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Weakly dependent random variables

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In probability, weak dependence of random variables is a generalization of independence that is weaker than the concept of a martingale. 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 { F d } d = 1 ∞ ∈ ∏ d = 1 ∞ ( S d → R ) {\displaystyle \{{\mathcal {F}}_{d}\}_{d=1}^{\infty }\in \prod _{d=1}^{\infty }{\left(S^{d}\to \mathbb {R} \right)}} , a decreasing sequence { θ δ } δ = 1 ∞ → 0 {\displaystyle \{\theta _{\delta }\}_{\delta =1}^{\infty }\to 0} , and a function ψ ∈ F 2 × ( Z + ) 2 → R + {\displaystyle \psi \in {\mathcal {F}}^{2}\times (\mathbb {Z} ^{+})^{2}\to \mathbb {R} ^{+}} . A sequence { X n } n = 1 ∞ {\displaystyle \{X_{n}\}_{n=1}^{\infty }} of random variables is ( { F d } d = 1 ∞ , { θ δ } δ , ψ ) {\displaystyle (\{{\mathcal {F}}_{d}\}_{d=1}^{\infty },\{\theta _{\delta }\}_{\delta },\psi )} -weakly dependent iff, for all j 1 ≤ j 2 ≤ ⋯ ≤ j d < j d + δ ≤ k 1 ≤ k 2 ≤ ⋯ ≤ k e {\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 ϕ ∈ F d {\displaystyle \phi \in {\mathcal {F}}_{d}} , and θ ∈ F e {\displaystyle \theta \in {\mathcal {F}}_{e}} , we have

| Cov ⁡ ( ϕ ( X j 1 , … , X j d ) , θ ( X k 1 , … , X k e ) ) | ≤ ψ ( ϕ , θ , d , e ) θ δ {\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.

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

E ⁡ [ X i ∣ X 1 , … , X i − 1 ] = 0 , E ⁡ [ X i 2 ∣ X 1 , … , X i − 1 ] ≤ R i E ⁡ [ X i 2 ] , E ⁡ [ X i k ∣ X 1 , … , X i − 1 ] ≤ 1 2 E ⁡ [ X i 2 ∣ X 1 , … , X i − 1 ] L k − 2 k ! {\displaystyle {\begin{aligned}\operatorname {E} \left&=0,\\\operatorname {E} \left&\leq R_{i}\operatorname {E} \left,\\\operatorname {E} \left&\leq {\tfrac {1}{2}}\operatorname {E} \leftL^{k-2}k!\end{aligned}}}

See also

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.

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