Keywords and phrases: 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
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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.
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