In probability, weak dependence of random variables is a generalization of independence that is weaker than the concept of a martingale. A (time) sequence of random variables is weakly dependent if distinct portions of the sequence have a covariance that asymptotically decreases to 0 as the blocks are further separated in time. Weak dependence primarily appears as a technical condition in various probabilistic limit theorems.
Formal definition Fix a set S, a sequence of sets of measurable functions { F d } d = 1 ∞ ∈ ∏ d = 1 ∞ ( S d → R ) {\displaystyle \{{\mathcal {F}}_{d}\}_{d=1}^{\infty }\in \prod _{d=1}^{\infty }{\left(S^{d}\to \mathbb {R} \right)}} , a decreasing sequence { θ δ } δ = 1 ∞ → 0 {\displaystyle \{\theta _{\delta }\}_{\delta =1}^{\infty }\to 0} , and a function ψ ∈ F 2 × ( Z + ) 2 → R + {\displaystyle \psi \in {\mathcal {F}}^{2}\times (\mathbb {Z} ^{+})^{2}\to \mathbb {R} ^{+}} . A sequence { X n } n = 1 ∞ {\displaystyle \{X_{n}\}_{n=1}^{\infty }} of random variables is ( { F d } d = 1 ∞ , { θ δ } δ , ψ ) {\displaystyle (\{{\mathcal {F}}_{d}\}_{d=1}^{\infty },\{\theta _{\delta }\}_{\delta },\psi )} -weakly dependent iff, for all j 1 ≤ j 2 ≤ ⋯ ≤ j d < j d + δ ≤ k 1 ≤ k 2 ≤ ⋯ ≤ k e {\displaystyle j_{1}\leq j_{2}\leq \dots \leq j_{d}<j_{d}+\delta \leq k_{1}\leq k_{2}\leq \dots \leq k_{e}} , for all ϕ ∈ F d {\displaystyle \phi \in {\mathcal {F}}_{d}} , and θ ∈ F e {\displaystyle \theta \in {\mathcal {F}}_{e}} , we have
| Cov ( ϕ ( X j 1 , … , X j d ) , θ ( X k 1 , … , X k e ) ) | ≤ ψ ( ϕ , θ , d , e ) θ δ {\displaystyle |\operatorname {Cov} {(\phi (X_{j_{1}},\dots ,X_{j_{d}}),\theta (X_{k_{1}},\dots ,X_{k_{e}}))}|\leq \psi (\phi ,\theta ,d,e)\theta _{\delta }}
Note that the covariance does not decay to 0 uniformly in d and e.
… excerpt ends here. Continue reading the full article.
