Super-polynomial accuracy of one dimensional randomized nets using the median-of-means

Let $f$ be analytic on $[0,1]$ with $|f^{(k)}(1/2)|\leq A\alpha^kk!$ for some constant $A$ and $\alpha<2$. We show that the median estimate of $\mu=\int_0^1f(x)\,\mathrm{d}x$ under random linear scrambling with $n=2^m$ points converges at the rate $O(n^{-c\log(n)})$ for any $c< 3\log(2)/\pi^2\approx 0.21$. We also get a super-polynomial convergence rate for the sample median of $2k-1$ random linearly scrambled estimates, when $k=\Omega(m)$. When $f$ has a $p$'th derivative that satisfies a $\lambda$-H\"older condition then the median-of-means has error $O( n^{-(p+\lambda)+\epsilon})$ for any $\epsilon>0$, if $k\to\infty$ as $m\to\infty$.