Weakly dependent random variables

In probability, weak dependence of random variables is a generalization of independence that is weaker than the concept of a martingale^{[citation needed]}. 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 ${\displaystyle \{{\mathcal {F}}_{d}\}_{d=1}^{\infty }\in \prod _{d=1}^{\infty }{\left(S^{d}\to \mathbb {R} \right)}}$, a decreasing sequence ${\displaystyle \{\theta _{\delta }\}_{\delta =1}^{\infty }\to 0}$, and a function ${\displaystyle \psi \in {\mathcal {F}}^{2}\times (\mathbb {Z} ^{+})^{2}\to \mathbb {R} ^{+}}$. A sequence ${\displaystyle \{X_{n}\}_{n=1}^{\infty }}$ of random variables is ${\displaystyle (\{{\mathcal {F}}_{d}\}_{d=1}^{\infty },\{\theta _{\delta }\}_{\delta },\psi )}$-weakly dependent iff, for all ${\displaystyle j_{1}\leq j_{2}\leq \dots \leq j_{d}<j_{d}+\delta \leq k_{1}\leq k_{2}\leq \dots \leq k_{e}}$, for all ${\displaystyle \phi \in {\mathcal {F}}_{d}}$, and ${\displaystyle \theta \in {\mathcal {F}}_{e}}$, we have^[1]: 315

${\displaystyle |\operatorname {Cov} {(\phi (X_{j_{1}},\dots ,X_{j_{d}}),\theta (X_{k_{1}},\dots ,X_{k_{e}}))}|\leq \psi (\phi ,\theta ,d,e)\theta _{\delta }}$

Note that the covariance does not decay to 0 uniformly in d and e.^[2]: 9

Common applications

Weak dependence is a sufficient weak condition that many natural instances of stochastic processes exhibit it.^[2]: 9 In particular, weak dependence is a natural condition for the ergodic theory of random functions.^[3]

A sufficient substitute for independence in the Lindeberg–Lévy central limit theorem is weak dependence.^[1]: 315 For this reason, specializations often appear in the probability literature on limit theorems.^{[2]: 153–197} These include Withers' condition for strong mixing,^[1][4] Tran's "absolute regularity in the locally transitive sense,"^[5] and Birkel's "asymptotic quadrant independence."^[6]

Weak dependence also functions as a substitute for strong mixing.^[7] Again, generalizations of the latter are specializations of the former; an example is Rosenblatt's mixing condition.^[8]

Other uses include a generalization of the Marcinkiewicz–Zygmund inequality and Rosenthal inequalities.^{[1]: 314, 319}

Martingales are weakly dependent ^{[citation needed]}, so many results about martingales also hold true for weakly dependent sequences. An example is Bernstein's bound on higher moments, which can be relaxed to only require^[9][10]

${\displaystyle {\begin{aligned}\operatorname {E} \left[X_{i}\mid X_{1},\dots ,X_{i-1}\right]&=0,\\\operatorname {E} \left[X_{i}^{2}\mid X_{1},\dots ,X_{i-1}\right]&\leq R_{i}\operatorname {E} \left[X_{i}^{2}\right],\\\operatorname {E} \left[X_{i}^{k}\mid X_{1},\dots ,X_{i-1}\right]&\leq {\tfrac {1}{2}}\operatorname {E} \left[X_{i}^{2}\mid X_{1},\dots ,X_{i-1}\right]L^{k-2}k!\end{aligned}}}$

See also

• Central limit theorem
• Independence of random variables
• Martingale (probability theory)