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 [math]\displaystyle{ \{\mathcal{F}_d\}_{d=1}^{\infty}\in\prod_{d=1}^{\infty}{\left(S^d\to\mathbb{R}\right)} }[/math], a decreasing sequence [math]\displaystyle{ \{\theta_{\delta}\}_{\delta=1}^{\infty}\to0 }[/math], and a function [math]\displaystyle{ \psi\in\mathcal{F}^2\times(\mathbb{Z}^+)^2\to\mathbb{R}^+ }[/math]. A sequence [math]\displaystyle{ \{X_n\}_{n=1}^{\infty} }[/math] of random variables is [math]\displaystyle{ (\{\mathcal{F}_d\}_{d=1}^{\infty},\{\theta_{\delta}\}_{\delta},\psi) }[/math]-weakly dependent iff, for all [math]\displaystyle{ j_1\leq j_2\leq\dots\leq j_d\lt j_d+\delta\leq k_1\leq k_2\leq\dots\leq k_e }[/math], for all [math]\displaystyle{ \phi\in\mathcal{F}_d }[/math], and [math]\displaystyle{ \theta\in\mathcal{F}_e }[/math], we have[1]:315
- [math]\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} }[/math]
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]
- [math]\displaystyle{ \begin{align} \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{align} }[/math]
See also
- Central limit theorem
- Independence of random variables
- Martingale (probability theory)
References
- ↑ 1.0 1.1 1.2 1.3 Doukhan, Paul; Louhichi, Sana (1999-12-01). "A new weak dependence condition and applications to moment inequalities" (in en). Stochastic Processes and Their Applications 84 (2): 313–342. doi:10.1016/S0304-4149(99)00055-1. ISSN 0304-4149.
- ↑ 2.0 2.1 2.2 Dedecker, Jérôme; Doukhan, Paul; Lang, Gabriel; Louhichi, Sana; Leon, José Rafael; José Rafael, León R.; Prieur, Clémentine (2007) (in en-gb). Weak Dependence: With Examples and Applications. Lecture Notes in Statistics. 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" (in en). Journal of Applied Probability 41 (2): 425–436. doi:10.1239/jap/1082999076. ISSN 0021-9002.
- ↑ Withers, C. S. (December 1981). "Conditions for linear processes to be strong-mixing" (in en). Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 57 (4): 477–480. doi:10.1007/bf01025869. ISSN 0044-3719.
- ↑ Tran, Lanh Tat (1990). "Recursive kernel density estimators under a weak dependence condition" (in en). Annals of the Institute of Statistical Mathematics 42 (2): 305–329. doi:10.1007/bf00050839. ISSN 0020-3157.
- ↑ Birkel, Thomas (1992-07-11). "Laws of large numbers under dependence assumptions" (in en). 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" (in en). Proceedings of the National Academy of Sciences 102 (40): 14150–14154. doi:10.1073/pnas.0506715102. ISSN 0027-8424. PMID 16179388. Bibcode: 2005PNAS..10214150W.
- ↑ Rosenblatt, M. (1956-01-01). "A Central Limit Theorem and a Strong Mixing Condition" (in en). Proceedings of the National Academy of Sciences 42 (1): 43–47. doi:10.1073/pnas.42.1.43. ISSN 0027-8424. PMID 16589813. Bibcode: 1956PNAS...42...43R.
- ↑ Fan, X.; Grama, I.; Liu, Q. (2015). "Exponential inequalities for martingales with applications". Electronic Journal of Probability 20: 1–22. doi:10.1214/EJP.v20-3496.
- ↑ Bernstein, Serge (December 1927). "Sur l'extension du théorème limite du calcul des probabilités aux sommes de quantités dépendantes" (in fr). Mathematische Annalen 97 (1): 1–59. doi:10.1007/bf01447859. ISSN 0025-5831.
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