Total variation bounds for Gaussian functionals

Let $X=\{X_t:0\le t\le 1\}$ be a centered Gaussian process with continuous paths, and $I_n=\frac{a_n}{2}\,\int_0^1 t^{n-1} (X_1^2-X_t^2)\,dt$ where the $a_n$ are suitable constants. Fix $\beta\in (0,1)$, $c_n>0$ and $c>0$ and denote by $N_c$ the centered Gaussian kernel with (random) variance $cX_1^2$. Under an Holder condition on the covariance function of $X$, there is a constant $k(\beta)$ such that \begin{gather*} \norm{P\bigl(\sqrt{c_n}\,I_n\in\cdot\bigr)-E\bigl[N_c(\cdot)\bigr]\,}\le k(\beta)\,\Bigl(\frac{a_n}{n^{1+\alpha}}\Bigr)^\beta+\frac{\abs{c_n-c}}{c}\quad\text{for all }n\ge 1, \end{gather*} where $\norm{\cdot}$ is total variation distance and $\alpha$ the Holder exponent of the covariance function. Moreover, if $\frac{a_n}{n^{1+\alpha}}\rightarrow 0$ and $c_n\rightarrow c$, then $\sqrt{c_n}\,I_n$ converges $\norm{\cdot}$-stably to $N_c$, in the sense that \begin{gather*} \norm{P_F\bigl(\sqrt{c_n}\,I_n\in\cdot\bigr)-E_F\bigl[N_c(\cdot)\bigr]\,}\rightarrow 0 \end{gather*} for every measurable $F$ with $P(F)>0$. In particular, such results apply to $X=$ fractional Brownian motion. In that case, they strictly improve the existing results in \cite{NNP16} and provide an essentially optimal rate of convergence.