Estimation of Additive Error in Mixed Spectra for Stable Processes

Rachid Sabre


Consider a symmetric α stable process having a spectral representation with an additive constant error. An estimator of that error and its rate of convergence are given. We study the rate of convergence when the spectral density have some behaviors at origin. Few long memory processes are taken here as example.


Spectral density; Jackson kernel; Stable processes

Full Text:

PDF (English)


N. AZZAOUI, L. CLAVIER, R. SABRE (2002). Path delay model based on stable distribution for the 60GHz indoor channel. Proceeding of Global Telecommunications Conference, 2002, Volume 3, pp.1638–1643 vol.3.

F. BRICE, F. PENE, M. WENDLER (2017). Stable limit theorem for U-statistic processes indexed by a random walk. Electronic Communications in Probability, 22, no. 9, pp. 1–12.

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, M. MAEJIMA (1989). Two classes of self-similar stable processes with stationary increments. Stochastic Processes and their Applications, 32, pp. 305–329.

L. CLAVIER, M. RACHDI, Y. DELIGNON, V. LETUC, C. GARNIER, P. A. ROLAND (2001).Wide band 60GHz indoor channel: characterization and statistical modelling. IEEE 54th VTC fall, Atantic City, NJ USA, 7–11 October, 2001.

N. DEMESH (1988). Application of the polynomial kernels to the estimation of the spectra of discrete stable stationary processes. (Russian) AkademiiNauk Ukrainskoj, Institute of Mathematics. Preprint 64, pp. 12–36.

A. JANICKI, A. WERON (1993). Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes. Series: Chapman and Hall/CRC Pure and Applied Mathematics, New York.

S. KOGON AND D.MANOLAKIS (1996). Signal modeling with self-similar alpha-stable processes: The fractional Levy motion model. IEEE Transactions on Signal Processing, 44, pp. 1006- ˝ U1010.

M. B. MARCUS, K. SHEN (1989). Bounds for the expected number of level crossings of certain harmonizable infinitely divisible processes. Stochastic Processes and their Applications, 76, no. 1, pp. 1–32.

E. MASRY, S. CAMBANIS (1984). Spectral density estimation for stationary stable processes. Stochastic Processes and their Applications, 18, pp. 1–31.

J. P. MONTILLET, YU. KEGEN (2015). Modeling geodetic processes with Levy alphastable distribution and FARIMA. Mathematical Geosciences, 47, no. 6, pp. 627–646.

C. L. NIKIAS, M. SHAO (1995). Signal Processing with Alpha-Stable Distributions and Applications.Wiley, New York.

K. PANKI, S. RENMING (2014). Stable process with singular drift. Stochastic Processes and their Applications, 124, no. 7, pp. 2479–2516.

K. PANKI, K. TAKUMAGAI, W. JIANG (2017). Laws of the iterated logarithm for symmetric jump processes. Bernoulli, 23, no. 4A, pp. 2330–2379.

M. PEREYRA, H. BATALIA (2012). Modeling ultrasound echoes in skin tissues using symmetric alpha-stable processes. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 59, no. 1, pp. 60–72.

R. SABRE (1994). Estimation de la densité de la mesure spectrale mixte pour un processus symétrique stable strictement stationnaire. Compte Rendu de lŠAcadémie des Sciences. Paris, 319, série I, pp. 1307–1310.

R. SABRE (1995). Spectral density estimate for stationary symmetric stable random field. Applicationes Mathematcaes, 23, 2, pp. 107–133.

R. SABRE (2012). Spectral density estimate for alpha-stable p-adic processes. Statistica, 72, no. 4, pp. 432–448.

G. SAMORODNITSKY, M. TAQQU (1994). Stable non Gaussian Processes. Chapman and Hall, New York.

M. SHAO, C. L. NIKIAS (1993). Signal processing with fractional lower order moments: Stable processes and their applications. Proceedings of IEEE, 81, pp. 986–1010.

M. SCHILDER (1970). Some structure theorems for the symmetric stable laws. The Annals of Mathematical Statistics, 41, no. 2, pp. 412–421.

E. SOUSA (1992). Performance of a spread spectrum packet radio network link in a Poisson field of interferences. IEEE Transactions on Information Theory, 38, pp. 1743–1754.

L.WU, Z.WANG (2015). Filtering and Control for Classes of Two-Dimensional Systems. Studies in Systems, Decision and Control, 18, Springer, Cham.

C. ZHEN-QING, W. LONGMIN (2016). Uniqueness of stable processes with drift. Proceedings of the American Mathematical Society, 144, pp. 2661–2675.

X. ZHONG, A. B. PREMKUMAR (2012). Particle filtering for acoustic source tracking in impulsive noise with alpha-stable process. IEEE Sensors Journal, 13, no. 2, pp. 589–600.

DOI: 10.6092/issn.1973-2201/7300