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 , 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
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]
See also
References
- ^ a b c d 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.
- ^ a b c 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.