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)

References

1. Doukhan, Paul; Louhichi, Sana (1999-12-01). "A new weak dependence condition and applications to moment inequalities". Stochastic Processes and Their Applications. 84 (2): 313–342. doi:10.1016/S0304-4149(99)00055-1. ISSN 0304-4149.
2. Dedecker, Jérôme; Doukhan, Paul; Lang, Gabriel; Louhichi, Sana; Leon, José Rafael; José Rafael, León R.; Prieur, Clémentine (2007). Weak Dependence: With Examples and Applications. Lecture Notes in Statistics. Vol. 190. doi:10.1007/978-0-387-69952-3. ISBN 978-0-387-69951-6.
3. Wu, Wei Biao; Shao, Xiaofeng (June 2004). "Limit theorems for iterated random functions". Journal of Applied Probability. 41 (2): 425–436. doi:10.1239/jap/1082999076. ISSN 0021-9002. S2CID 335616.
4. Withers, C. S. (December 1981). "Conditions for linear processes to be strong-mixing". Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 57 (4): 477–480. doi:10.1007/bf01025869. ISSN 0044-3719. S2CID 122082639.
5. Tran, Lanh Tat (1990). "Recursive kernel density estimators under a weak dependence condition". Annals of the Institute of Statistical Mathematics. 42 (2): 305–329. doi:10.1007/bf00050839. ISSN 0020-3157. S2CID 120632192.
6. Birkel, Thomas (1992-07-11). "Laws of large numbers under dependence assumptions". Statistics & Probability Letters. 14 (5): 355–362. doi:10.1016/0167-7152(92)90096-N. ISSN 0167-7152.
7. Wu, Wei Biao (2005-10-04). "Nonlinear system theory: Another look at dependence". Proceedings of the National Academy of Sciences. 102 (40): 14150–14154. Bibcode:2005PNAS..10214150W. doi:10.1073/pnas.0506715102. ISSN 0027-8424. PMC 1242319. PMID 16179388.
8. Rosenblatt, M. (1956-01-01). "A Central Limit Theorem and a Strong Mixing Condition". Proceedings of the National Academy of Sciences. 42 (1): 43–47. Bibcode:1956PNAS...42...43R. doi:10.1073/pnas.42.1.43. ISSN 0027-8424. PMC 534230. PMID 16589813.
9. Fan, X.; Grama, I.; Liu, Q. (2015). "Exponential inequalities for martingales with applications". Electronic Journal of Probability. 20: 1–22. arXiv:1311.6273. doi:10.1214/EJP.v20-3496. S2CID 119713171.
10. Bernstein, Serge (December 1927). "Sur l'extension du théorème limite du calcul des probabilités aux sommes de quantités dépendantes". Mathematische Annalen (in French). 97 (1): 1–59. doi:10.1007/bf01447859. ISSN 0025-5831. S2CID 122172457.

External links

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