Asymptotic properties for quadratic functionals of linear self-repelling diffusion process and applications

References

• Benaïm, M., Ledoux, M., Raimond, O. (2002). Self-interacting diffusions. Probabil. Theory Relat. Fields. 122(1):1–41. DOI: https://doi.org/10.1007/s004400100161.
   Google Scholar
• Benaïm, M., Ciotir, I., Gauthier, C. E. (2015). Self-repelling diffusions via an finite dimensional approach. Stoch. Pde. Anal. Comp. 3(4):506–530. DOI: https://doi.org/10.1007/s40072-015-0059-5.
   Google Scholar
• Bercu, B., Rouault, A. (2002). Sharp large deviations for the Ornstein-Uhlenbeck process. Theory Probab. Appl. 46(1):1–19. DOI: https://doi.org/10.1137/S0040585X97978737.
   Google Scholar
• Bercu, B., Coutin, L., Savy, N. (2012). Sharp large deviations for the non-stationary Ornstein-Uhlenbeck process. Stoch. Proc. Appl. 122(10):3393–3424. DOI: https://doi.org/10.1016/j.spa.2012.06.006.
   Google Scholar
• Bercu, B., Richou, A. (2017). Large deviations for the Ornstein-Uhlenbeck process without tears. Stat. Probabil. Lett. 123:45–55. DOI: https://doi.org/10.1016/j.spl.2016.11.030.
   Google Scholar
• Bishwal, J. P. N. (2003). Maximum likelihood estimation in partially observed stochastic differential system driven by a fractional Brownian motion. Stoch. Anal. Appl. 21(5):995–1007. DOI: https://doi.org/10.1081/SAP-120024701.
   Web of Science ®Google Scholar
• Chambeu, S., Kurtzmann, A. (2011). Some particular self-interacting diffusions: ergodic behavior and almost sure convergence. Bernoulli. 17(4):1248–1267. DOI: https://doi.org/10.3150/10-BEJ310.
   Google Scholar
• Chen, Y., Kuang, N.H., Li, Y. (2020). Berry-Esséen bound for the parameter estimation of fractional Ornstein-Uhlenbeck processes. Stoch. Dyn. 20(04):2050023. DOI: https://doi.org/10.1142/S0219493720500239.
   Google Scholar
• Cranston, M., Le Jan, Y. (1995). Self attracting diffusions: two case studies. Math. Ann. 303(1):87–93. DOI: https://doi.org/10.1007/BF01460980.
   Web of Science ®Google Scholar
• Cranston, M., Mountford, T. S. (1996). The strong law of large numbers for a Brownian polyer. Ann. Probab. 24(3):1300–1323. DOI: https://doi.org/10.1214/aop/1065725183.
   Google Scholar
• Du Roy de Chaumaray, M. (2017). Large deviations for the squated radial Ornstein-Uhlenbeck process. Theory Probab. Appl. 61(3):408–441. DOI: https://doi.org/10.1137/S0040585X97T988253.
   Google Scholar
• Durrett, R. T., Rogers, L. C. G. (1992). Asymptotic behavior of Brownian polymers. Probab. Theory Relat. Fields. 92(3):337–349. DOI: https://doi.org/10.1007/BF01300560.
   Google Scholar
• El Onsy, B., Es-Sebaiy, K., Viens, F. G. (2017). Parameter estimation for a partially observed Ornstein-Uhlenbeck process with long-memory noise. Stochastics. 89(2):431–468. DOI: https://doi.org/10.1080/17442508.2016.1248967.
   Web of Science ®Google Scholar
• Es-Sebaiy, K., Viens, F. G. (2019). Optimal rates for parameter estimation of stationary Gaussian processes. Stoch. Proc. Appl. 129(9):3018–3054. DOI: https://doi.org/10.1016/j.spa.2018.08.010.
   Web of Science ®Google Scholar
• Gan, Y. H., Yan, L. T. (2018). Least squares estimation for a linear self-repelling diffusion driven by fractional Brownian motion (in Chinese). Sci. Chin. Math. 48:1143–1158. DOI: https://doi.org/10.1360/SCM-2017-0387.
   Google Scholar
• Gao, F. Q., Xiong, J., Zhao, X. Q. (2018). Moderate deviations and nonparametric inference for monotone functions. Ann. Stat. 46(3):1225–1254. DOI: https://doi.org/10.1214/17-AOS1583.
   Google Scholar
• Gauthier, C. E. (2016). Self attracting diffusions on a sphere and application to a periodic case. Electron. Commun. Probabil. 21:1–12. DOI: https://doi.org/10.1214/16-ECP4547.
   Google Scholar
• Herrmann, S., Roynette, B. (2003). Boundedness and convergence of some self-attracting diffusions. Math. Annalen. 325(1):81–96. DOI: https://doi.org/10.1007/s00208-002-0370-0.
   Google Scholar
• Herrmann, S., Scheutzow, M. (2004). Rate of convergence of some self-attracting diffusions. Stoch. Proc. Appl. 111(1):41–55. DOI: https://doi.org/10.1016/j.spa.2003.10.012.
   Google Scholar
• Hu, Y., Nualart, D. (2010). Parameter estimation for fractional Ornstein-Uhlenbeck processes. Stat. Probabil. Lett. 80(11–12):1030–1038. DOI: https://doi.org/10.1016/j.spl.2010.02.018.
   Web of Science ®Google Scholar
• Hu, Y., Nualart, D., Zhou, H. (2019). Parameter estimation for fractional Ornstein-Uhlenbeck processes of general Hurst parameter. Stat. Inference Stoch. Proc. 22(1):111–142. DOI: https://doi.org/10.1007/s11203-017-9168-2.
   Google Scholar
• Jiang, H., Liu, H., Zhou, Y. Z. (2020). Asymptotic properties for the parameter estimation in Ornstein-Uhlenbeck process with discrete observations. Electron. J. Stat. 14(2):3192–3229. DOI: https://doi.org/10.1214/20-EJS1738.
   Google Scholar
• Jiang, H., Zhang, N. (2020). Cramér-type moderate deviations for the likelihood ratio process of Ornstein-Uhlenbeck process with shift. Stoch. Dyn. 92(3):478–496. DOI: https://doi.org/10.1142/S0219493721500271.
   Google Scholar
• Jiang, H., Zhang, N. (2020). Cramér-type moderate deviations for statistics in the non-stationary Ornstein-Uhlenbeck process. Stochastics. 92(3):478–496. DOI: https://doi.org/10.1080/17442508.2019.1637877.
   Web of Science ®Google Scholar
• Kleptsyn, V., Kurtzmann, A. (2012). Ergodicity of self-attracting motion. Electron. J. Probab. 17:1–37. DOI: https://doi.org/10.1214/EJP.v17-2121.
   Google Scholar
• Kutoyants, Y. A. (2019). On parameter estimation of the hidden Ornstein-Uhlenbeck process. J. Multivariate Anal. 169:248–269. DOI: https://doi.org/10.1016/j.jmva.2018.09.008.
   Google Scholar
• Kutoyants, Y. A. (2019). On parameter estimation of the hidden ergodic Ornstein-Uhlenbeck process. Electron. J. Stat. 13(2):4508–4526. DOI: https://doi.org/10.1214/19-EJS1631.
   Google Scholar
• Kutoyants, Y. A., Zhou, L. (2021). On parameter estimation of the hidden Gaussian process in perturbed SDE. Electron. J. Stat. 15(1):211–234. DOI: https://doi.org/10.1214/20-EJS1788.
   Google Scholar
• Liu, W., Shao, Q. M. (2010). Cramér-type moderate deviation for the maximum of the periodogram with application to simultaneous tests in gene expression time series. Ann. Statist. 38(3):1913–1935. DOI: https://doi.org/10.1214/09-AOS774.
   Google Scholar
• Liu, W., Shao, Q. M. (2013). A Cramér moderate deviation theorem for Hotellinga̧ŕs T2-statistic with applications to global tests. Ann. Stat. 41(1):296–322. DOI: https://doi.org/10.1214/12-AOS1082.
   Web of Science ®Google Scholar
• Major, P. (2007). On a multivariate version of Bernsteins inequality. Electron. J. Probab. 12:966–988. DOI: https://doi.org/10.1214/EJP.v12-430.
   Web of Science ®Google Scholar
• Mountford, T., Tarrés, P. (2008). An asymptotic result for Brownian polymers. Ann. Inst. H Poincaré Probab. Statist. 44(1):29–46. DOI: https://doi.org/10.1214/07-AIHP113.
   Google Scholar
• Nadarajah, S., Pogány, T. K. (2016). On the distribution of the product of correlated normal random variables. C.R. Acad. Sci. Paris Ser. I. 354(2):201–204. DOI: https://doi.org/10.1016/j.crma.2015.10.019.
   Google Scholar
• Nualart, D., Nualart, E. (2018). Introduction to Malliavin Calculus. Cambridge: Cambridge University Press.
   Google Scholar
• Pemantle, R. (1988). Phase transition in reinforced random walk. Ann. Probab. 16(3):1229–1241. DOI: https://doi.org/10.1214/aop/1176991687.
   Google Scholar
• Sun, X. C., Yan, L. T. (2021). Asymptotic behavior on the linear self-interacting diffusion driven by α-stable motion. Stochastics. 53:1–23. DOI: https://doi.org/10.1080/17442508.2020.1869239.
   Google Scholar
• Tarrés, P., Tóth, B., Valkó, B. (2012). Diffusivity bounds for 1D Brownian polymers. Ann. Probab. 40(2):695–713. DOI: https://doi.org/10.1214/10-AOP630.
   Google Scholar
• Yan, L. T., Sun, Y., Lu, Y. S. (2008). On the linear fractional self-attracting diffusion. J. Theor. Probab. 21(2):502–516. DOI: https://doi.org/10.1007/s10959-007-0113-y.
   Google Scholar