Skip to main content
Log in

Mittag-Leffler vector random fields with Mittag-Leffler direct and cross covariance functions

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

In terms of the two-parameter Mittag-Leffler function with specified parameters, this paper introduces the Mittag-Leffler vector random field through its finite-dimensional characteristic functions, which is essentially an elliptically contoured one and reduces to a Gaussian one when the two parameters of the Mittag-Leffler function equal 1. Having second-order moments, a Mittag-Leffler vector random field is characterized by its mean function and its covariance matrix function, just like a Gaussian one. In particular, we construct direct and cross covariances of Mittag-Leffler type for such vector random fields.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bapat, R. B., Raghavan, T. E. S. (1997). Nonnegative matrices and applications. Cambridge: Cambridge University Press.

  • Barndorff-Nielsen, O. E., Leonenko, N. N. (2005). Spectral properties of superpositions of Ornstein–Uhlenbeck type processes. Methodology and Computing in Applied Probability, 7, 335–352.

    Google Scholar 

  • Berberan-Santos, M. N. (2005). Properties of the Mittag-Leffler relaxation function. Journal of Mathematical Chemistry, 38, 629–635.

    Google Scholar 

  • Blumenfeld, R., Mandelbrot, B. B. (1997). Lévy dusts, Mittag-Leffler statistics, mass fractal lacunarity, and perceived dimension. Physical Review E, 56, 112–118.

    Google Scholar 

  • Cramér, H. (1940). On the theory of stationary random processes. Annals of Mathematics, 41, 215–230.

    Google Scholar 

  • Cramér, H., Leadbetter, M. R. (1967). Stationary and related stochastic processes: sample function properties and their applications. New York: Wiley.

  • Cressie, N. (1993). Statistics for spatial data (revised ed.). New York: Wiley.

  • Dattorro, J. (2005). Convex optimization and Euclidean distance geometry. California: Meboo Publishing.

  • Djrbashian, M. M. (1993). Harmonic analysis and boundary value problems in the complex domain. Basel: Birkhäuser Verlag.

    Book  MATH  Google Scholar 

  • Du, J., Leonenko, N., Ma, C., Shu, H. (2012). Hyperbolic vector random fields with hyperbolic direct and cross covariance functions. Stochastic Analysis and Applications, 30, 662–674.

    Google Scholar 

  • Du, J., Ma, C. (2011). Spherically invariant vector random fields in space and time. IEEE Transactions on Signal Processing, 59, 5921–5929.

    Google Scholar 

  • Erdélyi, A. (Ed.). (1955). Higher transcendental functions. Bateman project (Vol. 3). New York: McGraw-Hill.

  • Fujita, Y. (1993). A generalization of the results of Pillai. Annals of the Institute of Statistical Mathematics, 45, 361–365.

    Article  MathSciNet  MATH  Google Scholar 

  • Gikhman, I. I., Skorokhod, A. V. (1969). Introduction to the theory of random processes. Philadelphia: W. B. Saunders Co.

  • Glockle, W. G., Nonnenmacher, T. F. (1995). A fractional calculus approach to self-similar protein dynamics. Biophysical Journal, 68, 46–53.

    Google Scholar 

  • Haubold, H. J., Mathai, A. M., Saxena, R. K. (2011). Mittag-Leffler functions and their applications. Journal of Applied Mathematics, 2011, Article ID 298628.

  • Jayakumar, K. (2003). Mittag-Leffler process. Mathematical and Computer Modelling, 37, 1427–1434.

    Article  MathSciNet  MATH  Google Scholar 

  • Jayakumar, K., Pillai, R. N. (1993). The first-order autoregressive Mittag-Leffler process. Journal of Applied Probability, 30, 462–466.

    Google Scholar 

  • Jayakumar, K., Ristic, M. M., Mundassery, D. A. (2010). A generalization to bivariate Mittag-Leffler and bivariate discrete Mittag-Leffler autoregressive processes. Communication in Statistics—Theory and Methods, 39, 942–955.

    Google Scholar 

  • Jose, K. K., Uma, P. (2009). On Marshall-Olkin Mittag-Leffler distributions and processes. Far East Journal of Theoretical Statistics, 28, 189–199.

    Google Scholar 

  • Jose, K. K., Uma, P., Lekshmi, V. S., Haubold, H. J. (2010). Generalized Mittag-Leffler distributions and processes for applications in astrophysics and time series modeling. Proceedings of the third UN/ESA/NASA workshop on the international heliophysical year 2007 and basic space science (pp. 79–92). New York: Springer.

  • Kleiber, W., Nychka, D. (2012). Nonstationary modeling for multivariate spatial processes. Journal of Multivariate Analysis, 112, 76–91.

    Google Scholar 

  • Kneller, G. R., Hinsen, K. (2004). Fractional Brownian dynamics in proteins. The Journal of Chemical Physics, 12, Article ID 10278.

  • Lin, G. D. (1998). On the Mittag-Leffler distributions. Journal of Statistical Planning and Inference, 74, 1–9.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, C. (2005). Semiparametric spatio-temporal covariance models with the autoregressive temporal margin. Annals of the Institute of Statistical Mathematics, 57, 221–233.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, C. (2011a). Vector random fields with second-order moments or second-order increments. Stochastic Analysis and Applications, 29, 197–215.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, C. (2011b). Covariance matrices for second-order vector random fields in space and time. IEEE Transactions on Signal Processing, 59, 2160–2168.

    Article  MathSciNet  Google Scholar 

  • Ma, C. (2011c). Covariance matrix functions of vector \(\chi ^2\) random fields in space and time. IEEE Transactions on Communications, 59, 2554–2561.

    Article  Google Scholar 

  • Ma, C. (2011d). Vector random fields with long-range dependence. Fractals, 19, 249–258.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, C. (2013a). Student’s t vector random fields with power-law and log-law decaying direct and cross covariances. Stochastic Analysis and Applications, 31, 167–182.

    Article  MathSciNet  MATH  Google Scholar 

  • Ma, C. (2013b). K-distributed vector random fields in space and time. Statistics and Probability Letters, 83, 1143–1150.

    Article  MathSciNet  MATH  Google Scholar 

  • Matheron, G. (1989). The internal consistency of models in geostatistics. In M. Armstrong (Ed.), Geostatistics (Vol. 1, pp. 21–38). Netherlands: Kluwer.

  • Minozzo, M., Ferracuti, L. (2012). On the existence of some skew-normal stationary processes. Chilean Journal of Statistics, 3, 159–172.

    Google Scholar 

  • Paris, R. B., Kaminski, D. (2001). Asymptotics and Mellin-Barnes integrals. Cambridge: Cambridge University Press.

  • Pillai, R. N. (1990). Mittag-Leffler functions and related distributions. Annals of the Institute of Statistical Mathematics, 42, 157–161.

    Article  MathSciNet  MATH  Google Scholar 

  • Pollard, H. (1946). The representation of \(e^{-x^\lambda }\) as a Laplace integral. Bulletin of the American Mathematical Society, 52, 908.

    Article  MathSciNet  MATH  Google Scholar 

  • Pollard, H. (1948). The completely monotonic character of the Mittag-Leffler function \(E_\alpha (-x)\). Bulletin of the American Mathematical Society, 54, 1115–1116.

    Article  MathSciNet  MATH  Google Scholar 

  • Schneider, W. R. (1996). Completely monotone generalized Mittag-Leffler functions. Expositiones Matematicae, 14, 3–16.

    MATH  Google Scholar 

  • Stein, E. M., Weiss, G. (1971). Introduction to Fourier analysis on Euclidean spaces. Princeton: Princeton University Press.

  • Uma, B., Swaminathan, T. N., Ayyaswamy, P. S., Eckmann, D. M., Radhakrishnan, R. (2011). Generalized Langevin dynamics of nanoparticle using a finite element approach: Thermostating with correlated noise. The Journal of Chemical Physics, 135, Article ID 114104.

  • Viñales, A. D., Despósito, M. A. (2007). Anomalous diffusion induced by a Mittag-Leffler correlated noise. Physical Review E, 75, Article ID 042102.

  • Weron, K., Klauzer, A. (2010). Generalization of the Khinchin theorem to Lévy flights. Physics Review Letters, 105, Article ID 260603.

  • Yaglom, A. M. (1987). Correlation theory of stationary and related random functions. New York: Springer.

    Google Scholar 

Download references

Acknowledgments

This work is partly supported by U.S. Department of Energy under Grant DE-SC0005359. The author is grateful to Professor N.N. Leonenko for helpful discussions regarding the Mittag-Leffler function, and to an anonymous referee for valuable comments and suggestions that led to a considerably improved presentation of this paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Chunsheng Ma.

About this article

Cite this article

Ma, C. Mittag-Leffler vector random fields with Mittag-Leffler direct and cross covariance functions. Ann Inst Stat Math 65, 941–958 (2013). https://doi.org/10.1007/s10463-013-0398-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-013-0398-9

Keywords

Navigation