Investigation of sample paths properties for some classes of φ -sub-Gaussian stochastic processes
Volume 8, Issue 1 (2021), pp. 41–62
Pub. online: 26 January 2021
Type: Research Article
Open Access
Open Access
Received
7 September 2020
7 September 2020
Revised
21 December 2020
21 December 2020
Accepted
7 January 2021
7 January 2021
Published
26 January 2021
26 January 2021
Abstract
This paper investigates sample paths properties of φ-sub-Gaussian processes by means of entropy methods. Basing on a particular entropy integral, we treat the questions on continuity and the rate of growth of sample paths. The obtained results are then used to investigate the sample paths properties for a particular class of φ-sub-Gaussian processes related to the random heat equation. We derive the estimates for the distribution of suprema of such processes and evaluate their rate of growth.
References
Anh, V.V., Leonenko, N.N.: Non-Gaussian scenarios for the heat equation with singular initial data. Stoch. Process. Appl. 84, 91–114 (1999). MR1720100. https://doi.org/10.1016/S0304-4149(99)00053-8
Beghin, L., Kozachenko, Yu., Orsingher, E., Sakhno, L.: On the Solutions of Linear Odd-Order Heat-Type Equations with Random Initial Conditions. J. Stat. Phys. 127(4), 721–739 (2007). MR2319850. https://doi.org/10.1007/s10955-007-9309-x
Beghin, L., Orsingher, E., Sakhno, L.: Equations of Mathematical Physics and Compositions of Brownian and Cauchy Processes. Stoch. Anal. Appl. 29, 551–569 (2011). MR2812517. https://doi.org/10.1080/07362994.2011.581071
Buldygin, V.V., Kozachenko, Yu.V.: Metric Characterization of Random Variables and Random Processes. American Mathematical Society, Providence, RI (2000). 257 p. MR1743716. https://doi.org/10.1090/mmono/188
Giuliano Antonini, R., Kozachenko, Yu.V., Nikitina, T.: Spaces of φ-subgaussian random variables. Rendiconti Accademia Nazionale delle Scienze XL. Memorie di Matematica e Applicazioni 121, XXVII, 95–124 (2003). MR2056414
Dozzi, M., Kozachenko, Yu., Mishura, Yu., Ralchenko, K.: Asymptotic growth of trajectories of multifractional Brownian motion with statistical applications to drift parameter estimation. Stat. Inference Stoch. Process. 21(1), 21–52 (2018). MR3769831. https://doi.org/10.1007/s11203-016-9147-z
Dudley, R.M.: The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1, 290–330 (1967). MR0220340. https://doi.org/10.1016/0022-1236(67)90017-1
Dudley, R.M.: Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics, vol. 63. Cambridge University Press, Cambridge (1999). 452 p. MR1720712. https://doi.org/10.1017/CBO9780511665622
Fernique, X.: Régularité des trajectoires des fonctions aléatoires gaussiennes. In: Ecole d’Eté de Probabilités de Saint-Flour, IV-1974. Lecture Notes in Math., vol. 480, pp. 1–96. Springer, Berlin (1975). MR0413238
Kampé de Feriet, J.: Random solutions of the partial differential equations. In: Proceedings of the 3rd Berkeley Symposium on Mathematical Statistics and Probability, vol. III, pp. 199–208. University of California Press, Berkeley (1955). MR0084927
Kozachenko, Yu.V., Koval’chuk, Yu.A.: Boundary value problems with random initial conditions and series of functions of $Su{b_{\varphi }}(\Omega )$. Ukr. Math. J. 50(4), 572–585 (1998). MR1698149. https://doi.org/10.1007/BF02487389
Kozachenko, Yu., Melnikov, A., Mishura, Yu.: On drift parameter estimation in models with fractional Brownian motion. Statistics 49(1), 35–62 (2015). MR3304366. https://doi.org/10.1080/02331888.2014.907294
Kozachenko, Yu.V., Leonenko, G.M.: Large deviations type inequality for the supremum of the heat random field. Methods Funct. Anal. Topol. 8(3), 46–49 (2002). MR1926910
Kozachenko, Yu.V., Leonenko, G.M.: Extremal behavior of the heat random field. Extremes 8, 191–205 (2006). MR2275918. https://doi.org/10.1007/s10687-006-7967-8
Kozachenko, Yu.V., Ostrovskij, E.I.: Banach spaces of random variables of sub-Gaussian type. Theory Probab. Math. Stat. 32, 45–56 (1986). MR0882158
Kozachenko, Yu., Orsingher, E., Sakhno, L., Vasylyk, O.: Estimates for functionals of solutions to Higher-Order Heat-Type equations with random initial conditions. J. Stat. Phys. 172(6), 1641–1662 (2018). MR3856958. https://doi.org/10.1007/s10955-018-2111-0
Kozachenko, Yu., Orsingher, E., Sakhno, L., Vasylyk, O.: Estimates for distribution of suprema of solutions to higher-order partial differential equations with random initial conditions. Mod. Stoch. Theory Appl. 7(1), 79–96 (2020). MR4085677. https://doi.org/10.15559/19-vmsta146
Kozachenko, Yu., Rozora, I.: Sample continuity conditions with probability one for square-Gaussian stochastic processes. Theory Probab. Math. Stat. 101, 153–166 (2020). https://doi.org/10.1090/tpms/1118
Kozachenko, Yu.V., Slivka, G.I.: Justification of the Fourier method for hyperbolic equations with random initial conditions. Theory Probab. Math. Stat. 69, 67–83 (2004). MR2110906. https://doi.org/10.1090/S0094-9000-05-00615-0
Kozachenko, Yu.V., Slyvka-Tylyshchak, A.I.: On the increase rate of random fields from space ${\mathrm{Sub}_{\varphi }}(\Omega )$ on unbounded domains. Stat. Optim. Inf. Comput. 2(2), 79–92 (2014). MR3351372. https://doi.org/10.19139/45
Ledoux, M., Talagrand, M.: Probability in a Banach Space: Isoperimetry and Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 23. Springer, Berlin (1991). 482 p. MR1102015. https://doi.org/10.1007/978-3-642-20212-4
Ledoux, M.: Isoperimetry and Gaussian Analysis. Lecture Notes in Mathematics, vol. 1648, pp. 165–294. Springer (1996). MR1600888. https://doi.org/10.1007/BFb0095676
Leonenko, N.N., Woyczynski, W.A.: Scaling limits of solution of the heat equation for singular non-Gaussian data. J. Stat. Phys. 91(1–2), 423–438 (1998). MR1632518. https://doi.org/10.1023/A:1023060625577
Orsingher, E., Beghin, L.: Time-fractional equations and telegraph processes with Brownian time. Probab. Theory Relat. Fields 128, 141–160 (2004). MR2027298. https://doi.org/10.1007/s00440-003-0309-8
Orsingher, E., Beghin, L.: Fractional diffusion equations and processes with randomly-varying time. Ann. Probab. 37(1), 206–249 (2009). MR2489164. https://doi.org/10.1214/08-AOP401
Rosenblatt, M.: Remarks on the Burgers Equation. J. Math. Phys. 9, 1129–1136 (1968). MR0264252. https://doi.org/10.1063/1.1664687
Talagrand, M.: The Generic Chaining. Upper and Lower Bounds of Stochastic Processes. Springer Monographs in Mathematics. Springer, Berlin (2005). 222 p. MR2133757
Talagrand, M.: Upper and Lower Bounds for Stochastic Processes. Modern Methods and Classical Problems. Springer, Berlin, Heidelberg (2014). 626 p. MR3184689. https://doi.org/10.1007/978-3-642-54075-2
