×

Weak dependence, models and some applications. (English) Zbl 1433.60017

Summary: The paper is devoted to recall weak dependence conditions from J. Dedecker et al.’s monograph [Weak dependence. With examples and applications. New York, NY: Springer (2007; Zbl 1165.62001)]; the main basic results are recalled here and we go further in some new applications. We develop here several models of weakly dependent processes and random fields. Among them, an ARCH(\(\infty\)) model is considered with statistical applications to ordinary least squares. A last part aims at proving new asymptotic results for weakly dependent random fields. Such applications are indeed the main proof of the interest of this theoretical notion which measures the asymptotic decorrelation of a process.

MSC:

60F17 Functional limit theorems; invariance principles
60F05 Central limit and other weak theorems
60G07 General theory of stochastic processes
60G60 Random fields
62M09 Non-Markovian processes: estimation

Citations:

Zbl 1165.62001
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andrews D (1984) Non strong mixing autoregressive processes. J Appl Probab 21: 930–934 · Zbl 0552.60049
[2] Bardet JM, Doukhan P, León JR (2007) Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle’s estimate. J.T.S.A. (to appear) hal.archives-ouvertes.fr/hal-00126489/en/ · Zbl 1198.62087
[3] Bickel PJ, Wichura MJ (1971) Convergence criteria formultiparameter stochastic processes and some applications (English). Ann Math Stat 42: 1656–1670 · Zbl 0265.60011
[4] Billingsley P (1968) Convergence of probability measures. Wiley, New York · Zbl 0172.21201
[5] Bolthausen E (1982) On the C.L.T for stationary mixing random fields. Ann Probab 10(4): 1047–1050 · Zbl 0496.60020
[6] Bradley R (2007) Introduction to strong mixing conditions, vols 1–3. Kendricks Press, Heber City · Zbl 1134.60004
[7] Bulinski AV, Shashkin A (2006) Strong invariance principle for dependent random fields, IMS Lecture Notes–Monograph Series, vol 48, pp 128–143, Providence · Zbl 1130.60041
[8] Dedecker J, Doukhan P (2003) A new covariance inequality and applications. Stoch Process Appl 106: 63–80 · Zbl 1075.60513
[9] Dedecker J, Doukhan P, Lang G, León JR, Louhichi S, Prieur C (2007) Weak dependence, examples and applications. Lecture Notes in Statistics, vol 190 · Zbl 1165.62001
[10] Dedecker J, Prieur C (2005) New dependence coefficients. Examples and applications to statistics. Probab Theory Relat Fields 132: 203–236 · Zbl 1061.62058
[11] Doukhan P (1994) Mixing: properties and examples, Lecture Notes in Statistics, vol 85. Springer, New York · Zbl 0801.60027
[12] Doukhan P (2003) Models inequalities and limit theorems for stationary sequences. In: Doukhan et al (ed) Theory and applications of long range dependence. Birkhäuser, Basel, pp 43–101 · Zbl 1032.62081
[13] Doukhan P, Lang G (2002) Rates in the empirical central limit theorem for stationary weakly dependent random fields. Stat Inference Stoch Processes 5: 199–228 · Zbl 1061.60016
[14] Doukhan P, Louhichi S (1999) A new weak dependence condition and applications to moment inequalities. Stoch Process Appl 84: 313–342 · Zbl 0996.60020
[15] Doukhan P, Teyssière G, Winant P (2006) A LARCH( vector valued process. In: Bertail P, Doukhan P, Soulier P (eds) Lecture Note in Statistics, vol 187, Dependence in Probability and statistics, pp 245–258 · Zbl 1113.60038
[16] Doukhan P, Truquet L (2007) Weakly dependent random fields with infinite memory. Alea 2: 111–132 · Zbl 1140.60029
[17] Doukhan P, Wintenberger O (2007) A central limit theorem under non causal weak dependence and sharp moment assumptions. Probab Math Stat 27: 45–73
[18] Doukhan P, Wintenberger O (2008) Weakly dependent chains with infinite memory. Stoch Process Appl (to appear) · Zbl 1166.60031
[19] Engle RF (1982) Autoregressive conditional heteroskedasticity with estimate of the variance in the U.K. inflation. Econometrica 15: 286–301
[20] Giraitis L, Kokoszka P, Leipus R (2000) Stationary ARCH models: dependence structure and central limit theorem. Econom Theory 16: 3–22 · Zbl 0986.60030
[21] Giraitis L, Surgailis D (2002) ARCH-type bilinear models with double long memory. Stoch Process Appl 100: 275–300 · Zbl 1057.62070
[22] Guyon X (1991) Random fields on a network. Springer, New York · Zbl 0734.60054
[23] Rio E (2000) Théorie asymptotique pour des processus aléatoires faiblement dépendants. SMAI, Mathématiques et Applications 31. Springer, Heidelberg
[24] Rosenblatt M (1956) A central limit theorem and a strong mixing condition. Proc Nat Ac Sc USA 42:43–47 · Zbl 0070.13804
[25] Stein C (1972) A bound for the error in normal approximation of a sum of dependent random variables. In: Proceedings of Berkeley symposium, M.S.P.2, pp 583–603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.