IID functional data in the block bootstrap regime

The filter $f(s_0,s_1,\dots )=s_{0}e$ is measurable: it is the coordinate projection followed by a map from the finite innovation space to $H$. We have $E{X}_t=0$ and $E\mathrm{\parallel }{X}_t{\mathrm{\parallel }}_H^2=1$. For every $m\ge 1$, the retained current innovation gives $X_{t,m}=X_t$ exactly, so ${\delta }_m=0$. Thus both ${\sum }_{m\ge 1}{\delta }_m<\mathrm{\infty }$ and $m{\delta }_m\to 0$ hold. Independence and identical laws follow directly because each $X_t$ depends only on its own IID innovation.

The integer lengths satisfy $1\le \lceil n^{1\mathrm{/}3}\rceil \le n$, $b_n\to \mathrm{\infty }$ and $b_n\mathrm{/}\sqrt{n}\le n^{-1\mathrm{/}6}+n^{-1\mathrm{/}2}\to 0$. The window vanishes off $[0,1]$, takes values in $[0,1]$, is positive near $1\mathrm{/}2$, symmetric, and nondecreasing on $[0,1\mathrm{/}2]$. Every midpoint weight equals one, so $W_{b_n}=\sqrt{b_n}>0$ and $a_{b_n}=1$. All assumptions of both theorems are therefore met, giving their stated conditional law and covariance limits for this IID example.