CROSS-VARIATION OF YOUNG INTEGRAL WITH RESPECT TO
LONG-MEMORY FRACTIONAL BROWNIAN MOTIONS
Ivan Nourdin
Rola Zintout
Abstract: We study the asymptotic behaviour of the cross-variation of two-dimensional processes
having the form of a Young integral with respect to a fractional Brownian motion of index
. When is smaller than or equal to , we show asymptotic mixed normality.
When is stricly greater than , we obtain a limit that is expressed in terms of the
difference of two independent Rosenblatt processes.
2000 AMS Mathematics Subject Classification: Primary: 60F05, 60G15, 60G22;
Secondary: 60G18, 60H07.
Keywords and phrases: Fractional Brownian motion, Rosenblatt process, Young
integral, Breuer–Major theorem, Taqqu’s theorem, stochastic differential equations.