The fractional mixed fractional Brownian motion. (English) Zbl 1065.60034

Let \(B_1\) and \(B_2\) be two independent fractional Brownian motions of Hurst index \(H_1\) and \(H_2\), respectively. Given real numbers \(\lambda_1\) and \(\lambda_2\), the two-parameter process \(Z\) is defined by \[ Z(w,s):= \lambda_1\,s^{H_2}\,B_1(w) + \lambda_2\,s^{H_1}\,B_2(w),\quad 0\leq w\leq s. \] The investigated statistic is \(Y(t):= \sup_{0\leq s\leq t}\sup_{0\leq w\leq s}| Z(w,s)|\). The main theorem of the present paper states necessary conditions for a function \(f\) on \([0,\infty)\) in order to belong to the lower-lower class of \(Y\).


60G15 Gaussian processes
60G18 Self-similar stochastic processes
Full Text: DOI


[1] Cheridito, P., Mixed fractional Brownian motion, Bernoulli, 7, 913-934, (2001) · Zbl 1005.60053
[2] El-Nouty, C., On the lower classes of fractional Brownian motion, Studia sci. math. hungar., 37, 363-390, (2001) · Zbl 1006.60076
[3] El-Nouty, C., 2002. Lower classes of fractional Brownian motion under Hölder norms. In: Berkes, I., Csáki, E., Csörgő, M. (Eds.), Limit Theorems in Probability and Statistics, Balatonlelle, 1999. János Bolyai Mathematical Society, Budapest.
[4] El-Nouty, C., 2003a. Lower classes of integrated fractional Brownian motion. Studia, submitted for publication.
[5] El-Nouty, C., A note on the fractional integrated fractional Brownian motion, Acta appl. math., 78, 103-114, (2003) · Zbl 1030.60023
[6] Kuelbs, J.; Li, W.V.; Shao, Q.M., Small ball probabilities for Gaussian processes with stationary increments under Hölder norms, J. theoret. probab., 8, 361-386, (1995) · Zbl 0820.60023
[7] Li, W.V.; Shao, Q.M., Small ball estimates for Gaussian processes under Sobolev type norms, J. theoret. probab., 12, 699-720, (1999) · Zbl 0932.60039
[8] Révész, P., Random walk in random and non-random environments, (1990), World Scientific Publishing Co Teaneck, NJ · Zbl 0733.60091
[9] Stolz, W., Some small ball probabilities for Gaussian processes under nonuniform norms, J. theoret. probab., 9, 613-630, (1996) · Zbl 0855.60039
[10] Talagrand, M., Lower classes of fractional Brownian motion, J. theoret. probab., 9, 191-213, (1996) · Zbl 0840.60076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.