Self-similar processes with stationary increments from the space \(\text{SSub}_{\varphi}(\Omega)\). (Ukrainian, English) Zbl 1026.60037

Teor. Jmovirn. Mat. Stat. 65, 67-78 (2001); translation in Theory Probab. Math. Stat. 65, 77-88 (2002).
Summary: The authors deal with centered mean square integrable stochastic processes with stationary increments \(Z_{\alpha}=\{Z_{\alpha}(t)\colon t\geq 0\}\) with covariance function \[ R_{\alpha}(t,s)=\frac{1}{2}\left(t^{2\alpha}+ s^{2\alpha}+|t-s|^{2\alpha}\right),\quad \alpha\in(0,1), \] from the space \(\text{SSub}_{\varphi}(\Omega)\) [for more information on properties of stochastic processes from the space \(\text{Sub}_{\varphi}(\Omega)\) see, for example, V. V. Buldygin and Yu. V. Kozachenko, “Metric characterization of random variables and random processes” (1998; Zbl 0933.60031)]. This class of stochastic processes includes \(\alpha\)-self-similar processes \(Z_{\alpha}=\{Z_{\alpha}(t)\colon t\geq 0\}= \{x^{-\alpha}Z_{\alpha}(xt)\colon t\geq 0\},x\geq 0\), with stationary increments. The fractional Brownian motion belongs to this class of processes when the Orlicz \(N\)-function \(\varphi(t)=x^2/2\). It is proved that all separable modifications of the considered processes are continuous with probability one on any compact set. Estimates for the probability of large deviations are proposed. Conditions under which these processes are from the space \(C(\mathbb R^{+},c)\) are found.


60G07 General theory of stochastic processes
60G70 Extreme value theory; extremal stochastic processes