The Scale Set for the Van Trees Assembly: Eventual Validity of the Copy-Side Constraints, Lower Bounds for the Chernoff Exponents, and Vanishing of the Error Majorants

Data. Let $l\ge2$, $m\ge1$ and $\tilde{l}\ge1$ be natural numbers, and let $B\ge0$, $\tilde{B}\ge0$, $K\ge0$, $\tilde{K}\ge0$, $\underline{b}>0$, $T>0$, $s\in(0,T]$, $\Lambda\ge0$, $\mathsf{M}\ge0$, $\bar\Phi\ge0$, $\mathsf{c}_{0}\ge0$, $c_{Q}\ge0$, $\kappa^{\sharp}\ge1$, $C_{\mathrm{flw}}\ge0$, $C_{\mathrm{ctl}}\ge0$, $\mathsf{P}\ge0$ and $\mathsf{Q}\ge0$ be real numbers (the scalar data). Put

Conventions. Natural numbers are regarded as real numbers, real powers $t^{a}=\exp(a\log t)$ of a real $t>0$ are those of Real Power of a Positive Real Number, and $\exp$, $\log$, $\sqrt{\cdot}$, $\lfloor\cdot\rfloor$ and limits of sequences are as in Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities, whose claims are used freely. For a real $t\ge0$ we write $t^{1/2}=\sqrt{t}$ and $t^{1/4}=(t^{1/2})^{1/2}$, consistent with the real powers for $t>0$ by claim 1 of that lemma; in particular $N^{1/2}=\sqrt{N}$, $N^{1/4}=\sqrt{\sqrt{N}}$, $N^{-1/2}=1/\sqrt{N}$, $N^{-1/4}=1/N^{1/4}$ and $N^{-1}=1/N$. Write $a\wedge b$ for the smaller of two real numbers $a,b$. For real numbers $k>0$ and $x\ge0$ let

be the Chernoff exponent of that lemma, extended to $x=0$ by the same formula. All objects introduced below are real numbers depending on the scalar data and on a natural number $N$, and "$u_{N}\to u$" refers to the sequence $(u_{N})_{N\in\mathbb{N}}$. Notational cautions: $J_{N}$ (a natural number, the number of cells per clock) and $\mathsf{J}_{N}$ (an information majorant) are distinct, as are $\mathsf{w}_{N}$ and $w_{1,N},w_{2,N}$; $\mathsf{E}^{\star}_{N}$ and $\mathsf{e}^{\star}_{N}$; $\mathsf{m}_{N}$ (a move size) and the control dimension $m$; $\mathsf{M}$ and $M_{N}$; $D_{N}$ and $d_{N}$; $\mathsf{B}_{N}$ and $B$; $\mathsf{k}_{N}$ and $K,\tilde{K}$; $\mathsf{a}_{N}$ and $\alpha_{N}$; and $\mathsf{c}_{0}$, $\mathsf{c}_{\star}$ (scalar data) and $\mathsf{c}_{N}$ (a derived quantity indexed by $N$).

The scale set. For $N\in\mathbb{N}$ put

Derived quantities. For $N\in\mathbb{N}$ put

(the formula for $\mathsf{g}_{N}$ is read for those $N$ for which $\mu_{N}-\mathsf{m}_{N}\ge0$ and $D_{N}-2\ge0$; for the remaining $N$ put $\mathsf{g}_{N}=0$; the formula for $\bar\Pi_{N}$ is read for every $N$),

and the error majorants

Then the following hold.

1. (Eventual validity of the constraints.) There is a natural number $N_{\mathrm{c}}$ such that for every natural number $N\ge N_{\mathrm{c}}$: (i) $\mathsf{m}_{N}\in\mathbb{N}$ and $N^{1/16}<\mathsf{m}_{N}\le2N^{1/16}$; (ii) $R_{N}\in\mathbb{N}$ and $NBs<R_{N}$, $N\le R_{N}\le(Bs+2)N$; (iii) $J_{N}\in\mathbb{N}$, $\tfrac12N^{3/4}\le\mu_{N}\le N^{3/4}$, $\mu_{N}\ge2$, and $N^{1/4}\le d_{N}\le l(l-1)(Bs+3)N^{1/4}$; (iv) $D_{N}\le N^{1/16}<\mathsf{m}_{N}$, $A_{N}\le C_{A}\mathsf{m}_{N}\le2C_{A}N^{1/16}$, $0\le\Lambda_{1}sA_{N}<L_{N}$, $M_{N}\in\mathbb{N}$, $L_{N}<M_{N}\le L_{N}+1$ and $M_{N}+2\le M_{N}+3\le C_{M}N^{1/16}$; (v) $D_{N}\ge4$, $0<\eta_{N}\le1$, $0<\delta_{N}\le1$, $\zeta_{N}>0$, $\theta_{N}\in\mathbb{N}$ and $N^{1/4}<\theta_{N}\le2N^{1/4}$; (vi) $\mathsf{m}_{N}<\mu_{N}/2$; (vii) $\varepsilon_{0,N}\le\tfrac12$, $2\theta_{N}\varepsilon_{0,N}\le1$, $\bar{E}_{N}\le1$, $\mathsf{E}^{\star}_{N}\le1$, $\mathsf{E}^{\mathrm{ch}}_{N}\le1$, $\bar{\mathsf{j}}_{N}\le1$ and $\kappa^{\mathrm{mv}}_{N}\le1$; (viii) $\mathsf{m}_{N}(x_{N}+\mathsf{m}_{N})\le\delta_{N}\mu_{N}/2$. Such an $N_{\mathrm{c}}$ is fixed for the remainder of the statement.

2. (Chernoff exponents.) For all real $k>0$ the map $x\mapsto\varpi_{k}(x)$ is nondecreasing on $[0,\infty)$, and for every real $n\ge0$ and every $N\in\mathbb{N}$, the number $y=(n+2)^{1/2}N^{1/32}$ satisfies $\varpi_{n+2}(y)\ge N^{1/32}/4$. Moreover, for every $N\ge N_{\mathrm{c}}$,

3. (Limits.) (a) $\varepsilon_{S,N}\to0$, $\varepsilon_{\mathrm{ctl},N}\to0$, $\varepsilon_{0,N}\to0$, $\bar{E}_{N}\to0$, $\mathsf{c}_{N}\to0$, $\mathsf{e}^{\star}_{N}\to0$, $\kappa_{0,N}\to1$, $\bar{\mathsf{j}}_{N}\to0$, $\mathsf{j}^{\star}_{N}\to0$, $\bar\Pi_{N}\to0$, $\mathsf{g}_{N}\to0$, $\mathsf{e}_{F,N}\to0$, $\epsilon_{\psi,N}\to0$, $\kappa_{N}\to0$, $\mathsf{Q}_{N}\to\mathsf{Q}$, $\frac{2d_{N}\mathsf{w}_{N}^{2}\mathsf{B}_{N}}{N}\to0$ and $\mathsf{J}_{N}\to\mathsf{P}+\mathsf{Q}$. (b) $\mathsf{e}_{2,N}\to0$, $\mathsf{e}_{3,N}\to0$, $\mathsf{e}_{4,N}\to0$, $\mathsf{e}_{5,N}\to0$, $\mathsf{e}_{\mathrm{inj},N}\to0$ and $\mathsf{a}_{N}\to0$. (c) $\mathcal{I}_{N}\to\mathsf{P}+\mathsf{Q}$.

4. (Eventual smallness.) For every real $\epsilon>0$ there is a natural number $N_{\mathrm{sc}}\ge N_{\mathrm{c}}$ such that for every natural number $N\ge N_{\mathrm{sc}}$,