A joint fourth cumulant bound for the stationary solution of a fractional SDE and the LAN property for $0

Let $(\bar{X}_t)_{t\in\mathbb{R}}$ be the stationary solution for an additive stochastic differential equation driven by a fractional Brownian motion. We give an upper bound for a joint fourth cumulant of the form $\kappa_4(f_0(\bar{X}_{0}),f_1(\bar{X}_{t_1}),f_2( \bar{X}_{t_2}), f_3(\bar{X}_{t_3}))$. Our tool is the partial Malliavin calculus for the stationary solution for an SDE driven by fBM. As an application, we show that local asymptotic normality of a likelihood ratio field for the drift parameter of a fractional SDE holds even in the case $0