Spectral density estimation for symmetric stable p-adic processes
DOI:
https://doi.org/10.6092/issn.1973-2201/3657Abstract
Applications of p-adic numbers ar beming increasingly important espcially in the field of applied physics. The objective of this work is to study the estimation of the spectral of p-adic stable processes. An estimator formed by a smoothing periodogram is constructed. It is shwon that this estimator is asymptotically unbiased and consistent. Rates of convergences are also examined.References
G. BACHMAN, (1964), Element of abstract harmonic analysis. Academic Press, New York and London.
A KH. BIKULOV, A. VOLOVICHB, (1997), p-adic Brownian motion 61, N. 3, pp. 537-552.
D.R. BRILLINGER, (1991), Some asymptotics of finite Fourier transforms of a stationary p-adic process, “Journal of Combinatorics, Information and Systems”, 16, pp. 155-169.
S. CAMBANIS, (1983), Complex symetric stable variables and processes, in P.K. SEN, ed, “Contributions to Statistics: Essays in Honour of Norman L. Johnson” North-Holland. New York, pp. 63-79.
S. CAMBANIS, C.D. HARDIN, A. WERON, (1987), Ergodic properties of stationary stable processes, “Stochastic Process Appl.”, 24, pp. 1-18.
S. CAMBANIS, M. MAEJIMA, (1989), Two classes of self-similar stable processes with stationary increments, “Stoch. Proc. Appl.”, 32, pp. 305-329.
S. CAMBANIS, A.R. SOLTAN, (1984), Prediction of stable processes: Spectral and moving average representation, “Z. Wahrsch. Varw. Gebiete”, 66, pp. 593-612.
D. CLYDE, JR. HARDIN, (1982), On the spectral representation of symmetric stable processes, “Journal of Multivariate Analysis”, 12, pp. 385-401.
R. CIANCI, A. KHRENNIKOV, (1994), Canp-adic numbers be useful to regularize divergent expectation values of quantum observables, “International Journal of Theoretical Physics”, 33, N. 6, pp. 1217-1228.
E. DAGUM, (2010), Time Series Modelling and Decomposition, “Statistica” 70, I. 4, pp. 433-457.
N. DEMESH, (1988), Application of the polynomial kernels to the estimation of the spectra of discrete stable stationary processes, (in russian) “Akad.Nauk.Ukrain” S.S.R. Inst. Mat, 64, pp. 12-36.
B. DRAGOVICH, A.YU. DRAGOVICH, (2009), A p-adic model of DNA sequence and genetic code, “Mathematics and Statistics”, 1, N. 1, pp. 34-41.
D. DZYADIK, (1977), Introduction à la thèorie de l’approximation uniforme par fonctions polynomiales, Akad. Nauk Ukrain. S.S.R. Inst. Mat.
E. HEWITT, K.A. ROSS, (1963), Abstract harmonic analysis, volume 1. Academic Press, New York.
Y. HOSOYA, (1978), Discrete-time stable processes and their certain properties, “Ann. Probability”, 6, pp. 94-105.
Y. HOSOYA, (1982), Harmonizable stable processes, “Z. Wahrsch. Verw. Gebiete”, 60, pp. 517-533.
LI. HUA-CHIEH, (2001), On heights of p-adic dynamical systems “Proceeding of the American mathematical society”, 130, N. 2, pp. 379-386.
K. KAMIZONO, (2007), Symmetric stochastic integrals with respect to p-adic Brownian motion, “Probability and Stochastic Processes:” formerly Stochastics and Stochastics Reports, 79, N. 6, pp. 523-538.
A. KHRENNIKOV, (1993), p-Adic probability theory and its applications. The principle of statistical stabilization of frequencies, “Theoretical and Mathematical Physics”, 97, N. 3, pp. 1340-1348.
A. KHRENNIKOV, (1998), p-Adicbehaviour of Bernoulliprobabilities, “Statistics and Probability Letters”, 37, N. 4, pp. 375-379.
N. KOBLITZ, (1980), p-adic analysis: Ashort course on recent work, volume 46 of Lecture Notes, London Math. Soc., Combridge.
A. KLAPPER, M. GORESKY, (1994), 2-Adic shift registers, “Lecture Notes in Computer Science”, 809, pp. 174-178.
S.V. KOZYREV, (2008), Toward an ultrametric theory of turbulence, “Theoretical and Mathematical Physics”, 157, N. 3, pp. 1713-1722.
A. MAKAGON, V. MANDERKAR, (1990), The spectral representation of stable processes: Harmonizability and regularity, “Probability Theory and Related Fields”, 85, pp. 1-11.
P.A. MORETTIN, (1980), Homogeneous random processes on locally compact Abelian groups, “Anai Academia Brasileira de Ciências”, 52, pp. 1-6.
E. MASRY, S. CAMBANIS, (1984), Spectral density estimation for stationary stable processes, “Stochastic processes and their applications”, 18, pp. 1-31.
E. MASRY, (1978), Alias-free sampling: An alternative conceptualization and its applications, “IEEE Trans. Information theory”, 24, pp. 317-324.
A. PALLINI, (2007), Bernstein-type approximation using the beta-binomial distribution, “Statistica”, 67, N. 4, pp. 367-387.
P. PERLOV, (1989), Direct estimation of the spectrum of stationary stochastic processes, “Trans. From Problemy Perdacti Informatsii”, 25, N. 2, pp. 3-12.
MP. PRIESTLEY, (1981), Spectral analysis and time series, Probability and Mathematical Statistics, Academic Press, London.
M. RACHDI, V. MONSAN, (1999), Asymptoic properties of p-adic spectral estimates of second order, “J. of combinatoirics Information and System Sciences”, 24, N. 2, pp. 113-142.
M. RACHDI, R. SABRE, (2009), Mixed-spectra analysis for stationary random field, “Stat. Methods Appl.”, 18, pp. 333-358.
R. SABRE, (1994), Estimation de la densité de la mesure spectrale mixte pour un processus symétrique stable strictement stationaire, C. R. Acad. Sci. Paris, t. 319, Série I, pp. 1307-1310.
R. SABRE, (1995), Spectral density estimation for stationary stable random fields, “Applicationes Math.”, 23, N. 2, pp. 107-133.
R. SABRE, (2002), Aliasing free for symmetric stable random fields, “Egyp. Stat. J.”, 46, pp. 53-75.
R. SABRE, (2003), Nonparammetric estimation of the continuous part of mixed measure, “Statistica”, LXIII, N. 3, pp. 441-467.
G. SAMORODNITSKY, M. TAQQU, (1994), Stable non gaussian random processes, Stochastic Modeling Chapman and Hall, New York, London.
M. SAMUELIDES, L. TOUZILLIER, (1990), Analyse harmoniqu., vol. 1 and 2. CEPAD, Toulouse.
M. SCHILDER, (1970), Some structure theorems for the Symmetric Stable laws, “Ann. Math. Stat.”, 42, N. 2, pp. 412-421.
V.S. VLADIMIROV, (1988), Generalized functions over the field of p-adic numbers, “Russian Math. Surveys”, 10, pp. 19-64.
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