Weakly Dependent Random Variables
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In probability, weak dependence of random variables is a generalization of
independence Independence is a condition of a nation, country, or state, in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the status of ...
that is weaker than the concept of a martingale. A (time)
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s is weakly dependent if distinct portions of the sequence have a
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one ...
that
asymptotically In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
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 , a sequence of sets of
measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
s \_^\in\prod_^, a decreasing sequence \_^\to0, and a function \psi\in\mathcal^2\times(\mathbb^+)^2\to\mathbb^+. A sequence \_^ of random variables is (\_^,\_,\psi)-weakly dependent iff, for all j_1\leq j_2\leq\dots\leq j_d, for all \phi\in\mathcal_d, and \theta\in\mathcal_e, we have : , \operatorname, \leq\psi(\phi,\theta,d,e)\theta_ Note that the covariance does ''not'' decay to uniformly in and .


Common applications

Weak dependence is a sufficient weak condition that many natural instances of stochastic processes exhibit it. 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. For this reason, specializations often appear in the probability literature on limit theorems. 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. Martingales are weakly dependent , 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 \operatorname \left X_i \mid X_1, \dots, X_ \right &= 0, \\ \operatorname \left X_i^2 \mid X_1, \dots, X_ \right &\leq R_i \operatorname \left X_i^2 \right \\ \operatorname \left X_i^k \mid X_1, \dots, X_ \right &\leq \tfrac \operatorname \left X_i^2 \mid X_1, \dots, X_ \right L^ k! \end


See also

*
Central limit theorem In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
* Independence of random variables *
Martingale (probability theory) In probability theory, a martingale is a stochastic process in which the expected value of the next observation, given all prior observations, is equal to the most recent value. In other words, the conditional expectation of the next value, given ...


References

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External links


An example inspiring this idea

An application

A paper on an alternative to weak dependence
Stochastic processes