The fractional mixed fractional Brownian motion. (Q1423093)

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 \(Y\).