Cell-Count Form of a Compensated Counting-Path Functional Along a Time Change: Window-Discrepancy Bounds for the Time-Change and Partial-Cell Errors

Let $R>0$ and $s>0$ be real numbers, let $J\ge1$ and $l\ge1$ be natural numbers, and let $0=b_0<b_1<\dots<b_J=R$ be real numbers (the cell boundaries), with cell lengths $\mu_j=b_j-b_{j-1}$ ($1\le j\le J$) and $\mu_{\max}=\max_{1\le j\le J}\mu_j$. Let $p$ be a counting path. Define its cell counts $\mathsf{K}_j=p(b_j)-p(b_{j-1})$ ($1\le j\le J$), its compensated path $\bar{M}(x)=p(x)-x$ ($x\in[0,R]$), and, for a real number $w\ge0$, its window discrepancy

$$\mathrm{Disc}_w(p)=\sup\bigl\{|p(u')-p(u)-(u'-u)|:\ 0\le u\le u'\le R,\ u'-u\le w\bigr\},$$

the least upper bound of a nonempty set of real numbers bounded above by $p(R)+R$ (the number $R$ is fixed throughout and suppressed from the notation).

A map from $[0,s]$ into $\mathbb{R}$ or into Euclidean space $\mathbb{R}^l$ is measurable when its components are measurable with respect to the trace Borel $\sigma$-algebra on $[0,s]$ and the Borel $\sigma$-algebra of the real line, and bounded measurable when moreover its components are bounded; $\int_{[0,s]}\cdot\,du$ is the componentwise Lebesgue integral over the compact interval $[0,s]$, $|\cdot|$ the Euclidean norm, and $\mathbf{1}\{P\}$ equals $1$ if the condition $P$ holds and $0$ otherwise.

Let $w_1\ge0$ be a real number and let $\mathsf{C},\bar{\mathsf{C}}:[0,s]\to[0,R]$ be measurable maps with $|\mathsf{C}_u-\bar{\mathsf{C}}_u|\le w_1$ for every $u\in[0,s]$. Let $v\in\mathbb{R}^l$ and let $H:[0,s]\to\mathbb{R}^l$ be bounded measurable; put $\lVert H\rVert_1=\int_{[0,s]}|H_u|\,du$, the integrand being bounded measurable by Norm Bound for a Vector-Valued Lebesgue Integral over a Compact Interval. Products of a real number with a point of $\mathbb{R}^l$ are scalar multiples. For a measurable $\mathsf{C}':[0,s]\to[0,R]$ define the compensated functional

$$\Lambda(\mathsf{C}')=v\,\bar{M}(\mathsf{C}'_s)+\int_{[0,s]}H_u\,\bar{M}(\mathsf{C}'_u)\,du\in\mathbb{R}^l$$

(defined by claim 1), and define the cell coefficients

$$\alpha_j=v\,\mathbf{1}\{b_j\le\bar{\mathsf{C}}_s\}+\int_{[0,s]}\mathbf{1}\{b_j\le\bar{\mathsf{C}}_u\}\,H_u\,du\in\mathbb{R}^l\qquad(1\le j\le J).$$

1. (Measurability and the partial-cell decomposition.) For every measurable $\mathsf{C}':[0,s]\to[0,R]$ the map $u\mapsto p(\mathsf{C}'_u)$ is bounded measurable, so that $u\mapsto H_u\bar{M}(\mathsf{C}'_u)$ is bounded measurable and $\Lambda(\mathsf{C}')$ is defined; the maps $u\mapsto\mathbf{1}\{b_j\le\bar{\mathsf{C}}_u\}H_u$ are bounded measurable, so that the $\alpha_j$ are defined. Moreover, for every $x\in[0,R]$,

$$\Bigl|\bar{M}(x)-\sum_{j=1}^{J}(\mathsf{K}_j-\mu_j)\,\mathbf{1}\{b_j\le x\}\Bigr|\le\mathrm{Disc}_{\mu_{\max}}(p).$$

2. (Time-change error.) For every $u\in[0,s]$, $|\bar{M}(\mathsf{C}_u)-\bar{M}(\bar{\mathsf{C}}_u)|\le\mathrm{Disc}_{w_1}(p)$, and

$$|\Lambda(\mathsf{C})-\Lambda(\bar{\mathsf{C}})|\le\bigl(|v|+\lVert H\rVert_1\bigr)\,\mathrm{Disc}_{w_1}(p).$$

3. (Cell-count form.)

$$\Bigl|\Lambda(\bar{\mathsf{C}})-\sum_{j=1}^{J}(\mathsf{K}_j-\mu_j)\,\alpha_j\Bigr|\le\bigl(|v|+\lVert H\rVert_1\bigr)\,\mathrm{Disc}_{\mu_{\max}}(p).$$

4. (Combined bound.)

$$\Bigl|\Lambda(\mathsf{C})-\sum_{j=1}^{J}(\mathsf{K}_j-\mu_j)\,\alpha_j\Bigr|\le\bigl(|v|+\lVert H\rVert_1\bigr)\bigl(\mathrm{Disc}_{w_1}(p)+\mathrm{Disc}_{\mu_{\max}}(p)\bigr).$$