A Note on Physical Dependence and Mixing Conditions for Triangular Arrays

Under mild structural assumptions and regularity conditions on the marginal and conditional densities, an explicit bound on the $β$-mixing coefficients in terms of the physical dependence measure is provided. Consequently, weak physical dependence implies $β$-mixing and strong mixing for triangular arrays, complementing Hill (2025), who proved the converse implication under moment assumptions.

💡 Research Summary

The paper investigates the relationship between physical dependence and classical mixing conditions for triangular arrays of random variables. While many dependence measures exist—mixing coefficients (α, β, ρ, ψ) and the physical dependence measure δₚ introduced by Wu (2005)—their interconnections have been only partially understood. Hill (2025) showed that strong mixing implies weak physical dependence under moment assumptions, but the converse direction was missing.

The authors consider a locally stationary triangular array X = {X_{i,n}}{i=1,…,n} indexed by n, assuming that each observation can be written as X{i,n}=G(i/n,F_i), where G is a measurable filter, F_i = σ(ε_k, k≤i) is generated by i.i.d. innovations ε, and a copy ε* is used to define the physical dependence measure
δ₁(G,i)=E|G(i/n,F_i)−G(i/n,F_i^*)|.
Weak physical dependence of order 1 means that the tail sum Θ_k = Σ_{h≥k} δ₁(G,h) tends to zero as k→∞.

Two structural assumptions are imposed.

1. Assumption 1 (existence of the filter and bounded order‑1 physical dependence).
2. Assumption 2 (regularity of marginal and conditional densities). Specifically, the joint marginal density p_{j+k,n} of (X_{j+k,n},…,X_{n,n}) and the conditional density q_{j,k,n} of the same block given the past σ‑algebra must possess L¹ partial derivatives. Moreover, weighted sums of the L¹ norms of these derivatives, with positive weights w_m satisfying Σ w_m ≤ 1, are uniformly bounded by constants D₁ and D₂.

Under these conditions, Theorem 1 establishes an explicit bound for the β‑mixing coefficients:
β(k) ≤ C·q·D·Θ_k, where D = max{D₁,D₂}, C is an absolute constant, and q is a universal factor arising from the proof. Consequently, if the physical dependence measure decays (Θ_k → 0), the β‑mixing coefficients also decay, implying strong α‑mixing (since α(k) ≤ ½β(k)). This result complements Hill’s converse theorem, completing the equivalence between weak physical dependence and mixing for triangular arrays under the stated regularity.

The proof proceeds in four main steps:

• Coupling construction – Define a copy \tilde X that replaces the past innovations by independent copies while keeping the same marginal distribution. Using the definition of δ₁, the expected weighted ℓ¹ distance between X and \tilde X is bounded by Θ_k.
• Kantorovich–Rubinstein duality – Relate the Wasserstein‑1 distance W₁,d between the marginal law P_{j+k,n} and the conditional law Q_{j,k,n} to the expected coupling distance, yielding W₁,d(P,Q) ≤ Θ_k.
• Mollification and density comparison – Introduce a product mollifier Φ_{w,ε} to smooth both densities. Proposition 1 bounds the smoothing error by ε times the weighted L¹ norms of the derivatives; Proposition 2 bounds the L¹ distance of the smoothed densities by ε·W₁,d(P,Q). Optimizing ε leads to ∥p−q∥₁ ≤ C·D·W₁,d(P,Q).
• From total variation to β‑mixing – Using Scheffé’s identity (∥p−q∥₁ = 2 d_{TV}) and the definition β(k)=sup_j E