Electronic Journal of Probability

Sample path properties of permanental processes

Michael B. Marcus and Jay Rosen

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Abstract

Let $X_{\alpha }=\{X_{\alpha }(t),t\in{\cal T} \}$, $\alpha >0$, be an $\alpha $-permanental process with kernel $u(s,t)$. We show that $X^{1/2}_{\alpha }$ is a subgaussian process with respect to the metric \[ \sigma (s,t)= (u(s,s)+u(t,t)-2(u(s,t)u(t,s))^{1/2})^{1/2} .\nonumber \] This allows us to use the vast literature on sample path properties of subgaussian processes to extend these properties to $\alpha $-permanental processes. Local and uniform moduli of continuity are obtained as well as the behavior of the processes at infinity. Examples are given of permanental processes with kernels that are the potential density of transient Lévy processes that are not necessarily symmetric, or with kernels of the form \[ \widetilde{u} (x,y)= u(x,y)+f(y),\nonumber \] where $u$ is the potential density of a symmetric transient Borel right process and $f$ is an excessive function for the process.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 58, 47 pp.

Dates
Received: 3 November 2017
Accepted: 30 May 2018
First available in Project Euclid: 11 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1528704076

Digital Object Identifier
doi:10.1214/18-EJP183

Mathematical Reviews number (MathSciNet)
MR3814252

Zentralblatt MATH identifier
06924670

Subjects
Primary: 60K99: None of the above, but in this section 60G15: Gaussian processes 60G17: Sample path properties 60G99: None of the above, but in this section

Keywords
permanental processes subgaussian processes moduli of continuity of permanental processes permanental processes at infinity

Rights
Creative Commons Attribution 4.0 International License.

Citation

Marcus, Michael B.; Rosen, Jay. Sample path properties of permanental processes. Electron. J. Probab. 23 (2018), paper no. 58, 47 pp. doi:10.1214/18-EJP183. https://projecteuclid.org/euclid.ejp/1528704076


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References

  • [1] N. Bourbaki, Functions of a real variable, elementary theory, Springer-Verlag, New York, (2004).
  • [2] R. Blumenthal, and R. Getoor, Markov Processes and Potential Theory. New York: Academic Press, (1968).
  • [3] K. L. Chung, and J. B. Walsh, Markov Processes, Brownian motion and time symmetry, second edition, Springer-Verlag, New York, (2005).
  • [4] J. L. Doob, Classical potential theory and its probabilistic counterpart, Springer, NY, (1984).
  • [5] N. Eisenbaum and H. Kaspi, On permanental processes, Stochastic Processes and their Applications, 119, (2009), 1401–1415.
  • [6] P. J. Fitzsimmons and J. Rosen, Markovian loop soups: permanental processes and isomorphism theorems, Electron. J. Probab. 19 (2014), no. 60, 1–30
  • [7] I. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Products, Academic Press, Oxford, (1980).
  • [8] N. Jain and M. B. Marcus, Continuity of Subgaussain Processes, in Probability on Banach Spaces, Advances in Probability Vol. 4, 81–196, Marcel Dekker, New York, (1978).
  • [9] O. Kallenberg, Foundations of modern probability theory, Second Edition, Springer, NY, (2002).
  • [10] H. Kogan and M. B. Marcus, Permanental Vectors, Stochastic Processes and their Applications, 122, (2012), 1226–1247.
  • [11] H. Kogan, M. B. Marcus and J. Rosen, Permanental processes Communications on Stochastic Analysis, 5, (2011), 81–102.
  • [12] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer-Verlag, New York, (1991).
  • [13] M. B. Marcus, Multivariate gamma distributions, Electron. Commun. Probab. 19 (2014), no. 86, 1–10.
  • [14] M. B. Marcus, Hölder conditions for Gaussian processes with stationary increments, TAMS, 134, (1968), 29–52.
  • [15] M. B. Marcus and J. Rosen, Markov Processes, Gaussian Processes and Local Times, Cambridge University Press, New York, (2006).
  • [16] M. B. Marcus and J. Rosen, A sufficient condition for the continuity of permanental processes with applications to local times of Markov processes, Annals of Probability, 41, (2013), 671–698.
  • [17] M. B. Marcus and J. Rosen, Permanental random variables, $M$-matrices and $\alpha $-permanents, High Dimensional Probability VII, The Cargèse Volume, Progress in Probability 71 (2016), 363–379, Birkhauser, Boston.
  • [18] M. B. Marcus and J. Rosen, Conditions for permanental processes to be unbounded, Annals of Probability, 45, (2017), 2059–2086.
  • [19] M. B. Marcus and J. Rosen, A non-symmetric generalization of a Markov chain example of Kolmogorov and its relation to permanental processes, in preparation.
  • [20] D. Vere-Jones, Alpha-permanents, New Zealand J. of Math., 26, (1997), 125–149.