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.
Fix a set S, a sequence of sets of measurable functions , a decreasing sequence , and a function . A sequence of random variables is -weakly dependent iff, for all , for all , and , we have [1]: 315
Note that the covariance does not decay to 0 uniformly in d and e. [2]: 9
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]
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.
Fix a set S, a sequence of sets of measurable functions , a decreasing sequence , and a function . A sequence of random variables is -weakly dependent iff, for all , for all , and , we have [1]: 315
Note that the covariance does not decay to 0 uniformly in d and e. [2]: 9
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]