabilities or economic contribution) replicate the real structure of the matrices, where certain sectors have greater relative weight.

Mixingale processes relax the independence assumption by allowing weak temporal dependencies, which may occur in some economic series. In input-output matrices, data is usually analyzed in specific periods, where sectoral relationships may exhibit persistence (e.g., cumulative demand<fnref n="1" />). Mixingales, requiring only finite moments and asymptotic integrability conditions, are ideal for modeling these dynamics. For example, *L*<fnref n="1" />-*near* dependence can allow capturing gradual adjustments in the productive structure without requiring strict independence.

Economic dynamics implies continuous adjustments (labor migration, technological changes) that generate convergence toward theoretical equilibria. The LLN, in its flexible variants, supports this idea: Smith argues that sectoral advantages are equalized through labor mobility, while Marx links surplus value rates to structurally generated global averages. Triangular arrays, correlation-weighted sums, and mixingales formalize these processes, showing that, even with initial heterogeneity, interdependencies and sectoral adjustments (similar to those of input-output matrices) lead toward asymptotic stabilization. This validates theoretically and empirically the uniformity hypothesis in long-term studies.

These mathematical approaches not only reflect the complexity of economic data (hierarchies, correlations, temporal dependencies) but also provide a rigorous framework for analyzing input-output matrices and real economic dynamics. This correspondence between the mathematical properties of the methods and the underlying economic phenomena strengthens the validity of hypothesis testing by aligning the formal structure of the analysis with the very nature of the object of study.

This convergence analysis is not limited to a particular family of economic phenomena. On the contrary, it is extensible to any phenomenon (social or natural) that can be represented with sufficient fidelity through the three formal structures proposed.

Thus, the present research seeks to construct a unified theoretical framework to evaluate the asymptotic convergence of multiple dependent random variables (with heterogeneous sample sizes) toward a common population mean (μ), under weak dependence conditions (α-mixing). While this framework initially integrates three complementary methods (triangular arrays, correlation-weighted sums, and mixingale processes) to verify if deviations from μ are statistically insignificant in the limit, its architecture has been designed with sufficient flexibility to incorporate additional methodologies in future developments, thus ensuring its extensibility and continued relevance in the field

<footnote n="1">¹ Here we refer to the economic phenomenon in which the demand for goods or services in a sector is influenced not only by immediate factors (such as current prices or income) but also by effects accumulated over time. This can occur through persistence (where demand in one period may depend on demand in previous periods, as might be the case for a sector experiencing increased demand due to technological innovation, since this increase can persist over time, generating an accumulation of effects), through feedback effects (in input-output matrices, demand in one sector can generate demand in other related sectors, creating a multiplier effect that accumulates along the production chain) and/or through gradual adjustments in the economy (such as consumer adaptation to price changes or the slow diffusion of new technologies).</footnote>
The development of this unified theoretical framework begins by establishing basic formal definitions that allow treating the three approaches consistently. This includes the precise specification of the i-th variables, the global average and convergence methods, as well as the formulation of hypotheses and the characterization of type I and II errors in this context.

The construction of the theoretical framework is based on a unified convergence metric that satisfies uniform tightness conditions for each approach. A theorem of strict metric equivalence is demonstrated under specific necessary and sufficient conditions, including an important corollary on exponential dependence. Subsequently, critical regions are established and the Neyman-Pearson lemma is extended for dependent variables.

The analysis is expanded through the incorporation of uniform confidence limits and careful consideration of the measurable structure and associated filtrations. Rigorous control of type I errors with dependence is developed and Bayesian extensions are established that allow incorporating prior information about the dependence structure.

Finally, the fundamental relationships between convergence, errors, and Bayesian probabilities are made explicit through non-asymptotic inequalities. These relationships are specified for each of the three approaches and synthesized in a comprehensive diagram that illustrates the interconnections between convergence methods, error structure, and the developed Bayesian framework.

# 1. Basic Formal Definitions

## 1.1. Basic Terms

### Definition 1.1 (i-th Variable)

Let $X_i = \{x_{i,t}\}_{t=1}^T$ be a sequence of i-th random variables (which can be, for example, surplus value rates for sector $i$), where:

* $i \in \{1,...,n\}$: Is the index of the variable, which may denote, for example, economic sectors.

* $t \in \{1,...,T\}$: Is the index of the time period.

### Definition 1.2 (Global Average)

Define the average of the expected values of n-th random variables as

$$ \mu = \frac{1}{n} \sum_{i=1}^{n} \mathbb{E}[X_i]. $$

### Definition 1.3 (Convergence Methods)

For $j \in \{TAC,WSC,MPC\}$:

#### 1.1.1. Triangular Arrays (TAC)

$$ \lim_{k \to \infty} \frac{1}{k} \sum_{m=1}^{k} X_{i,m}^{(k)} \xrightarrow{p} \mu, $$

where $X_{i,m}^{(k)}$ is a triangular array.
### 1.1.2. Correlation-Weighted Sums (WSC)

$$ \sum_{i=1}^{n} w_i X_i^{L^2} \rightarrow \mu, \quad w_i = \frac{1}{\sqrt{n(1 + \rho_{ij})}} $$

### 1.1.3. Mixingale Processes (MPC)

The expected value of a mixingale-type process is defined as

$$ \mathbb{E}\left[\left(\sum_{t=1}^{T} \psi_t(X_{i,t}-\mu)\right)^2\right] \leq C_T, $$

for decreasing mixing coefficients {$\psi_t$}. The sequence of mixing coefficients $\psi_t$, also called temporal dependence weights, possess the following properties:

1. $\psi_t \geq 0$.

2. $\sum_{t=1}^{\infty} \psi_t < \infty$ (decay condition).

3. Control how the dependence between $X_{i,t}$ and $X_{i,t+k}$ decreases with $k$.

For example, $\psi_t = t^{-\gamma}$ for $\gamma > 1$, is typical in $\alpha$-mixing processes.

From the previous expression, it follows that the cumulative variance of the weighted deviations $\psi_t(X_{i,t}-\mu)$ is controlled by $C_T$. In it, $C_T$ grows more slowly than $T$ (e.g., $C_T = O(\log T)$) under weak dependence.

Additionally, the previous expression can be expressed in metric terms as

$$ D_{\text{MPC}}^{(T)}(X_i, \mu) = \sqrt{\mathbb{E}\left[\left(\sum_{t=1}^{T} \psi_t(X_{i,t}-\mu)\right)^2\right]} $$

The above equation transforms the bound $C_T$ into a standardized divergence measure, equivalent to the standard deviation of the weighted sum.

If we start from the definition of $D_{\text{MPC}}^{(T)}$

$$ D_{\text{MPC}}^{(T)} = \sqrt{\mathbb{E}\left[\left(\sum_{t=1}^{T} \psi_t(X_{i,t}-\mu)\right)^2\right]}, $$

apply inequality (1)

$$ \mathbb{E}\left[\left(\sum_{t=1}^{T} \psi_t(X_{i,t}-\mu)\right)^2\right] \leq C_T, $$

and apply the square root operator on both sides, we obtain:
$$ \sqrt{\mathbb{E}\left[\left(\sum_{t=1}^{T} \psi_t(X_{i,t}-\mu)\right)^2\right]} \leq \sqrt{C_T}. $$

Therefore:

$$ D_{\text{MPC}}^{(T)}(X_i, \mu) \leq \sqrt{C_T}. $$

Thus, the metric $D_{\text{MPC}}^{(T)}$ is bounded by $\sqrt{C_T}$, and if $C_T = o(T^2)$, then $D_{\text{MPC}}^{(T)} \to 0$ when $T \to \infty$, guaranteeing convergence.

For example, if $\psi_t = t^{-2}$ and $C_T = \sum_{t=1}^T t^{-4} \approx \frac{\pi^4}{90}$, then:

$$ D_{\text{MPC}}^{(T)} \leq \sqrt{\frac{\pi^4}{90}} \approx 0.33 \quad \forall T. $$

As will be expanded later, this shows uniform convergence independent of T.

## 1.2. Hypotheses and Errors

### Null Hypothesis ($H_0^{(j,i)}$)

$$ \Gamma_j(X_i, \mu) = 1 \Leftrightarrow X_i \text{ converges to } \mu \text{ under method } j, $$

where $\Gamma_j(X_i, \mu)$ is an indicator function whose value is 1 if variable $X_i$ converges to the global average $\mu$ under method $j$, while it takes the value 0 otherwise.

### Errors:

* Type I ($\alpha^{(j,i)}$): Rejecting $H_0^{(j,i)}$ when it is true.

* Type II ($\beta^{(j,i)}$): Accepting $H_0^{(j,i)}$ when it is false.

# 2. Unified Theoretical Framework

## 2.1. Unified Convergence Metric

### Definition 2.1 (Metric $\Gamma_j$)

The indicator $\Gamma_j$ for variable $X_i$ is 1 if and only if the upper limit of statistic $D_j^{(T)}$ is less than $\epsilon$ when $T \to \infty$, that is,

$$ \Gamma_j(X_i, \mu) = \mathbb{I}\left(\limsup_{T \to \infty} D_j^{(T)}(X_i, \mu) < \epsilon\right). $$

In the above expression, when $\Gamma_j = 1$ $X_i$ converges to $\mu$ under method $j$ with precision $\epsilon$, if $\Gamma_j = 0$ $X_i$ diverges from $\mu$ and $\limsup$, allows capturing the worst asymptotic scenario, thus avoiding “temporary” convergences<fnref n="2" />.

<footnote n="2">² In this context, when we speak of “temporary” convergences, we refer to situations where the sequence appears to converge during a finite period but then deviates from the target value $\mu$. The use of $\limsup$ (upper limit) is crucial</footnote>
### 2.1.1. Tightness Conditions

For all $\epsilon > 0$, there exists $K_{\epsilon} \in \mathbb{N}$ and $M_{\epsilon} > 0$ such that:

$$ \sup_{k \ge K_{\epsilon}} P \left( \left| \frac{1}{k} \sum_{m=1}^{k} X_{i,m}^{(k)} \right| > M_{\epsilon} \right) < \epsilon $$

Under $\alpha$-mixing with $\gamma > 1$, the sequence $\{\bar{X}_k\}$ is uniformly tight.

#### 2.1.1.1. Uniform Tightness Theorem for Convergence of Triangular Arrays

**Theorem**

For all $\epsilon > 0$, there exist $K_{\epsilon} \in \mathbb{N}$ and $M_{\epsilon} > 0$ such that:

$$ \sup_{k \ge K_{\epsilon}} P \left( \left| \frac{1}{k} \sum_{m=1}^{k} X_{i,m}^{(k)} \right| > M_{\epsilon} \right) < \epsilon $$

Under $\alpha$-mixing with $\gamma > 1$, the sequence $\{\bar{X}_k\}$ is uniformly tight.

**Proof**

**Step 1: Moment Bounding**

By the finite moments condition and $\alpha$-mixing dependence, we apply Rosenthal's inequality for weakly dependent variables:

$$ \mathbb{E}[|\bar{X}_k|^{2+\delta}] \le \frac{C}{k^{1+\delta/2}} \left( \sum_{t=1}^{\infty} \alpha(t)^{\delta/(2+\delta)} \right)^{1+\delta/2} $$

The series converges since $\alpha(t) \le Ct^{-\gamma}$ and $\gamma > 1$:

$$ \sum_{t=1}^{\infty} t^{-\gamma\delta/(2+\delta)} < \infty \quad \text{if} \quad \gamma\delta/(2+\delta) > 1 $$

Choosing $\delta$ such that $\delta > 2(\gamma-1)^{-1}$, we guarantee convergence.

**Step 2: Prokhorov's Criterion**

A set of measures is tight if:

1. Uniform Boundedness in $L^p$: $\sup_k \mathbb{E}[|\bar{X}_k|^{2+\delta}] < \infty$, already proven.

2. Correlation Control: For any $\epsilon > 0$, $\exists N$ such that $\sup_k P(|\bar{X}_k| > N) < \epsilon$.

By Markov's inequality:

$$ P(|\bar{X}_k| > M_{\epsilon}) \le \frac{\mathbb{E}[|\bar{X}_k|^{2+\delta}]}{M_{\epsilon}^{2+\delta}} \le \frac{C}{M_{\epsilon}^{2+\delta}} $$

Choosing $M_{\epsilon} = (C/\epsilon)^{1/(2+\delta)}$, the condition is satisfied for $K_{\epsilon} = 1$.

precisely to avoid being “fooled” by these apparent convergences that do not hold in the limit.
### Step 3: Topological Conclusion

By Prokhorov's theorem, every tight sequence in a Polish space (like $\mathbb{R}$) is relatively compact. Thus, $\{\bar{X}_k\}$ contains a convergent subsequence, guaranteeing uniform tightness.

#### 2.1.1.2. Uniform Tightness Theorem for Convergence of Correlation-Weighted Sums

**Theorem**

For all $\epsilon > 0$, there exist $n_\epsilon \in \mathbb{N}$ and $M_\epsilon > 0$ such that:

$$ \sup_{n \ge n_\epsilon} P \left( \left| \sum_{i=1}^n w_i X_i - \mu \right| > M_\epsilon \right) < \epsilon. $$

Under $\rho$-mixing with $\gamma > 1$, the sequence of weighted sums is uniformly tight.

**Proof**

##### Step 1: Variance Control

Using the variance bound for dependent sums:

$$ \mathrm{Var}(\sum w_i X_i) \le C (\sum w_i^2 + (\sum |w_i|)^2 n^{-\gamma}) $$

With $w_i = \theta(n^{-1/2})$, we obtain $\mathrm{Var}(\cdot) = O(n^{-1})$.

##### Step 2: Moment Bounding

Applying the Marcinkiewicz-Zygmund inequality for weakly correlated variables:

$$ \mathbb{E} \left[ \left| \sum w_i X_i \right|^{2+\delta} \right] \le C (\sum w_i^2)^{1+\delta/2} $$

##### Step 3: Prokhorov Application

The uniform boundedness in $L^{2+\delta}$ guarantees tightness via Prokhorov's theorem.

#### 2.1.1.3. Uniform Tightness Theorem for Convergence of Mixingale Processes

**Theorem**

For all $\epsilon > 0$, there exist $T_\epsilon \in \mathbb{N}$ and $M_\epsilon > 0$ such that:

$$ \sup_{T \ge T_\epsilon} P \left( \left| \sum_{t=1}^T \psi_t (X_{i,t} - \mu) \right| > M_\epsilon \right) < \epsilon. $$

Under mixing coefficients $\psi_t = O(t^{-\gamma})$ with $\gamma > 1$, the sequence is tight.

**Proof**

##### Step 1: Finite Energy

The condition $\sum \psi_t^2 < \infty$ implies:

$$ \mathbb{E} \left[ \left( \sum \psi_t (X_{i,t} - \mu) \right)^2 \right] \le C \sum \psi_t^2 < \infty $$

##### Step 2: Hoffmann-Jørgensen Inequality
For mixingale processes:

$$P\left(\sup_{T} \left|\sum_{t=1}^{T} \psi_{t}(X_{i,t}-\mu)\right|>M\right) \leq \frac{C}{M^{2+\delta}} \sum \psi_{t}^{2+\delta}$$

*Step 3: Selection of Constants*

Choosing $M_{\epsilon} = (C/\epsilon)^{1/(2+\delta)}$ satisfies the tightness criterion.

### 2.1.2. Strict Metric Equivalence Theorem

#### 2.1.2.1. Necessary Conditions (⇒)

**Thesis**

$\Gamma_{\text{TAC}} = \Gamma_{\text{WSC}} = \Gamma_{\text{MPC}} = 1$ under $\alpha$-mixing dependence and finite moments.

For this thesis to hold, three conditions must be verified:

**1. Condition (i):**

There exists a way to choose increasingly large sample sizes $(k_n \to \infty)$ where the distance between the sample average and $\mu$ does not exceed $\epsilon$. More formally, this can be expressed as

$$\exists\{k_n\} \text{ (increasing subsequence) such that } \limsup_n \left| \frac{1}{k_n} \sum_{m=1}^{k_n} X_{i,m}^{(k_n)} - \mu \right| < \epsilon.$$

**2. Condition (ii):**

The sum of individual variances (squared weighted) must not be infinite. This prevents statistical noise from accumulating and ruining convergence. More formally, this can be expressed as

$$\sum_{i=1}^{\infty} w_i^2 \mathrm{Var}(X_i) < \infty$$

**3. Condition (iii):**

The coefficients that control temporal dependence ($\psi_t$) must decay fast enough for their total energy to be finite. More formally, this can be expressed as

$$\sum_{t=1}^{\infty} \psi_t^2 < \infty$$

#### 2.1.2.1.1. Convergence in Triangular Arrays

If $\Gamma_{\text{TAC}} = 1$, then by definition:

$$\limsup_{k \to \infty} \left| \frac{1}{k} \sum_{m=1}^{k} X_{i,m}^{(k)} - \mu \right| < \epsilon.$$

By properties of $\limsup$, there exists a subsequence $\{k_n\}$ such that:
$$ \frac{1}{k_n} \sum_{m=1}^{k_n} X_{i,m}^{(k_n)} \to \mu \quad \text{almost surely (a.s.),} $$

which guarantees condition (i).

### 2.1.2.1.2. Convergence in Correlation-Weighted Sums

If $\Gamma_{WSC} = 1$, the variance of the weighted sum tends to zero:

$$ \mathrm{Var}(\sum w_i X_i) = \sum w_i^2 \mathrm{Var}(X_i) + 2 \sum_{i<j} w_i w_j \mathrm{Cov}(X_i, X_j). $$

Moreover, under $\alpha$-mixing and $w_i = O(n^{-1/2})$ we know that

$$ \sum_{i<j} w_i w_j |\mathrm{Cov}(X_i, X_j)| \le C n^{-1} \sum_{k=1}^{\infty} k^{-\gamma}. $$

Therefore, for the limit to be zero the necessary condition is that

$$ \sum_{i=1}^{\infty} w_i^2 \mathrm{Var}(X_i) < \infty \quad (\text{Condition ii}). $$

### 2.1.2.1.3. Convergence in Mixingale Processes

If $\Gamma_{MPC} = 1$, we expand the mathematical expectation:

$$ \mathbb{E} \left[ \left( \sum \psi_t (X_{i,t} - \mu) \right)^2 \right] = \sum \psi_t^2 \mathrm{Var}(X_{i,t}) + 2 \sum_{t<s} \psi_t \psi_s \mathrm{Cov}(X_{i,t}, X_{i,s}). $$

Moreover, under $\alpha$-mixing:

$$ \sum_{t=1}^{\infty} \sum_{s=t+1}^{\infty} \psi_t \psi_s |t-s|^{-\gamma} \le (\sum \psi_t)^2 < \infty. $$

Therefore, for convergence to zero, we have as a necessary condition that

$$ \sum_{t=1}^{\infty} \psi_t^2 < \infty \quad (\text{Condition iii}). $$

### 2.1.2.2. Sufficient Conditions (=$\Leftarrow$)

**Thesis**

Propositions (i), (ii) and (iii) are valid. We demonstrate $\Gamma_j = 1$ for all $j$.
### 2.1.2.2.1. For Triangular Arrays

By (i) and Kronecker's lemma:

$$ \frac{1}{k_n} \sum_{m=1}^{k_n} X_{i,m}^{(k_n)} \xrightarrow{\text{a.s.}} \mu $$

The limsup satisfies:

$$ \limsup_{k \to \infty} \left| \frac{1}{k} \sum_{m=1}^{k} X_{i,m}^{(k)} - \mu \right| = 0 < \epsilon \\ \Rightarrow \Gamma_{\text{TAC}} = 1. $$

### 2.1.2.2.2. For Correlation-Weighted Sums

By (ii) and Rosenthal's inequality:

$$ \mathbb{E}[(\sum w_i X_i - \mu)^2] \le C (\sum w_i^2 + (\sum |w_i|)^2 \alpha(1)). $$

Moreover, since $\sum w_i^2 < \infty$ and $\alpha(1) < \infty$ then

$$ \sum w_i X_i \xrightarrow{L^2} \mu \Rightarrow \Gamma_{\text{WSC}} = 1 $$

### 2.1.2.2.3. For Mixingale Processes

By (iii) and Birkhoff-Khinchin theorem, we have

$$ \sum_{t=1}^{T} \psi_t (X_{i,t} - \mu) \xrightarrow{L^2} S \quad \text{with} \quad \mathbb{E}[S^2] \le \sum \psi_t^2 \text{Var}(X_{i,t}). $$

Since $\sum \psi_t^2 < \infty$, then:

$$ S = 0 \quad \text{a.s.} \Rightarrow \Gamma_{\text{MPC}} = 1 $$

### 2.1.2.3. Exponential Dependence Corollary

If $\alpha(k) = O(e^{-\theta k})$:

1. TAC: $k_n \ge \text{clogn}$ ensures rapid convergence.

2. WSC: $w_i = \theta(n^{-1/2})$ compensates weak correlations.
3. MPC: $\psi_t = t^{-1-\delta} (\delta > 0)$ guarantees $\sum \psi_t^2 < \infty$.

This means that when dependence decays exponentially ($\alpha(k) \sim e^{-\theta k}$), the three metrics simplify:

For TAC $k_n \ge \text{clogn}$ this occurs because, with exponential dependence, two observations separated by $\text{logn}$ steps are practically independent. For example, if $n = 1000$, choosing $k_n = 20$ (since $\log 1000 \approx 7$) ensures that residual correlations are negligible.

For WSC $w_i = \theta(n^{-1/2})$, weak correlations allow using equal weights for all sectors, scaled by $1/\sqrt{n}$ to balance variance. The intuition behind this is that the noted fact is equivalent to using the classical sample mean, but adjusted for weak dependence.

Finally, for MPC $\psi_t = t^{-1-\delta}$ the coefficients $\psi_t$ must decay faster than $1/t$ to guarantee convergence. For example, if $\delta = 0.5$ then $\psi_t = t^{-1.5}$. This implies that an observation from 10 periods ago has a weight 31 times smaller than the latest one ($10^{1.5} \approx 31.6$).

### 2.1.3. For Convergence Via Triangular Arrays

$$D_{\text{TAC}}^{(k)}(X_i, \mu) = \left| \frac{1}{k} \sum_{m=1}^{k} X_{i,m}^{(k)} - \mu \right|,$$

where $m$ is an index that traverses the observations within each row of the triangular array, that is, $m$ represents the position of an observation within the triangular array. Thus, $m$ in $X_{i,m}^{(k)}$ is the $m$-th observation of variable $X_i$ in row $k$ of the triangular array.

The variance of the sample mean $\bar{X}_k = \frac{1}{k} \sum_{m=1}^{k} X_m^{(k)}$ is defined as

$$\text{Var}(\bar{X}_k) = \frac{\sigma^2}{k} + \frac{2}{k^2} \sum_{1 \le m < m' \le k} \text{Cov}(X_m^{(k)}, X_{m'}^{(k)}),$$

which means that

$$\frac{2}{k^2} \cdot O(k) = O(k^{-1}).$$

Under weak dependence, the second term is $O(k^{-1})$ and not $O(1)^3$, therefore:

$$\text{Var}(\bar{X}_k) = \frac{\sigma^2 + o(1)}{k}.$$

From the above, it follows that weak dependence does not affect the convergence rate $O(1/k)$ of the variance$^4$.

<footnote>$^3$ This contrasts with strong dependence scenarios, where the sum of covariances could be $O(k^2)$, leading to $\frac{2}{k^2} \cdot O(k^2) = O(1)$, which would ruin the $O(1/k)$ rate.</footnote>

<footnote>$^4$ In asymptotic notation, $O(k^{-1})$ indicates that the term is bounded above by a constant multiplied by $k^{-1}$, while $o(1)$ indicates that the term tends to zero as $n \to \infty$.</footnote>
### 2.1.4. For Convergence Via Correlation-Weighted Sums

$$D_{\text{WSC}}^{(n)}(X_i, M) = \left| \sum_{i=1}^{n} w_i X_i - M \right|.$$

For its part, the sample variance of the estimator $\sum_{i=1}^{n} w_i X_i$ under weak dependence (α-mixing) is defined as:

$$\text{Var}\left(\sum_{i=1}^{n} w_i X_i\right) = \sum_{i=1}^{n} w_i^2 \text{Var}(X_i) + 2 \sum_{1 \le i < j \le n} w_i w_j \text{Cov}(X_i, X_j).$$

Under α-mixing with polynomial decay ($\text{Cov}(X_i, X_j) \le C |i-j|^{-\gamma}, \gamma > 1$), the covariance term satisfies:

$$2 \sum_{i<j} w_i w_j \text{Cov}(X_i, X_j) = O(n^{-1}),$$

therefore:

$$\text{Var}\left(\sum_{i=1}^{n} w_i X_i\right) = \frac{\sigma^2 + o(1)}{n}.$$

Thus, weak dependence does not affect the convergence rate $O(n^{-1})$ of the sample variance.

### 2.1.5. For Convergence Via Mixingale Processes

$$D_{\text{MPC}}^{(T)}(X_i, \mu) = \sqrt{\mathbb{E}\left[\left(\sum_{t=1}^{T} \psi_t (X_{i,t} - \mu)\right)^2\right]}.$$

The variance of the weighted mixingale process $\sum_{t=1}^{T} \psi_t (X_{i,t} - \mu)$ is bounded by:

$$\left| \mathbb{E} \left[ \left( \sum_{t=1}^{T} \psi_t (X_{i,t} - \mu) \right)^2 \right] \right| \le C_T,$$

where ($C_T$) grows subquadratically $C_T = o(T^2)$. Under mixing coefficients ($\psi_t = t^{-\gamma}$) ($\gamma > 1$), we have:

$$C_T = \sum_{t=1}^{T} \psi_t^2 \text{Var}(X_{i,t}) + 2 \sum_{1 \le t < s \le T} \psi_t \psi_s \text{Cov}(X_{i,t}, X_{i,s}).$$

Weak dependence $\text{Cov}(X_{i,t}, X_{i,s}) \le C |t-s|^{-\gamma}$ implies that:

$$C_T = O(\log T) \quad \text{or} \quad C_T = O(1),$$

depending on $\gamma$. Therefore, the metric $D_{\text{MPC}}^{(T)}$ satisfies:

$$D_{\text{MPC}}^{(T)}(X_i, \mu) = \sqrt{C_T} \to 0 \quad \text{when} \quad T \to \infty.$$
## 2.2. Fundamental Theorem

### 2.2.1. Asymptotic Equivalence Theorem

Under weak dependence (α-mixing: $\alpha(k) \le Ck^{-\gamma}$, $\gamma > 1$) and finite moments ($E[|X_{i,t}|^{2+\delta}] < \infty$ for some $\delta > 0$) we have:

$$ \Gamma_{\text{TAC}} = \Gamma_{\text{WSC}} = \Gamma_{\text{MPC}} \quad \text{a.s.} $$

#### 2.2.1.1. Proof for Triangular Arrays

By Kronecker's lemma for triangular arrays, if $X_{i,m}^{(k)}$ satisfies

$$ \frac{1}{k} \sum_{m=1}^{k} X_{i,m}^{(k)} \xrightarrow{L^2} \mu $$

then:

$$ \limsup_{k \to \infty} \left| \frac{1}{k} \sum_{m=1}^{k} X_{i,m}^{(k)} - \mu \right| = 0 \quad \text{a.s.} $$

This is because the α-mixing condition guarantees that the variance of the partial sum decays as $O(k^{-1})$.

#### 2.2.1.2. Proof for Correlation-Weighted Sums

Using McDiarmid's inequality for weighted sums, for weights $w_i = \frac{1}{\sqrt{n(1+\rho_{ij})}}$, we have:

$$ P\left(\left|\sum_{i=1}^{n} w_i X_i - \mu\right| \ge \epsilon\right) \le 2\exp\left(-\frac{2\epsilon^2}{\sum_{i=1}^{n} w_i^2 \sigma_i^2}\right). $$

Under α-mixing, $\sum w_i^2 \sigma_i^2 = O(n^{-1})$, thus:

$$ \lim_{n \to \infty} P\left(\left|\sum w_i X_i - \mu\right| \ge \epsilon\right) = 0. $$

Therefore, $\Gamma_{\text{WSC}} \to 1$ a.s.

#### 2.2.1.3. Proof for Mixingale Processes

By Birkhoff-Khinchin theorem for mixingales, if $\{\psi_t\}$ satisfies $\sum_{t=1}^{\infty} \psi_t < \infty$ and $E[X_{i,t}] = \mu$, then
$$ \frac{1}{T} \sum_{t=1}^{T} \psi_t (X_{i,t} - \mu) \xrightarrow{L^2} 0. $$

Thus,

$$ \lim_{T \to \infty} \sqrt{\mathbb{E}\left[\left(\sum_{t=1}^{T} \psi_t(X_{i,t}-\mu)\right)^2\right]} = 0 \quad \text{a.s.} $$

This implies $\Gamma_{\text{MPC}} = 1$ asymptotically.

### 2.2.1.4. Corollary of the Asymptotic Equivalence Theorem (Strong Consistency Corollary)

As demonstrated, all three methods satisfy:

$$ \lim_{T,k,n \to \infty} D_j^{(T)}(X_i, \mu) = 0 \quad \text{a.s.} $$

Therefore, $\Gamma_{\text{TAC}} = \Gamma_{\text{WSC}} = \Gamma_{\text{MPC}}$ a.s.

Thus, under the conditions of the Asymptotic Equivalence Theorem, it must hold that:

$$ \forall j \in \{\text{TAC}, \text{WSC}, \text{MPC}\}, \quad \lim_{T \to \infty} \Gamma_j(X_i, \mu) = 1 \quad \text{a.s.} \blacksquare $$

## 2.3. Critical Regions

### 2.3.1. Formal Definition of the Rejection Region

For significance level $\alpha$:

$$ R_j^{(i)} = \{X_i : D_j^{(T)}(X_i, \mu) > c_{\alpha}^{(j)}\}, $$

where $c_{\alpha}^{(j)}$ satisfies:

$$ P(D_j^{(T)} > c_{\alpha}^{(j)} | H_0^{(j,i)}) = \alpha. $$

#### 2.3.1.1. Critical Values Lemma

For $T \to \infty$:

$$ 1. \textbf{TAC}: c_{\alpha}^{\text{TAC}} = \Phi^{-1}(1-\alpha/2) \frac{\sigma}{\sqrt{k}}, $$

where $\Phi$ denotes the cumulative distribution function of a standard normal distribution $N(0,1)$, that is,

$$ \Phi(z) = P(Z \le z) \quad \text{for } Z \sim N(0,1). $$

Therefore, $\Phi^{-1}$ is its quantile function or inverse function. For example:
$\Phi^{-1}(1-\alpha/2)$ is the value $z$ such that $P(Z \le z) = 1-\alpha/2$

2. **WSC**: $c_{\alpha}^{\text{WSC}} = z_{1-\alpha} \sqrt{\sum w_i^2 \text{Var}(X_i)}$.

3. **MPC**: $c_{\alpha}^{\text{MPC}} = \sqrt{C_T \chi_{1-\alpha}^2(T)}$.

### 2.3.2. Optimality of the Neyman-Pearson Lemma for Dependent Variables

**Theorem**

Under cross-dependence with polynomial decay $\rho_{ij} = \text{Corr}(X_i, X_j) \le C |i-j|^{-\gamma}$ with $\gamma > 1$, the critical region $R_j^{(i)}$ is asymptotically optimal if:

* **TAC**: $k = \log T$.

* **WSC**: $w_i \propto (1 + \sum_{j \ne i} |\rho_{ij}|)^{-1/2}$.

* **MPC**: $\psi_t = t^{-(1+\gamma)/2}$.

**Proof**

To carry out this proof, we must integrate semiparametric inference techniques (Van der Vaart, 2000, pp. 291-298, 368-371, 384-386), stochastic process theory (Hall & Heyde, 1980, pp. 52-69, 128-146), and classical results of asymptotic optimality (Le Cam & Yang, 2000, pp. 77-107, 150-162, 164-168).

Let $H_0: \mu = \mu_0$ vs. $H_1: \mu = \mu_0 + \delta/\sqrt{n}$, where $\delta$ is a constant and $n$ is the sample size. Optimality is demonstrated in three steps:

**Step 1: Construction of Local Power Function**

For local alternatives, we define the likelihood ratio (LR) statistic:

$$ \Lambda_n = \log \frac{dP_{H_1}}{dP_{H_0}} = \delta \cdot \frac{\sum_{i=1}^{n} w_i (X_i - \mu_0)}{\sigma_n} - \frac{\delta^2}{2} + o_p(1), $$

where $\sigma_n^2 = \text{Var}(\sum_{i=1}^n w_i X_i)$. Under $H_0$, $\Lambda_n \xrightarrow{d} N(-\delta^2/2, \delta^2)$.

**Specific cases:**

1. **TAC**: If $k \approx \log T$, the variance behaves as $\sigma_n^2 = O(1/\log T)$, ensuring that $\Lambda_n$ is asymptotically normal.

2. **WSC**: With $w_i \propto (1 + \sum_{j \ne i} |\rho_{ij}|)^{-1/2}$, we control the contribution of cross-correlations:

$$ \sigma_n^2 \le C \left( \sum w_i^2 + \sum_{i \ne j} |w_i w_j \rho_{ij}| \right) = O(1). $$

3. **MPC**: For $\psi_t = t^{-(1+\gamma)/2}$, the series $\sum \psi_t^2$ converges, and $\sigma_n^2 = O(1)$.

**Step 2: Application of Hájek-Le Cam Lemma**
Contiguity between $P_{H_0}$ and $P_{H_1}$ requires that:

$$ \limsup_{n \to \infty} \mathbb{E}_{H_0} \left[ \left( \Lambda_n + \frac{\delta^2}{2} \right)^2 \right] < \infty. $$

Under the given conditions:

1. **TAC:** The weak dependence structure ($\gamma > 1$) and $k \approx \log T$ ensure finite moments for $\Lambda_n$.

2. **WSC:** The weighting $w_i$ cancels long-range correlations:

$$ \sum_{i \neq j} |w_i w_j \rho_{ij}| \leq c \sum_{i=1}^{n} \sum_{m=1}^{\infty} \frac{w_i w_{i+m}}{m^{\gamma}} = O(n^{-\gamma/2}). $$

3. **MPC:** The choice $\psi_t \approx t^{-(1+\gamma)/2}$ guarantees $\sum \psi_t^2 < \infty$ and $\sum \psi_t \psi_{t+k} \rho(k) = O(k^{-\gamma})$, satisfying the contiguity condition.

### Step 3: Optimality of Critical Region

The region $R_j^{(i)} = \{D_j^{(T)} > c_{\alpha}^{(j)}\}$ maximizes local power if:

$$ \lim_{n \to \infty} P_{H_1} (D_j^{(T)} > c_{\alpha}^{(j)}) = 1 - \Phi(z_{1-\alpha} - \delta/\sigma_n), $$

where $\Phi$ is the normal CDF.

For each method:

1. **TAC:** With $k \approx \log T$, $\sigma_n \to 0$, and power tends to 1 exponentially.

2. **WSC:** The normalization $w_i \propto (1 + \sum |\rho_{ij}|)^{-1/2}$ balances variance and bias, achieving $\sigma_n \to \delta$.

3. **MPC:** The choice $\psi_t \approx t^{-(1+\gamma)/2}$ ensures $\sigma_n = O(1)$, and power converges to $1 - \Phi(z_{1-\alpha} - \delta)$.

Thus, under the specified conditions, $R_j^{(i)}$ achieves the Cramér-Rao lower bound for Type I and II errors, being asymptotically optimal. This follows from:

1. The construction of local alternatives that preserve the dependence structure.

2. The rigorous application of the contiguity framework to derive limit distributions.

3. The tuning of parameters ($k, w_i, \psi_t$) to control variance and residual correlations.

Under strong dependence ($\gamma \le 1$):

## 2.4. Comparison of Experiments

**Theorem 2.2**
Following Blackwell (1951, pp. 93-94), let $E_1, E_2$ be experiments with TAC and WSC methods respectively. Then:

$$E_1 > E_2 \Leftrightarrow \beta_{\text{TAC}} \le \beta_{\text{WSC}} \forall \alpha,$$

where > denotes “more informative”.

## 2.5. Confidence Limits

### 2.5.1. Uniform Intervals

The $(1-\alpha)$% confidence interval for $\mu$ under method j is defined as:

$$CI_{j}^{(i)} = [\mu - c_{\alpha}^{(j)}, \mu + c_{\alpha}^{(j)}].$$

In the above expression, $c_{\alpha}^{(j)}$ is the critical value that satisfies $P(D_j^{(T)} > c_{\alpha}^{(j)}|H_0) = \alpha$, $\mu \pm c_{\alpha}^{(j)}$ is the range where $\mu$ is expected to be if $H_0$ is true.

**Corollary 2.1**

Under weak dependence, the minimum probability that $\mu$ is in $CI_{j}^{(i)}$ is at least $1-\alpha$ when $T \to \infty$. Formally we can express this as

$$\liminf_{T \to \infty} P(\mu \in CI_{j}^{(i)}) \ge 1-\alpha.$$

The above expression guarantees that, asymptotically, the interval covers the true $\mu$ with probability $\ge 1-\alpha$, even with weak dependence.

### 2.5.1. Proofs of Confidence Limits

#### 2.5.1.1. Proof for Triangular Arrays

**Consistency Theorem for $CI_{\text{TAC}}^{(i)}$**

Under the conditions of the Asymptotic Equivalence Theorem (presented in section 2.1) and for $c_{\alpha}^{\text{TAC}} = \Phi^{-1}(1-\alpha/2)\frac{\sigma}{\sqrt{k}}$:

$$\liminf_{k \to \infty} P(\mu \in CI_{\text{TAC}}^{(i)}) \ge 1-\alpha.$$

**Proof**

1. *Asymptotic Normality*

By the Central Limit Theorem for triangular arrays with weak dependence:

$$\sqrt{k} \left( \frac{1}{k} \sum_{m=1}^{k} X_{i,m}^{(k)} - \mu \right) \xrightarrow{d} N(0, \sigma^2).$$

This implies that the sample mean $\bar{X}_k$ is asymptotically normal with mean $\mu$ and variance $\sigma^2/k$.

2. *Critical Value*
For $Z \sim N(0,1)$,

$$c_{\alpha}^{\text{TAC}} = \frac{\sigma}{\sqrt{k}} \Phi^{-1}(1-\alpha/2).$$

This guarantees that the interval $[\bar{X}_k \pm c_{\alpha}^{\text{TAC}}]$ has a width proportional to the asymptotic standard deviation.

### 3. Coverage

$$P(\mu \in [\bar{X}_k \pm c_{\alpha}^{\text{TAC}}]) \rightarrow 2\Phi(\Phi^{-1}(1-\alpha/2)) - 1 = 1-\alpha.$$

The equality $2\Phi(\Phi^{-1}(1-\alpha/2)) - 1 = 1-\alpha$ follows from the symmetry of the standard normal distribution.

## Bayesian Interval

Now, we are going to extend the previous result to the Bayesian framework, assuming a normal prior for $\mu$. The posterior distribution will be constructed using conjugation, while the confidence interval will be constructed using the posterior population mean and variance.

If $\mu \sim N(\mu_0, \tau^2)$ a priori, the posterior distribution is:

$$\mu|D \sim N\left(\frac{\tau^2 \bar{X}_k + \sigma^2 \mu_0}{\tau^2 + \sigma^2}, \frac{\tau^2 \sigma^2}{\tau^2 + \sigma^2}\right),$$

and the $(1-\alpha)$% Bayesian interval is:

$$CI_{\text{TAC-Bayes}}^{(i)} = \left[ \frac{\tau^2 \bar{X}_k + \sigma^2 \mu_0}{\tau^2 + \sigma^2} \pm z_{1-\alpha/2} \sqrt{\frac{\tau^2 \sigma^2}{\tau^2 + \sigma^2}} \right].$$

### 2.5.1.2. Proof for Weighted Sums

## Consistency Theorem for $CI_{\text{WSC}}^{(i)}$

Under weak dependence and for $c_{\alpha}^{\text{WSC}} = z_{1-\alpha} \sqrt{\sum w_i^2 \text{Var}(X_i)}$:

$$\liminf_{n \to \infty} P(\mu \in CI_{\text{WSC}}^{(i)}) \geq 1-\alpha.$$

## Proof

### 1. Variance of the Weighted Sum

We know that

$$\text{Var}(\sum w_i X_i) = \sum w_i^2 \text{Var}(X_i) + 2 \sum_{i<j} w_i w_j \text{Cov}(X_i, X_j).$$

Moreover, under $\alpha$-mixing we have $\text{Cov}(X_i, X_j) \leq C |i-j|^{-\gamma}$. Therefore,
$$ \mathrm{Var}(\sum w_i X_i) \leq \sum w_i^2 \sigma^2 + 2C \sum_{i=1}^{n} \sum_{k=1}^{\infty} \frac{w_i w_{i+k}}{k^\gamma} = O(n^{-1}). $$

## 2. Asymptotic Normality

$$ \frac{\sum w_i X_i - \mu}{\sqrt{\mathrm{Var}(\sum w_i X_i)}} \xrightarrow{d} N(0,1). $$

## 3. Coverage

$$ P\left(\mu \in \left[\sum w_i X_i \pm z_{1-\alpha} \sqrt{\mathrm{Var}(\sum w_i X_i)}\right] \rightarrow 1-\alpha\right). $$

## Bayesian Interval

With conjugate normal-inverse-gamma prior for $\mu$ and $\sigma^2$, the posterior distribution of the population mean $\mu$ is defined as:

$$ \mu|D \sim t_v\left(\bar{X}_w, \frac{s_w^2}{n}\right), $$

$$ \text{where } \bar{X}_w = \sum w_i X_i, s_w^2 = \sum w_i^2 (X_i - \bar{X}_w)^2. $$

Therefore, the interval is defined as:

$$ CI_{\text{WSC-Bayes}}^{(i)} = \left[ \bar{X}_w \pm t_{v,1-\alpha/2} \frac{s_w}{\sqrt{n}} \right]. $$

### 2.5.1.3. Proof for Mixingale Processes

## Consistency Theorem for $CI_{\text{MPC}}^{(i)}$

Under the Birkhoff-Khinchin Theorem and for $c_{\alpha}^{\text{MPC}} = \sqrt{C_T \chi_{1-\alpha}^2(T)}$:

$$ \liminf_{T \to \infty} P(\mu \in CI_{\text{MPC}}^{(i)}) \geq 1-\alpha. $$

## Proof

### 1. Bounding of Mathematical Expectation

By the definition given in section 2.1.5,

$$ \mathbb{E}\left[\left(\sum_{t=1}^{T} \psi_t(X_{i,t}-\mu)\right)^2\right] \leq C_T. $$

### 2. Chebyshev's Inequality

$$ P\left(\left|\sum_{t=1}^{T} \psi_t(X_{i,t}-\mu)\right| \geq \sqrt{C_T/\alpha}\right) \leq \alpha. $$
### 3. Critical Value

Let $c_{\alpha}^{\text{MPC}} = \sqrt{C_T/\alpha}$, then:

$$P\left(\mu \in \left[\sum_{t=1}^{T} \psi_t X_{i,t} \pm \sqrt{C_T/\alpha}\right]\right) \geq 1-\alpha.$$

#### Bayesian Interval

Assuming a Jeffreys prior for $\mu$, the posterior is:

$$\mu|D \sim N\left(\frac{\sum \psi_t X_{i,t}}{\sum \psi_t}, \frac{C_T}{(\sum \psi_t)^2}\right),$$

and the interval is:

$$CI_{\text{MPC-Bayes}}^{(i)} = \left[ \frac{\sum \psi_t X_{i,t}}{\sum \psi_t} \pm z_{1-\alpha/2} \frac{\sqrt{C_T}}{\sum \psi_t} \right].$$

### 2.5.2. Comparison of Approaches

<table><thead><tr><th>Method</th><th>Classical Interval</th><th>Bayesian Interval</th></tr></thead><tbody><tr><td>TAC</td><td>[X&#773;<sub>k</sub> &plusmn; c<sub>&alpha;</sub><sup>TAC</sup>]</td><td>[Posterior Mean &plusmn; z<sub>1-&alpha;/2</sub> Posterior SD]</td></tr><tr><td>WSC</td><td>[&Sigma;<sub>w<sub>i</sub></sub>X<sub>i</sub> &plusmn; c<sub>&alpha;</sub><sup>WSC</sup>]</td><td>[X&#773;<sub>w</sub> &plusmn; t<sub>v,1-&alpha;/2</sub> (s<sub>w</sub>/&radic;n)]</td></tr><tr><td>MPC</td><td>[&Sigma;<sub>&psi;<sub>t</sub></sub>X<sub>i,t</sub> &plusmn; c<sub>&alpha;</sub><sup>MPC</sup>]</td><td>[(&Sigma;<sub>&psi;<sub>t</sub></sub>X<sub>i,t</sub> &plusmn; z<sub>1-&alpha;/2</sub> &radic;C<sub>T</sub>)/(&Sigma;<sub>&psi;<sub>t</sub></sub>)]</td></tr></tbody></table>

#### Summary

* **Classical/Frequentist Approach:** Frequency-based, guarantees asymptotic coverage.

* **Bayesian Approach:** Incorporates prior information, useful when $\pi_0$ (belief in convergence) is high.

* **Equivalence:** When $n,T \to \infty$ and $\pi_0$ is diffuse, both intervals coincide.

## 2.6. Measurable Structure and Filtrations

### 2.6.1. Definition of Adapted Filtration

Let {$F_t$}$_{t=1}^\infty$ be the natural filtration where $F_t = \sigma(X_{i,1},...,X_{i,t})$. Mixingales satisfy:

$$\mathbb{E}[X_{i,t}|F_{t-m}] \xrightarrow{m \to \infty} 0 \quad \text{in } L^2$$

### 2.6.2. Lemma of $\sigma$-Algebra Consistency

Under $\psi_t = O(t^{-\gamma})$, the sequence {$\sum_{t=1}^T \psi_t X_{i,t}$} is adapted to {$F_T$} and predictable with respect to $G_T = F_T \otimes \sigma(\mu).*$

#### Proof
1. Adaptability: Each term $\psi_t X_{i,t}$ is $\mathcal{F}_T$-measurable for $t \le T$, since $X_{i,t}$ is generated by $\mathcal{F}_t \subseteq \mathcal{F}_T$. The finite sum preserves measurability.

2. Predictability in $\mathcal{G}_T$: The augmented filtration $\mathcal{G}_T$ contains information about $\mu$. The coefficients $\psi_t$ are deterministic and the AR structure allows us to write:

$$ \sum_{t=1}^{T} \psi_t X_{i,t} = \sum_{t=1}^{T} \psi_t \left( \sum_{s=1}^{p} \phi_s X_{i,t-s} + \epsilon_t \right). $$

Each term in the expansion is $\mathcal{G}_{T-1}$-measurable, as it depends on $X_{i,t-s}$ (in $\mathcal{F}_{T-1}$) and $\epsilon_t$ independent future.

3. Decay Control: The condition $\psi_t = O(t^{-\gamma})$ guarantees that the series $\sum \psi_t X_{i,t}$ converges in $L^2(\mathcal{G}_T)$ by Weierstrass criterion, since:

$$ \mathbb{E}\left[\left(\sum_{t=1}^{\infty} |\psi_t X_{i,t}|\right)^2\right] \le \left(\sum_{t=1}^{\infty} |\psi_t| \sqrt{\mathbb{E}[X_{i,t}^2]}\right)^2 < \infty $$

# 3. Error Control and Bayesian Extensions

## 3.1. Type I Error Control with Dependence

**Theorem for Bound on $\alpha^{(j,i)}$**

Let $\rho_{kl} = \text{Corr}(X_k, X_l)$ with polynomial decay:

$$ |\rho_{kl}| < \frac{C}{|k-l|^\gamma}, \quad \gamma > 1, C > 0. $$

Then for any method j:

$$ \alpha^{(j,i)} \le \frac{1}{n} + \exp\left(-\frac{n\epsilon^2}{2\sigma^2(1 + 2C\zeta(\gamma))}\right), $$

where $\zeta(\gamma) = \sum_{m=1}^{\infty} \frac{1}{m^\gamma}$ is the Riemann zeta function.

**Conceptual Justification**

1. **Correlation Decay**: Ensures that dependence between distant sectors is negligible ($\gamma > 1$).

2. **Term $\zeta(\gamma)$**: Quantifies the cumulative impact of all cross-correlations.

3. **Exponential Bound**: Arises from applying Bernstein's inequality for weakly dependent variables.

## 3.2. Extension of the Neyman-Pearson Lemma

**Definition 3.1 (Generalized Likelihood Function)**

For $m$ methods and $n$ variables:
$$ \Lambda(\mathbf{X}) = \prod_{j=1}^{m} \prod_{i=1}^{n} \frac{\sup P(D|H_{0}^{(j,i)})}{\sup P(D|H_{1}^{(j,i)})}, $$

where $D = \{X_{i,t}\}_{i=1,t=1}^{n,T}$ is the sample space of all analyzed time series (in this case, the sectoral surplus value rates corresponding to the economic activities recorded in the input-output matrix).

### Generalized Decision Rule

$$ \text{Reject } H_{0}^{(j,i)} \text{ if } \Lambda(\mathbf{X}) < k_{\alpha} \prod_{j=1}^{m} \left( 1 + \frac{\beta^{(j,i)}}{\alpha^{(j,i)}} \right), $$

with $k_\alpha$ chosen so that:

$$ \sum_{j=1}^{m} \sum_{i=1}^{n} \alpha^{(j,i)} \le \alpha_{\text{total}}. $$

### Interpretation

* Product: Captures dependencies between methods and variables.

* Factor $\beta/\alpha$: Penalizes methods with high Type II error probability.

## 3.3. Bayes' Theorem for Type I and II Errors

### Definition for Bayesian Probabilities)

Let:

* $D$: Observed data ($D \in \mathcal{D}$ sample space).

* $\pi_0 = P(H_0^{(j,i)})$: Prior probability of convergence.

* $\pi_1 = 1-\pi_0$: Prior probability of divergence.

### Type I Error (False Positive)

$$ \alpha^{(j,i)} = P(\text{Reject } H_{0}^{(j,i)} | H_{0}^{(j,i)}, D) = \frac{\int_{R_{j}^{(i)}} P(D|H_{0}^{(j,i)}) dD}{\int_{D} P(D|H_{0}^{(j,i)}) dD}. $$

### Type II Error (False Negative)

$$ \beta^{(j,i)} = P(\text{Accept } H_{0}^{(j,i)} | H_{1}^{(j,i)}, D) = \frac{\int_{D \setminus R_{j}^{(i)}} P(D|H_{1}^{(j,i)}) dD}{\int_{D} P(D|H_{1}^{(j,i)}) dD}. $$

### Bayesian Relationship

$$ \alpha^{(j,i)} = \frac{(1-\pi_0) \cdot \text{Power}^{(j,i)}}{\pi_0 \cdot (1-\text{Power}^{(j,i)}) + (1-\pi_0) \cdot \text{Power}^{(j,i)}}, $$
where Power<sup>(j,i)</sup> = 1-β<sup>(j,i)</sup> is the power of method j for variable i.

**Explanation**

* D: Represents all empirical evidence (e.g., sectoral time series).

* $\pi_0$: Initial credibility in the convergence hypothesis (e.g., based on Marxist theory, previous research, etc.).

## 3.4. Connection with Martingale Theorems

### 3.4.1. Posterior Convergence Theorem as Martingale

**Theorem**

The sequence of posterior distributions {$\pi_T(\mu|D)$} forms a reverse martingale with respect to {$G_T$}, with:

$$ \lim_{T \to \infty} \pi_T (\mu \in A | G_T) = \mathbb{I}_A (\mu_0) \quad \text{a.s.} $$

**Proof**

*Step 1: Martingale Construction*

We define $M_T = \mathbb{E}[\mathbb{I}_A(\mu) | G_T]$. By the tower property of expectations:

$$ \mathbb{E}[M_{T+1} | G_T] = \mathbb{E}[\mathbb{E}[\mathbb{I}_A(\mu) | G_{T+1}] | G_T] = M_T. $$

Thus, {$M_T, G_T$} is a reverse martingale.

*Step 2: Convergence by Doob's Theorem*

Every reverse martingale bounded in $L^1$ converges almost surely. Since $|M_T| \le 1$, we apply the theorem:

$$ \exists M_{\infty} \in L^1(G_{\infty}) \quad \text{such that} \quad M_T \xrightarrow{a.s.} M_{\infty}. $$

*Step 3: Limit Identification*

By Bayesian consistency under mixing:

$$ \pi_T (\mu \in A | D) \xrightarrow{T \to \infty} \delta_{\mu_0} (A) \quad \text{a.s.,} $$

where $\delta_{\mu_0}$ is the Dirac measure at the true $\mu_0$. Thus, $M_{\infty} = \mathbb{I}_A(\mu_0)$.

## 3.5. Comparison of Mixing

**Lemma**

For stationary processes we have:

$$ \beta \text{- mixing} \Rightarrow \alpha \text{- mixing} \Rightarrow \rho \text{- mixing} $$

**Proof**

1. $\beta \Rightarrow \alpha$:

By definition, the $\beta$ coefficient measures:
$$ \beta(F_{-\infty}^0, F_k^\infty) = \frac{1}{2} \sup \sum \| P(A_i \cap B_j) - P(A_i) P(B_j) \|, $$

while $\alpha(k) \le \beta(k)$. The inequality follows directly from the definitions (Bradley, 2005, pp. 95-96).

2. $\alpha \Rightarrow \rho$:

Using the covariance bound for stationary processes:

$$ |\mathrm{Cov}(X_0, X_k)| \le 4\alpha(k)^{1/2} \|X_0\|_2 \|X_k\|_2. $$

The $\rho$-mixing coefficient satisfies $\rho(k) \le 2\pi\alpha(k)$ (Ibragimov, 1975, p. 2), thus decaying at least as fast as $\alpha(k)$.

<caption>**Table 3.1: Types of Mixing**</caption>

# 4. Convergence, Errors, and Bayesian Probabilities

## 4.1 Fundamental Relationship: $\Gamma_j$, $\alpha^{(j,i)}$ and $\beta^{(j,i)}$

**Definition 4.1 (Convergence Indicator)**

Given the indicator:

$$ \Gamma_j(X_i, \mu) = \mathbb{I} \left( \limsup_{T \to \infty} D_j^{(T)}(X_i, \mu) < \epsilon \right), $$

we have:

$$ \Gamma_j = 1 \Leftrightarrow H_0^{(j,i)} \text{ is true} $$

$$ \Gamma_j = 0 \Leftrightarrow H_1^{(j,i)} \text{ is true} $$

**Connection with Errors**

1. *Type I Error* ($\alpha^{(j,i)}$)

$$ \alpha^{(j,i)} = P(\Gamma_j = 0 | \Gamma_j = 1), $$

is the probability that the indicator signals divergence ($\Gamma_j = 0$) when there is actually convergence ($\Gamma_j = 1$).

2. *Type II Error* ($\beta^{(j,i)}$)

$$ \beta^{(j,i)} = P(\Gamma_j = 1 | \Gamma_j = 0), $$

is the probability that the indicator signals convergence ($\Gamma_j = 1$) when there is actually divergence ($\Gamma_j = 0$).

## 4.5. Triangular Arrays

**Definition (Row Generation)**

Each row $k$ is generated by:
$$X_{i,m}^{(k)} = \sum_{j=1}^{p} \phi_j X_{i,m-j}^{(k)} + \epsilon_{i,m}^{(k)},$$

where:

* $\{\epsilon_{i,m}^{(k)}\}$ i.i.d. with $E[\epsilon] = 0$, $\text{Var}(\epsilon) = \sigma^2$.

* Decay coefficients: $|\phi_j| \le Cj^{-(1+\gamma)}$, $\gamma > 1$.

### 4.3. Interpretation of Bayesian Integrals

#### 4.3.1. For $\alpha^{(j,i)}$

$$\alpha^{(j,i)} = \frac{\int_{R_j^{(i)}} P(D|H_0^{(j,i)}) dD}{\int_{D} P(D|H_0^{(j,i)}) dD}.$$

* **Numerator:** Probability mass of data that, despite satisfying $H_0$, falls in the rejection region $R_j^{(i)}$ (false positives).

* **Denominator:** Total probability mass under $H_0$ (normalization).

* **Interpretation:** Proportion of scenarios where method $j$ incorrectly rejects the convergence of $X_i$.

#### 4.3.2. For $\beta^{(j,i)}$

$$\beta^{(j,i)} = \frac{\int_{D \setminus R_j^{(i)}} P(D|H_1^{(j,i)}) dD}{\int_{D} P(D|H_1^{(j,i)}) dD}.$$

* **Numerator:** Probability mass of data that, satisfying $H_1$, does not fall in $R_j^{(i)}$ (false negatives).

* **Denominator:** Total probability mass under $H_1$.

* **Interpretation:** Proportion of scenarios where method $j$ incorrectly accepts the convergence of $X_i$.

### 4.4. Bayes' Theorem for Error Control

#### Fundamental Relationship

$$\alpha^{(j,i)} = \frac{(1-\pi_0) \cdot \text{Power}^{(j,i)}}{\pi_0 \cdot (1-\text{Power}^{(j,i)}) + (1-\pi_0) \cdot \text{Power}^{(j,i)}}.$$

#### 4.4.1. Theorem on the Posterior Rate of Error Convergence under Dependence

Under $\alpha$-mixing dependence and prior concentration conditions, Bayesian errors decay exponentially, that is,

$$P(\Gamma_j = 0 | D^{(n)}) \le e^{-n\epsilon_n^2}, \quad \alpha^{(j,i)}, \beta^{(j,i)} = O(e^{-n\epsilon_n^2}).$$
### 4.4.1.1. Proof

The above thesis must be demonstrated starting from:

1. **Dependent data:** $D^{(n)} = \{X_{i,t}\}$ with $\alpha(k) \le Ck^{-\gamma}, \gamma > 1$

2. **Structural Power:** Power$^{(j,i)} \ge 1 - e^{-n\epsilon_n^2}$

3. **Concentrated prior:** $\pi_0 = P(H_0^{(j,i)}) \ge e^{-n\epsilon_n^2}$,

#### Step 1: KL Divergence and Donsker-Varadhan Inequality

Let $P_0 = P(\cdot | H_0)$ and $P_1 = P(\cdot | H_1)$. The KL divergence satisfies:

$$D_{\text{KL}}(P_0 \parallel P_1) \le \frac{n}{2}\epsilon_n^2 + C\sum_{k=1}^{\infty} \alpha(k).$$

Under $\alpha(k) \le Ck^{-\gamma}$, the series converges and:

$$D_{\text{KL}}(P_0 \parallel P_1) \le n\epsilon_n^2.$$

#### Step 2: Bayesian Borel-Cantelli Lemma

The posterior probability of error is:

$$P(\Gamma_j = 0 | D^{(n)}) = \frac{\pi_1 \int_{R_j^{(i)}} P(D|H_1) dD}{\pi_0 \int_{D} P(D|H_0) dD + \pi_1 \int_{D} P(D|H_1) dD}$$

Using the KL bound and $\pi_0 \ge e^{-n\epsilon_n^2}$:

$$P(\Gamma_j = 0 | D^{(n)}) \le \frac{\pi_1 e^{n\epsilon_n^2}}{\pi_0} \le e^{-n\epsilon_n^2}$$

#### Step 3: Frequentist Error Rate

By the relationship between Bayesian and frequentist errors:

$$\alpha^{(j,i)} = \mathbb{E}_{H_0} [P(\Gamma_j = 0 | D^{(n)})] \le e^{-n\epsilon_n^2}$$

Similarly, $\beta^{(j,i)} \le e^{-n\epsilon_n^2}$ by Neyman-Pearson duality.

#### 4.4.1.2. Corollary

If $\alpha(k) \asymp k^{-\gamma}$, then:

$$\epsilon_n \asymp n^{\frac{\gamma-1}{2\gamma}}.$$

Thus, for $\gamma = 2$ we verify that $\epsilon_n \asymp n^{-1/4}$, implying rapid polynomial convergence.
### 4.4.2. Variables Involved

* $\pi_0 = P(H_0^{(j,i)})$: Prior probability of convergence (e.g., based on Marxist theory, previous research, etc.).

* $\text{Power}^{(j,i)} = 1 - \beta^{(j,i)}$: Capability of method j to detect divergence when it exists.

* $\epsilon_n$: Error decay rate, linked to the dependence structure ($\gamma$).

* $\pi_0 \propto e^{-n\epsilon_n^2}$: Connection between prior concentration and dependence complexity.

### 4.4.3. Conceptual Derivation

1. **Bayes' Theorem:**

$$P(H_0|D) = \frac{P(D|H_0)P(H_0)}{P(D)}.$$

2. **Type I Error as Bayesian False Positive:**

o If $D \in R_j^{(i)}$ (rejection), but $H_0$ is true:

$$\alpha^{(j,i)} = P(H_0|D \in R_j^{(i)}) .$$

3. **Substituting:**

$$\alpha^{(j,i)} = \frac{P(D \in R_j^{(i)} | H_0) \pi_0}{P(D \in R_j^{(i)} | H_0) \pi_0 + P(D \in R_j^{(i)} | H_1) (1 - \pi_0)}$$

where:

* $P(D \in R_j^{(i)} | H_0) = \alpha^{(j,i)}$ (by classical definition).

* $P(D \in R_j^{(i)} | H_1) = \text{Power}^{(j,i)}$.

Additionally, it should be noted that this theorem shows that:

1. **Dependence $\leftrightarrow$ Model Complexity:** Higher $\gamma$, lower $\epsilon_n$ (faster decay). This is because the relationship $\gamma \leftrightarrow \epsilon_n$ captures how the dependence structure affects complexity:

1.1. A high $\gamma$ (weak dependence) denotes that distant observations are almost independent, and that the effective model has “fewer hidden parameters” (low complexity). Consequently, a small $\epsilon_n \Rightarrow$ errors decay quickly ($\epsilon_n \approx n^{-0.4}$ for $\gamma = 3$).

1.2. A low $\gamma$ (strong dependence) denotes that correlations persist at long distances and that the model has “more hidden degrees of freedom” (high complexity). Therefore, a large $\epsilon_n$ implies that errors decay slowly ($\epsilon_n \approx n^{-0.1}$ for $\gamma = 1.1$).

2. **Informative Prior:** If $\pi_0$ reflects knowledge of $\gamma$, it accelerates posterior concentration. An informative prior that incorporates knowledge of $\gamma$ (dependence decay rate) accelerates posterior concentration because the dependence structure translates into initial knowledge (prior) as follows: when we know that $\gamma$ is large (dependence decays quickly), we assign high probability to $H_0$ ($\pi_0 \approx 1$), which reduces initial uncertainty, allowing the posterior to concentrate faster on the true hypothesis. For example, if the data comes
from a system with $\gamma = 3$, using $\pi_0 \propto e^{-n\epsilon_n^2}$ with $\epsilon_n \asymp n^{-1/3}$ accelerates convergence compared to a uniform prior.

3. **KL Balance:** The Donsker-Varadhan inequality quantifies how dependence limits the optimal rate.

### 4.4.4. Implications

* **Dependence on $\pi_0$:** If $\pi_0 \to 1$ (strong belief in convergence), $\alpha^{(j,i)} \to 0$.

* **Power-Type I Error Trade-off:** For fixed $\pi_0$:

$$ \alpha^{(j,i)} \propto \frac{\text{Power}^{(j,i)}}{1 - \text{Power}^{(j,i)}}. $$

* **Cost of Dependence:** Errors decay as $e^{-n^{1-1/\gamma}}$ vs. $e^{-n}$ in i.i.d.

* **Experimental Design:** To guarantee $\alpha^{(j,i)} < 0.05$, we need:

$$ n \ge \left( \frac{\log(20)}{\epsilon_n^2} \right)^{\frac{\gamma}{\gamma-1}} $$

* **Stein's Paradox:** Low dependence ($\gamma \to \infty$) recovers i.i.d. rates, but high dependence ($\gamma \to 1$) makes inference impossible. This paradox shows that, in estimating 3 or more parameters, joint estimators outperform individual ones in precision. In our context, when there is low dependence ($\gamma \to \infty$) the variables are almost independent (i.i.d.), allowing estimation of each parameter separately without loss of efficiency. This independence recovers the classical scenario where separate methods are optimal because strong correlations ($\gamma \to 1$) prevent treating parameters in isolation, requiring joint estimators that “steal information” between parameters, but extreme dependence makes even this impossible. Therefore, the paradox manifests as a theoretical limit establishing that in scenarios of weak dependence (large $\gamma$) we can use i.i.d. methods, while in those of strong dependence (small $\gamma$) no method (neither joint nor individual) works well.

## 4.5. Relationship between Convergence, Errors, and Bayesian Probabilities (Explicit Inequalities)

### 4.5.1. Non-Asymptotic Probabilistic Bounds

#### 4.5.1.1. Bernstein's Inequality for Mixingales

For $\{\psi_t X_{i,t}\}$ with $\|\psi_t X_{i,t}\|_{\infty} \le B$ and $\sum_{t=1}^T \psi_t^2 \le S_T^2$:

$$ P\left(\left|\sum_{t=1}^{T} \psi_t(X_{i,t}-\mu)\right| > \epsilon\right) \le 2\exp\left(-\frac{\epsilon^2}{2(S_T^2 + B\epsilon/3)}\right) + C_T\alpha(k_T), $$

where $k_T = \lfloor (\epsilon/B)^{1/\gamma} \rfloor$ and $C_T = O(T^{1/(1+\gamma)})$.
### 4.5.1.1.1. Specification of Triangular Arrays

#### Definition (Row Generation)

Each row $k$ is generated by:

$$X_{i,m}^{(k)} = \sum_{j=1}^{p} \phi_j X_{i,m-j}^{(k)} + \epsilon_{i,m}^{(k)},$$

where:

* $\{\epsilon_{i,m}^{(k)}\}$ i.i.d. with $E[\epsilon] = 0$, $Var(\epsilon) = \sigma^2$.

* Decay coefficients: $|\phi_j| \le Cj^{-(1+\gamma)}$, $\gamma > 1$.

#### Verification of the Lindeberg Condition

For all $\delta > 0$:

$$\lim_{k \to \infty} \frac{1}{k^{1+\delta/2}} \sum_{m=1}^{k} \mathbb{E} \left[ \left| X_{i,m}^{(k)} \right|^{2+\delta} \right] = 0.$$

The above is justified as a result of

1. *Moment Recursion*:

The AR structure implies

$$\| X_{i,m}^{(k)} \|_{2+\delta} \le C \sum_{j=1}^{\infty} |\phi_j| \| X_{i,m-j}^{(k)} \|_{2+\delta} + \| \epsilon \|_{2+\delta}.$$

2. *Geometric Decay*:

The condition $|\phi_j| \le Cj^{-(1+\gamma)}$ guarantees absolute convergence of the series.

3. *Lindeberg Rate*:

The factor $k^{-\delta/2}$ dominates the finite sum:

$$\frac{1}{k^{1+\delta/2}} \sum_{m=1}^{k} C \le Ck^{-\delta/2} \xrightarrow{k \to \infty} 0.$$

### 4.5.1.1.2. Correlation-Weighted Sums

**Bernstein's Theorem for Correlation-Weighted Sums** For the estimator $\sum_{i=1}^{n} w_i X_i$ with $|w_i X_i| \le B/n^{1/2}$ and $Var(\sum w_i X_i) \le \sigma^2/n$, we have
$$P(|\sum w_i X_i - \mu| > \epsilon) \le 2\exp\left(-\frac{n\epsilon^2}{2(\sigma^2 + B\epsilon/3)}\right) + Cn^{-\gamma}.$$

### Proof

#### Step 1: Block Stationarization

First, let's begin by dividing the series {$X_i$} into $m = \lfloor n^{1/(1+\gamma)} \rfloor$ blocks of size $b = \lfloor n^{\gamma/(1+\gamma)} \rfloor$. By $\alpha$-mixing with decay $O(k^{-\gamma})$, the blocks are approximately independent with residual correlation $O(m^{-\gamma})$.

#### Step 2: Apply Bernstein to Blocks

For each block $B_k$, define $Y_k = \sum_{i \in B_k} w_i X_i$. By construction, the {$Y_k$} are weakly dependent:

$$\mathbb{E}[Y_k] = O(b/n^{1/2}), \quad \mathrm{Var}(Y_k) = O(b/n)$$

Applying Bernstein to the sum $\sum_{k=1}^m Y_k$:

$$P(|\sum Y_k| > \epsilon) \le 2\exp\left(-\frac{\epsilon^2}{2(\sigma^2 + B\epsilon/3)}\right)$$

#### Step 3: Dependence Correction

The error due to inter-block dependence is bounded using the $\alpha$-mixing bound:

$$|P(|\sum X_i| > \epsilon) - P(|\sum Y_k| > \epsilon)| \le C\alpha(b) = Cn^{-\gamma}$$

#### Step 4: Combination of Terms

Adding the bounds for independent blocks and the approximation error:

$$P(|\sum w_i X_i - \mu| > \epsilon) \le 2\exp\left(-\frac{n\epsilon^2}{2(\sigma^2 + B\epsilon/3)}\right) + Cn^{-\gamma}.$$

### 4.5.1.1.3. Mixingale Processes

#### McDiarmid's Theorem for Mixingale Processes

For the process $D_T = \sum_{t=1}^T \psi_t (X_{i,t} - \mu)$ with $\| \psi_t (X_{i,t} - \mu) \|_{\infty} \le B_t$, we have:

$$P(|D_T| > \epsilon) \le 2\exp\left(-\frac{\epsilon^2}{2\sum_{t=1}^T B_t^2}\right) + C\sum_{t=1}^T \alpha(t).$$

### Proof

#### Step 1: Martingale Decomposition

Let's begin by constructing the martingale $M_t = \mathbb{E}[D_T | \mathcal{F}_t]$ with differences $d_k = M_k - M_{k-1}$. By the mixingale property, we have:
$$ \| d_k \|_{\infty} \le 2B_k \psi_k \quad \text{and} \quad \sum_{k=1}^{T} \mathbb{E}[d_k^2] \le 4 \sum_{k=1}^{T} B_k^2 \psi_k^2 $$

### Step 2: Apply Azuma-Hoeffding Inequality

For the martingale $M_T$ with bounded differences:

$$ P(|M_T| > \epsilon) \le 2\exp\left(-\frac{\epsilon^2}{8\sum B_k^2 \psi_k^2}\right) $$

### Step 3: Control Bias Term

The residual bias $D_T - M_T$ satisfies:

$$ \mathbb{E} |D_T - M_T| \le C \sum_{t=1}^{T} \alpha(t) $$

By Markov's inequality:

$$ P(|D_T - M_T| > \epsilon) \le \frac{C}{\epsilon} \sum_{t=1}^{T} \alpha(t) $$

### Step 4: Union of Bounds

Combining with triangle inequality:

$$ P(|D_T| > \epsilon) \le 2\exp\left(-\frac{\epsilon^2}{8\sum B_k^2 \psi_k^2}\right) + \frac{C}{\epsilon} \sum \alpha(t) $$

Optimizing $\epsilon$ yields the final form. •

## Conceptual Foundation

1. *Blocking in WSC*: Yu's (1994, pp. 101, 103-104) technique allows treating weak dependencies through partitions that "isolate" the process memory. The balance between block size (b) and number of blocks (m) optimizes the error rate.

2. *Martingale Strategy in MPC*: By incrementally conditioning on the filtration $\mathcal{F}_t$, the mixingale is transformed into a martingale with controlled differences. The bias term captures the residual dependence not absorbed by the martingale.

3. *Role of Mixing Coefficients*: In both cases, the decay rate of $\alpha(k)$ determines the convergence speed of the error term. For $\gamma > 1$, the series $\sum \alpha(k)$ converge, allowing non-asymptotic bounds.
## 4.6. Summary Diagram of the Relationships Between Convergence Methods, Errors, and Bayesian Framework

<figure><img src="image_2.png" /></figure>
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Gómez Julián, J. M. (2023). *Marxist Average Rate of Profit for United States (1960-2020) with Documentation* [Dataset]. OSF. https://doi.org/10.17605/OSF.IO/7J8KF

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https://doi.org/10.31219/osf.io/k435g_v1

Gómez Julián, J. M. (2025b). *Sectorial Exclusion Criteria in the Marxist Analysis of the Average Rate of Profit: The United States Case (1960-2020)* (Version 1). arXiv.
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Hall, P., & Heyde, C. C. (1980). *Martingale Limit Theory and Its Application*. Academic Press.

Ibragimov, I.A. (1975). A Note on the Central Limit Theorems for Dependent Random Variables. *Theory of Probability & Its Applications*, Vol. 20.

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