WAVELET ESTIMATION OF THE COVARIANCE OF ALMOST PERIODICALLY CORRELATED PROCESSES AND STUDY OF ASYMPTOTIC PROPERTIES IN A CONTEXT OF WEAK DEPENDENCE
We construct a multiresolution estimator and study its asymptotic properties. The estimation of the coefficients of the covariance decomposition of an almost periodically correlated process on a wavelet basis is dealt with. It is found that the covariance relates to random variables satisfying a weak dependence structure of quasi-association type. In this context, we first recall a method for constructing a wavelet base, with the decomposition of the covariance function in this base and obtain a set of coefficients to be estimated. We then construct an estimator of the coefficients obtained, under specific sampling conditions (jitter or delay).
Following are the three main results obtained in the paper:
quasi-association, wavelet transform, covariance, almost periodically correlated process, wavelet basis, multiresolution estimation, asymptotic normality.
Received: October 17, 2022; Revised: November 22, 2022; Accepted: December 26, 2022; Published: January 24, 2023
How to cite this article: Moussa Koné and Vincent Monsan, Wavelet estimation of the covariance of almost periodically correlated processes and study of asymptotic properties in a context of weak dependence, Far East Journal of Theoretical Statistics 67(1) (2023), 49-94. http://dx.doi.org/10.17654/0972086323004
This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License
References:
[1] BOIIR, Einige Sâtze liber Fourierreihen fastperiodischen Funktionen, Math. Zeitschrift 23 (1925), 28-44.[2] C. Charles, Introduction aux ondelettes, Note de cours à l’Université de Liège, Gembloux Agro-bio Tech (Unité de Statistique, Informatique et Mathématique Appliquées à la Bioingénierie), 2011. https://orbi.uliege.be/bitstream/2268/87186/1/Intro_ond1_v2.pdf.[3] C. Corduneanu, Almost Periodic Functions, Wiley, New York, 1968.[4] P. Doukhan and J. Leon, Déviation quadratique d’estimateurs d’une densité par projection orthogonales, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 425-430.[5] Dominique Dehay and Vincent Monsan, Discrete periodic sampling with jitter and almost periodically correlated processes, Stat. Inference Stoch. Process. 10 (2007), 223-253. DOI 10.1007/s11203-006-0004-3.[6] E. Gladyshev, Periodically and almost periodically correlated random processes with continuous time parameter, Theory Probab. Appl. 8 (1963), 173-177. [7] N. Herrndorff, An example on the central limit theorem for associated sequences, Ann. Prob. 12(3) (1984), 912-917.[8] H. L. Hurd and J. Léskow, Strongly consistent and asymptotically normal estimation of the covariance for almost periodically correlated processes, Stat. Decisions 10 (1992), 201-225.[9] H. L. Hurd, Correlation theory for the almost periodically correlated processes with continuous time parameter, J. Multivariate Anal. 37(1) (1991), 24-45.[10] G. Kerkyacharian and D. Picard, Density estimation by kernel and wavelet methods: optimality of Besov spaces, Statist. Probab. Lett. 18 (1993), 327-336.[11] D. Khoshnevisan and Thomas M. Lewis, A law of the iterated logarithm for stable processes in random scenery, Stochastic Process. Appl. 74(1) (1998), 89-121.[12] M. Loève, Probability Theory, Van Nostrand Reinhold, New York, 1965.[13] C. M. Newman and A. L. Wright, An invariance principle for certain dependent sequences, Ann. Probab. 9 (1981), 671-675.[14] C. M. Newman, Asymptotic independence and limit theorems for positively and negatively dependent random variables, Inequalities in Statistics and Probability, Y. L. Tong, ed., IMS Lecture Notes-Monograph Ser., Inst. Math. Statist., Hayward, CA, Vol. 5, 1984, pp. 127-140.[15] G. Strang, Wavelet transforms versus Fourier transforms, Bulletin (New Series) of the American Mathematical Society 28 (1993), 288-305.[16] Stephane Mallat, Une exploration des signaux en ondelettes, Editions de l’École Polytechnique, 2000.[17] T. M. Lewis, Limit theorems for partial sums of quasi-associated random variables, Asymptotic Methods in Probability and Statistics, B. Szyszkowicz, ed., Elsevier, Amsterdam, 1998, pp. 31-48.[18] G. Walter, Approximation of the delta function by wavelets, J. Approx. Theory 71 (1992), 329-343.[19] P. Billingsley, Convergence of Probability Measures, John Wiley and Sons, New York, 1967.[20] C. M. Newman, Normal fluctuations and the FKG inequalities, Commun. Math. Phys. 74 (1980), 119-128.
