Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-17T16:06:48.007Z Has data issue: false hasContentIssue false
  • English
  • Français

On Moduli of Continuity for Gaussian and l2-Norm Squared Processes Generated by Ornstein-Uhlenbeck Processes

Published online by Cambridge University Press:  20 November 2018

Miklós Csörgö  and
Zhengyan Lin
Miklós Csörgö
Affiliation:
Carleton University, Ottawa, Ontario
Zhengyan Lin
Affiliation:
Hangzhou University, People's Republic of China
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let be a sequence of independent Ornstein-Uhlenbeck processes with coefficients lk and ƛk,i.e., Xk(.)is a Gaussian process with EXk(t) =0 and

The process Y(.)was first studied by Dawson (1972) as the stationary solution of the infinite array of stochastic differential equations

where are independent Wiener processes (cf. also [6],[19],and [1]).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 1 , 01 February 1990 , pp. 141 - 158
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Antoniadis, A. and Carmona, R., Eigenfunction expansions for infinite dimensional Ornstein-Uhlenbeck processes, Probab. Theory Related Fields 74 (1987), 31–54.Google Scholar
2. Belyaev, K. Yu., Continuity and Hölder's conditions for sample functions of stationary Gaussian processes, in: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley and Los Angeles (University of California Press, 1960), 23–34.Google Scholar
3. Csörgo, M. and Lin, Z.Y., On moduli of continuity for Gaussian and x2 processes generated by Ornstein-Uhlenbeck processes, C.R. Math. Rep. Acad. Sci. Canada 10 (1988), 203–207.Google Scholar
4. Csörgö, M. and Révész, P., Strong approximations in probability and statistics (Akadémiai Kiadô, Budapest - Academic Press, New York, 1981).Google Scholar
5. Dawson, D.A., Stochastic evolution equations, Math. Biosciences 75 (1972), 287–316.Google Scholar
6. Stochastic evolution equations and related measure processes, J. Multivariate Anal. 5 (1975), 1–52.Google Scholar
7. Fernique, X., La régularité des fonctions aléatoires d'Ornstein-Uhlenbeck à valeurs dans l2; le cas diagonal, Manuscript (1989).Google Scholar
8. Iscoe, I., Marcus, M., McDonald, D., Talagrand, M. and Zinn, J., Continuity of I2 -valued Ornstein-Uhlenbeck processes, Manuscript (1989).Google Scholar
9. Iscoe, I. and McDonald, D., Continuity of I2-valued Ornstein-Uhlenbeck processes, Tech. Rep. Ser. Lab. Res. Stat. Probab. 58, Carleton University-University of Ottawa (1986).Google Scholar
10. Jain, N.C. and Marcus, M.B., Continuity of subgaussian processes, in: Probability on Banach spaces, Advances in probability and related topics 4 (1978), 81–196.Google Scholar
11. Marcus, M.B., Hölder conditions for Gaussian processes with stationary increments, Trans. Amer. Math. Soc. 134 (1968), 29–52.Google Scholar
12. Marcus, M.B., Hölder conditions for continuous Gaussian processes, Osaka J. Math. 7 (1970), 483- 494.Google Scholar
13. Marcus, M.B. and Shepp, L.A., Gaussian processes, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley and Los Angeles (University of California Press, 1971), 423–442.Google Scholar
14. Nisio, M., On the continuity of stationary Gaussian processes, Nagoya Math. J. 34 (1969), 89–104.Google Scholar
15. Schmuland, B., Moduli of continuity for some Hilbert space valued Ornstein-Uhlenbeck processes, C.R. Math. Rep. Acad. Sci. Canada 10 (1988), 197–202.Google Scholar
16. Schmuland, B., Some regularity results on infinite dimensional diffusions via Dirichlet forms, Stoch. Anal, and Applications 6 (1988), 327–348.Google Scholar
17. Sirao, T. and Watanabe, H., On the Holder continuity of stationary Gaussian processes, Proc. Japan Acad. 44 (1968), 482–484.Google Scholar
18. Slepian, D., The one sided barrier problem for Gaussian noise, Bell. Syst. Tech. J. 41 (1962), 463–501.Google Scholar
19. Walsh, J.B., A stochastic model of neural response, Adv. Appl. Probab. 13 (1981), 231–281.Google Scholar