Profile-to-Hilbert bootstrap interface

Let $H=L^2([0,1],\mathcal{B},\lambda ;\mathbb{R})$ be the real Hilbert space of almost-everywhere classes of square-integrable real profiles, with $⟨u,v{⟩}_H={\int }_0^{1}u(\xi )v(\xi )d\xi$, where $\mathcal{B}$ is generated by intervals with rational endpoints and $\lambda$ is Lebesgue measure restricted to $\mathcal{B}$. Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ and $(S,\mathcal{S},\nu )$ be probability spaces, let $({\delta }_t{)}_{t\in \mathbb{Z}}$ be independent identically distributed measurable $S$-valued random elements on $\mathrm{\Omega }$ with law $\nu$, and let $f:S^{{\mathbb{N}}_0}\to H$ be measurable from the product sigma-algebra ${\mathcal{S}}^{\otimes {\mathbb{N}}_0}$ to the Borel sigma-algebra of $H$. The associated causal profile process is $X_t=f({\delta }_t,{\delta }_{t-1},\dots )$. Assume $X_0$ is Bochner integrable and has Bochner mean ${\mathbb{E}}_B{X}_0:={\int }_{\mathrm{\Omega }}{X}_0(\omega )d\mathbb{P}(\omega )=0$, with $\mathbb{E}\mathrm{\parallel }{X}_0{\mathrm{\parallel }}_H^2<\mathrm{\infty }$. Assume Countable Choice and Dependent Choice and take an independent product extension carrying the countable replacement-innovation family.

For every $t\in \mathbb{Z}$, $m\ge 1$, and $k\le t-m$, let ${\delta }_{t,m,k}^{\mathrm{\prime }}$ have law $\nu$, with all these copies mutually independent and independent of the original innovations. Set

$X_{t,m}=f({\delta }_t,\dots ,{\delta }_{t-m+1},{\delta }_{t,m,t-m}^{\mathrm{\prime }},{\delta }_{t,m,t-m-1}^{\mathrm{\prime }},\dots ),{\delta }_m=(\mathbb{E}\mathrm{\parallel }{X}_0-X_{0,m}{\mathrm{\parallel }}_H^2{)}^{1\mathrm{/}2}\mathrm{.}$

The profile process is $L^2$-$m$-approximable when ${\sum }_{m=1}^{\mathrm{\infty }}{\delta }_m<\mathrm{\infty }$. For data $X_1,\dots ,X_n\in H$, $n\ge 1$, and an integer block length $1\le b\le n$, moving-block resampling independently draws starts uniformly from $\{1,\dots ,n-b+1\}$, concatenates the corresponding ordered length-$b$ blocks, and truncates the last block to obtain exactly $n$ resampled elements $X_1^{\ast },\dots ,X_n^{\ast }$. Define ${\bar{X}}_n^{\ast }=n^{-1}{\sum }_{j=1}^n{X}_j^{\ast }$, and let ${\mathbb{E}}^{\ast }$ denote expectation over these draws with the observations fixed. For any bounded real-linear functional $L:H\to \mathbb{R}$, let $\mathrm{\parallel }L\mathrm{\parallel }={\sup }_{\mathrm{\parallel }u{\mathrm{\parallel }}_H\le 1}\mathrm{\mid }L(u)\mathrm{\mid }$; the centered resampled scalar profile summary is $\sqrt{n}L({\bar{X}}_n^{\ast }-{\mathbb{E}}^{\ast }{\bar{X}}_n^{\ast })$. This definition specifies the data space and resampling law; it does not assert asymptotic validity.

Causal Bernoulli shifts, replacement coupling, and $L^2$-$m$-approximability

Moving-block resampling and its conditional center

Countable Choice for countable selections

Dependent Choice for the product extension

Bochner integrability and its integral

When a measured or simulated profile $q(z)$ on a physical segment $z\in (a,a+\mathrm{\ell })$ is represented in $H$, use $X(\xi )=\sqrt{\mathrm{\ell }}q(a+\mathrm{\ell }\xi )$, so ${\int }_a^{a+\mathrm{\ell }}\mathrm{\mid }q(z){\mathrm{\mid }}^{2}dz={\int }_0^1\mathrm{\mid }X(\xi ){\mathrm{\mid }}^{2}d\xi$. If $z$ has units of length and $q$ has units $Q$, then $X$ has units $Q\sqrt{\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}$, since $\xi$ is dimensionless. Any physical scalar sensor must be transformed with the profile and carry its corresponding units. Resolution, instrument averaging, shared calibration error, and finite-data uncertainty are not removed by this mathematical representation.

Affine-coordinate $L^2$ isometry and Jacobian