Renormalized self-intersection local time of bifractional Brownian motion.

Let B H , K = { B H , K ( t ) , t ≥ 0 } be a d-dimensional bifractional Brownian motion with Hurst parameters H ∈ ( 0 , 1 ) and K ∈ ( 0 , 1 ] . Assuming d ≥ 2 , we prove that the renormalized self-intersection local time ∫ 0 T ∫ 0 t δ ( B H , K ( t ) - B H , K ( s ) ) d s d t - E ( ∫ 0 T ∫ 0 t δ ( B H , K ( t ) - B H , K ( s ) ) d s d t ) exists in L 2 if and only if H K d < 3 / 2 , where δ denotes the Dirac delta function. Our work generalizes the result of the renormalized self-intersection local time for fractional Brownian motion.

Weak convergence to isotropic complex [Formula: see text] random measure.

In this paper, we prove that an isotropic complex symmetric α-stable random measure ([Formula: see text]) can be approximated by a complex process constructed by integrals based on the Poisson process with random intensity.