On the exponential integrability of the derivative of intersection and self-intersection local time for fractional Brownian motion and a limit theorem related to the self-intersection local time for fractional Brownian motion

Abstract

We give the correct condition for existence of the $k$-th derivative of the intersection local time for fractional Brownian motion, which was originally discussed in [Guo, J., Hu, Y., and Xiao, Y., Higher-order derivative of intersection local time for two independent fractional Brownian motions, Journal of Theoretical Probability 32, (2019), pp. 1190-1201]. We also show that the $k$-th derivative of the intersection and self-intersection local times of fractional Brownian motion are exponentially integrable for certain parameter values. In addition, we show convergence in distribution when the existence condition is violated for the $k$-th derivative of self-intersection local time of fractional Brownian motion under scaling.