Repeated observation of a uniform motion: prediction map, adopted IID sampling hypotheses and the resulting sample-mean bound on declared data

The declared model. In the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance take $N=1$ label of mass $m>0$, time window $I_0=\mathbb{R}$, configuration domain $Q={\mathbb{R}}^3$ and the zero force map $F(t,x,v):=0$. It is continuous and state-Lipschitz with constant $0$, so it satisfies the standing hypotheses of the branch. With $e_1$ the first standard basis vector, let $\mathrm{\Theta }=Z=\mathbb{R}$, with both labels read in velocity units $\mathrm{\ell }{\tau }^{-1}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). Use the constant family $F_{\theta }=0$ for every $\theta \in \mathrm{\Theta }$ and the preparation map $\zeta :Z\to \mathbb{R}\times {\mathbb{R}}^3\times {\mathbb{R}}^3$, $\zeta (z):=(0,0,z{e}_1)$ at time $t_0=0$. Then $x_{\theta ,z}(t)=(zt,0,0)$ is the global motion for every pair. Fix an observation time $t_1>0$ and the position readout $R(\theta ,z,(x,w)):=x\cdot e_1$, so the full prediction map is $\mathrm{\Phi }(\theta ,z)=z{t}_1$. For $v\in \mathbb{R}$ choose the diagonal pair $(\theta ,z)=(v,v)$ and abbreviate $x_v:=x_{v,v}$ and $\mathrm{\Phi }(v):=\mathrm{\Phi }(v,v)=x_v(t_1)\cdot e_1$. This explicit choice permits a one-label formula without identifying the parameter argument with the preparation argument of Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error. The readout has dimension $\mathrm{\ell }$ and is read in metres; the force family does not depend on its parameter label.

The motion. The curve $x_v(t):=(vt,0,0)$, $t\in \mathbb{R}$, is a motion of the declared model with datum $\zeta (v)$: its coordinate velocity is $w_v(t)=(v,0,0)$ (Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors), its coordinate acceleration vanishes, and $m{\ddot{x}}_v=0=F$ identically (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model). Hence the prediction map is $\mathrm{\Phi }(v)=v{t}_1$, of dimension $\mathrm{\ell }$ with $[t_1]=\tau$.

The repeated protocol. Fix $n\ge 1$ and adopt the IID sampling hypothesis and the second-moment hypothesis of Repeated-observation protocols of the force branch: adopted IID and second-moment sampling hypotheses and the sample mean for the errors ${\epsilon }_1,\dots ,{\epsilon }_n$ of $n$ repetitions of the protocol at the fixed pair $(\theta ,z)=(v,v)$, on a common probability space. Suppose additionally that $E[{\epsilon }_1]=0$ and $\mathrm{Var}({\epsilon }_1)={\sigma }^2\in [0,+\mathrm{\infty })$; these two statements are extra model hypotheses, not part of either definition. Then the observations are $Y_k=v{t}_1+{\epsilon }_k$ and $\mu :=E[Y_1]=v{t}_1,{\sigma }^2=\mathrm{Var}(Y_1),\bar{Y}=\frac{1}{n}{\sum }_{k=1}^n{Y}_k\mathrm{.}$ Consequently, by Finite IID sample-mean variance and Chebyshev bound, $E\mid \bar{Y}-v{t}_1{\mid }^2=\frac{{\sigma }^2}{n},\mathbb{P}(\mid \bar{Y}-v{t}_1\mid \ge a)\le \frac{{\sigma }^2}{n{a}^2}(a>0)\mathrm{.}$

A declared illustration. With the declared model data $v:=2\mathrm{m}{\mathrm{s}}^{-1}$, $t_1:=3\mathrm{s}$, ${\sigma }^2:=4{\mathrm{m}}^2$ and $n:=16$, the nominal prediction is $v{t}_1=6\mathrm{m}$ and the bound reads $\mathbb{P}(\mid \bar{Y}-6\mathrm{m}\mid \ge 1\mathrm{m})\le \frac{4}{16\cdot 1}=\frac{1}{4}$. These numbers are hypothetical illustration of the formula; they are not measured values, and no apparatus is asserted to realize the protocol.