Weakly dependent random variables

In probability, weak dependence of random variables is a generalization of independence that is weaker than the concept of a martingaleTemplate: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 Template:Mvar, a sequence of sets of measurable functions $\{{ℱ}_d{\}}_{d=1}^{\mathrm{\infty }}\in \prod _{d=1}^{\mathrm{\infty }}(S^d\to ℝ)$, a decreasing sequence $\{{\theta }_{\delta }{\}}_{\delta =1}^{\mathrm{\infty }}\to 0$, and a function $\psi \in {ℱ}^2\times ({\mathbb{Z}}^{+}{)}^2\to {ℝ}^{+}$. A sequence $\{X_n{\}}_{n=1}^{\mathrm{\infty }}$ of random variables is $(\{{ℱ}_d{\}}_{d=1}^{\mathrm{\infty }},\{{\theta }_{\delta }{\}}_{\delta },\psi )$-weakly dependent iff, for all $j_1\le j_2\le \dots \le j_d<j_d+\delta \le k_1\le k_2\le \dots \le k_e$, for all $\varphi \in {ℱ}_d$, and $\theta \in {ℱ}_e$, we haveTemplate:Rp

$|\mathrm{Cov}(\varphi (X_{j_1},\dots ,X_{j_d}),\theta (X_{k_1},\dots ,X_{k_e}))|\le \psi (\varphi ,\theta ,d,e){\theta }_{\delta }$

Note that the covariance does not decay to Template:Math uniformly in Template:Mvar and Template:Mvar.Template:Rp

Common applications

Weak dependence is a sufficient weak condition that many natural instances of stochastic processes exhibit it.Template:Rp In particular, weak dependence is a natural condition for the ergodic theory of random functions.

A sufficient substitute for independence in the Lindeberg–Lévy central limit theorem is weak dependence.Template:Rp For this reason, specializations often appear in the probability literature on limit theorems.Template:Rp These include Withers' condition for strong mixing, Tran's "absolute regularity in the locally transitive sense," and Birkel's "asymptotic quadrant independence."

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

Other uses include a generalization of the Marcinkiewicz–Zygmund inequality and Rosenthal inequalities.Template:Rp

Martingales are weakly dependent Template: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

$\begin{array}{cc}\mathrm{E}[X_i\mid X_1,\dots ,X_{i-1}]& =0,\\ \mathrm{E}[X_i^2\mid X_1,\dots ,X_{i-1}]& \le R_i\mathrm{E}\left[X_i^2\right],\\ \mathrm{E}[X_i^k\mid X_1,\dots ,X_{i-1}]& \le \frac{1}{2}\mathrm{E}[X_i^2\mid X_1,\dots ,X_{i-1}]L^{k-2}k!\end{array}$

See also

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

External links

• An example inspiring this idea
• An application
• A paper on an alternative to weak dependence