M-Dependent Random Variables

• M-dependent random variables are defined by a finite-range dependence structure where variables separated by more than m indices are independent.
• They are central to establishing limit theorems and sharp error bounds in time series, spatial statistics, and applied probability.
• Analytic techniques such as Stein’s method, characteristic function expansions, and block partitioning provide practical tools for handling m-dependence.

An m-dependent (or M-dependent) sequence of random variables is a fundamental object in probability theory and its applications, characterized by a local, finite-range dependence structure. Specifically, a sequence {Xₖ} is m-dependent if every pair of subcollections separated by more than m indices are independent. This local structure arises naturally in time series, spatial statistics, combinatorics, and applied probability, and acts as a generalization bridging the gap between independent and globally dependent processes. m-dependence enables the adaptation of powerful limit theorems and approximation methods with explicit control over dependence effects. The topic encompasses discrete and continuous settings, strong and weak limit theorems, sharp error bounds, and a spectrum of methodologies including characteristic function expansions, Stein’s method, and sub-linear expectations.

1. Formal Definition and Structural Properties

A sequence {Xₖ, k ∈ ℤ} is m-dependent if, for any s and t with t – s > m, the σ-algebras σ(X₁, …, Xₛ) and σ(Xₜ, Xₜ₊₁, …) are independent (Čekanavičius et al., 2013, Cekanavicius et al., 2014, Hoang et al., 20 Aug 2025). Equivalently, any two subsets of random variables separated by more than m indices are independent. For m = 0, this reduces to the classical notion of independence.

Key points:

• Local dependence: Only variables within distance m may be dependent; those further apart are independent.
• Reduction to 1-dependence: Grouping consecutive variables (e.g., via blocking) can reduce m-dependent sequences to 1-dependent ones, facilitating analysis using methods optimized for 1-dependence (Čekanavičius et al., 2013, Vellaisamy et al., 2013).
• Blockwise m-dependence: In some generalizations, such as partitioned or blockwise dependence, the sequence is divided into blocks each of which may exhibit internal dependence up to m, but blocks beyond m apart remain independent (Fu, 10 Apr 2025).

2. Limit Theorems and Degeneracy Criteria

Central Limit Theorems:

Classical central limit theorems (CLT) extend to m-dependent processes under mild conditions. For stationary m-dependent {Xₖ}, with moment controls, the partial sum Sₙ = X₁ + ⋯ + Xₙ satisfies

$$\frac{Sₙ - nμ}{\sqrt{n(\mathrm{Var}(X₁) + 2 \sum_{j=1}^m \mathrm{Cov}(X₁, X_{1+j}))}} \Longrightarrow \mathcal{N}(0,1),$$

where μ = E[X₁], and the variance includes all nonzero covariances within m steps (Islak, 2013).

Random sums (where the summation range is itself random and independent) also admit CLTs, with variances modified to incorporate both m-dependent covariance structure and the randomness in the number of summands (Islak, 2013).

Degeneracy in CLT:

The normalized sum of a stationary m-dependent sequence can exhibit degenerate behavior (i.e., asymptotic variance zero), even if each Xₖ is non-degenerate. Degeneracy occurs if and only if the centered variable admits the difference representation

$Xₖ - E[Xₖ] = Yₖ - Y_{k-1},$

for some stationary (m–1)-dependent sequence {Yₖ} (Janson, 2013). In this case, the partial sum Sₙ – E[Sₙ] = Yₙ – Y₀ does not “spread out” with n, and the normalization to obtain a nontrivial limit fails. This result yields a straightforward criterion: if Sₙ is not almost surely constant as a function of some subset of the underlying variables, the limit is nondegenerate (Janson, 2013).

3. Discrete Approximations and Error Bounds

Discrete m-dependent sums admit sharp distributional approximations by classical discrete families: Poisson, compound Poisson, negative binomial, and binomial distributions. Key technical results are:

• Characteristic function expansion: The law of the sum Sₙ is compared to a target measure G with characteristic function

$$\exp \left\{ \Gamma_1 (e^{it}-1) + \Gamma_2 (e^{it})^2 - 1 + \cdots \right\},$$

where the Γᵢ are factorial cumulants (Čekanavičius et al., 2013).

• Moment-matching approximations: The negative binomial distribution is used when $\mathrm{Var}(Sₙ) > E(Sₙ)$, the binomial when $\mathrm{Var}(Sₙ) < E(Sₙ)$. Explicit formulas for parameter selection ensure first and second moments are matched (Vellaisamy et al., 2013).
• Total variation and local bounds: The error between the true law of Sₙ and the approximation is bounded in total variation and local metrics. For example,

$$\| Fₙ - \mathrm{NB}(r, \bar{q}) \|_{TV} \leq C\, \min\left(1, \Gamma_1^{-3/2}\right) (R + (\Gamma_1/\Gamma_2 - 1))$$

with R a remainder related to higher moments and covariance decay (Čekanavičius et al., 2013).

Applications:

Applications include:

• Run statistics and pattern counts: 2-runs (counts of adjacent successes) and (k₁, k₂)-event statistics (blocks of k₁ failures followed by k₂ successes) are m-dependent. The distributions of these statistics are sharply approximated using negative binomial or binomial laws, with explicit O(n^{–1/2}) error terms (Čekanavičius et al., 2013, Vellaisamy et al., 2013).
• Poisson binomial approximations: The general framework recovers and sharpens results for sums of independent (m = 0) or Bernoulli-like (m = 1) variables with rare events (Vellaisamy et al., 2013).

4. Strong Laws, Law of the Iterated Logarithm, and Large Deviations

Strong Law of Large Numbers (SLLN):

The SLLN holds for m-dependent and stationary random variables under both linear and sublinear expectations, provided an appropriate integrability condition is met (e.g., finite Choquet integral in the sublinear case). The convergence of sample averages persists under m-dependence and can be formulated in capacity terms (Gu et al., 1 Apr 2024, Fu, 10 Apr 2025).

Law of the Iterated Logarithm (LIL):

For m-dependent stationary sequences under sublinear expectations, the normalized partial sums have their limsup contained in an interval determined by the limiting upper and lower “variances,” provided a Choquet integral condition holds: $$C_V\left[\frac{X_1^2}{\log\log|X_1|}\right] < \infty.$$ The law mirrors the form of the classical LIL, with limsup scaling as $\sqrt{2n\log\log n}$ and cluster set $[\underline{\sigma}, \overline{\sigma}]$, where $\underline{\sigma}^2$ and $\overline{\sigma}^2$ are the limits of lower and upper variances (Gu et al., 12 Jun 2025).

Large Deviations:

Sharp asymptotic ratios are obtained for point and tail probabilities of m-dependent sums relative to Poisson, negative binomial, or binomial approximations. Asymptotic expansions of the form

$$\frac{P(S_n = x)}{P(Z_n = x)} = \exp\{\Lambda(y)\}\left(1 + o(1)\right)$$

(where $Z_n$ is an appropriate reference distribution and $\Lambda(y)$ encodes Cramér-type corrections) describe the rare-event probabilities and show the importance of moment-matched approximations for extending “zones of equivalence” (Čekanavičius et al., 2019).

5. Berry–Esseen Bounds, High-dimensional CLTs, and Self-normalization

Optimal Berry–Esseen Rates:

Self-normalized sums (e.g., Studentized statistics) of m-dependent variables admit optimal Berry–Esseen bounds of order $O(n^{-1/2})$, with error constants depending polynomially on m and the local dependence structure (Zhang, 2021). Notably, only a finite third moment is required, which relaxes standard fourth moment assumptions.

Quantitative High-dimensional CLTs:

For m-dependent sequences of high-dimensional random vectors, the Kolmogorov–Smirnov distance between the sum’s law and that of the Gaussian approximation over rectangles admits a uniform rate of $(n/m)^{-1/2}$, where n is the sample size and m the dependence range (Bong et al., 2022). Proofs utilize telescoping and anti-concentration methods, adapting tools from independent to dependent settings, and only minimal covariance and third moment conditions are assumed.

Wasserstein and Total Variation Distances:

Explicit non-asymptotic bounds are given for Wasserstein and total variation distances in the CLT for m-dependent sequences. Representative bounds are

$$d_W\left(\frac{S_n}{\sigma_n}, Z \right) \leq 30 \left\{ c^{1/3} + 12\, U_n(c/2)^{1/2} \right\}$$

with $U_n(c)$ a Lindeberg-type sum over the tails of individual summands (Janson et al., 2022).

6. Ruin Probabilities and Applied Models

In discrete-time risk models, m-dependent random variables are incorporated as premium and claim processes. The classical Lundberg exponential inequality is generalized: $$\Psi(u) \leq (m+1)\exp\left(-\frac{R u}{m+1}\right)$$ for the ultimate ruin probability Ψ(u), reflecting the “cost” of dependency in the exponential decay rate. Partitioning techniques split the process into (m+1) independent subsequences, permitting the adaptation of supermartingale-based proofs (Hoang et al., 20 Aug 2025). This explicit treatment is directly applicable in insurance mathematics, actuarial science, and queueing models with local dependencies.

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Summary Table: Key Limit Results for m-Dependent Random Variables

Theorem/Result	Main Statement/Rate/Condition	Reference
CLT for Sums	Normalized sum converges to N(0,1) w/ var includes $\sum_{j=1}^m \mathrm{Cov}(X_1,X_{1+j})$	(Islak, 2013)
Degeneracy Criterion	Asymptotic variance = 0 iff $X_k - E[X_k] = Y_k - Y_{k-1}$ (Yₖ (m–1)-dep.)	(Janson, 2013)
SLLN (Sublinear)	$\lim_{n\to\infty} S_n/n$ exists in capacity, $C_{\mathbb V}(|X_1|)<\infty$	(Gu et al., 1 Apr 2024, Fu, 10 Apr 2025)
Berry–Esseen (Self-norm.)	Optimal $O(n^{-1/2})$ rate, error ∝ m^a, only 3rd moment needed	(Zhang, 2021)
High-dim. CLT Rate	$(n/m)^{-1/2}$ rate for Kolmogorov–Smirnov distance	(Bong et al., 2022)
Ruin Probability Bound	$\Psi(u) \leq (m+1) \exp(-R u/(m+1))$	(Hoang et al., 20 Aug 2025)

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7. Methodological Tools and Key Techniques

Several analytic methods underpin the modern theory of m-dependent random variables:

• Heinrich’s characteristic function method: Expansion of $\prod_k \phi_k(t)$, preserving the product structure under m-dependence, and matching cumulant terms order by order for sharp approximation (Čekanavičius et al., 2013, Čekanavičius et al., 2019).
• Convolution smoothing: Convolution with “nice” discrete measures increases smoothness and localization in error bounds, crucial for obtaining local and total variation bounds (Čekanavičius et al., 2013).
• Stein’s Method: Construction of analytic operators (Stein operators), both for classical and power series distributions, and solving corresponding Stein equations with explicit gradient bounds. Used for total variation and Wasserstein bounds (Kumar et al., 2020, Zhang, 2021, Anastasiou, 2016).
• Block partitioning: Grouping variables into blocks of length m+1 to restore effective independence or 1-dependence, facilitating use of independent-process results in both theoretical and applied contexts (Čekanavičius et al., 2013, Fu, 10 Apr 2025, Hoang et al., 20 Aug 2025).
• Capacity and sublinear expectation analysis: Development of limit theorems (SLLN, LIL, CLT) under sublinear expectation frameworks, using Choquet integrals and capacity-theoretic probability (Gu et al., 1 Apr 2024, Gu et al., 12 Jun 2025, Gu et al., 2023).

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8. Applications and Implications

m-dependent models encapsulate realistic local correlations in observed data, including in time series (short range dependencies), run or pattern statistics in sequences, reliability analysis, and actuarial science (e.g., risk and ruin theory). The analytic framework provides explicit error quantification for distributional approximations, extends classical probabilistic results to settings of uncertainty (sublinear expectations), and enables robust statistical inference even under mild deviations from independence. The dependency parameter m directly enters error constants and exponential rates, allowing practitioners to interpret, quantify, and, where appropriate, trade off between model realism (allowing dependence) and statistical tractability.