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

El-Nouty, Charles

Stat. Probab. Lett. 65, No. 2, 111-120 (2003).

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 \(Y\).

Reviewer: Werner Linde (Jena)

MSC:

60G15	Gaussian processes
60G18	Self-similar stochastic processes

Keywords:

fractional mixed fractional Brownian motion; lower classes

References:

[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