Moment Toolkit for the Synthetic Copy: Square-Integrable Majorants of the Window Discrepancies of the Copy Clocks and the Fourth Moment of a Weighted Centred Cell-Count Sum

Adopt the setting and notation of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record (as also adopted by The Copy Clocks Have Independent Poisson Increments on the Clock Interval: Law Identity with the Uniform Poisson Path and Applicability of the Window Discrepancy Bound): the probability space $(\Omega,\mathcal{F},P)$ with expectation $\mathbb{E}$, the natural numbers $N\ge1$, $l\ge2$, $m\ge1$, $\tilde{l}\ge1$, the real numbers $B\ge0$, $T>0$ and $R>0$ with $R\ge NBT$, the transition labels $c$, the cells $I_{c,j}=(b^{c}_{j-1},b^{c}_j]$ ($1\le j\le J_c$) of lengths $|I_{c,j}|$, indexed by the finite set $\mathsf{L}$ of pairs $q=(c,j)$ with $d$ elements and identified with $\{1,\dots,d\}$ by the fixed bijection, so that points of Euclidean space $\mathbb{R}^d$ have coordinates indexed by $\mathsf{L}$; the cell-count vector $\mathsf{K}=(\mathsf{K}_q)_{q\in\mathsf{L}}$; and the copy clocks $\mathsf{P}^{\sharp}=(\mathsf{P}^{\sharp,c})_c$, all of whose paths $u\mapsto\mathsf{P}^{\sharp,c}_u(\omega)$ are counting paths. Recall $\mathbb{N}_0=\mathbb{N}\cup\{0\}$ from the adopted setting, and write $\mu_q=|I_{c,j}|>0$ for $q=(c,j)$, $\mu=(\mu_q)_{q\in\mathsf{L}}\in\mathbb{R}^d$ (the vector of cell lengths; unrelated to the copy measure $\mu^{\sharp}$, which is not used here) and $\mu_{\max}=\max_{q\in\mathsf{L}}\mu_q$. Adopt from Cell-Count Form of a Compensated Counting-Path Functional Along a Time Change: Window-Discrepancy Bounds for the Time-Change and Partial-Cell Errors, for a counting path $p$ and a real number $w\ge0$, the 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 bounded above by $p(R)+R$, formed with the present $R$; and from Series Formula, Exponential Moments, and Chernoff Tail Bounds for the Poisson Distribution the Chernoff exponent $\varpi_{k}(x)=\min\bigl(\tfrac{x^{2}}{4k},\tfrac{x}{2}\bigr)$ ($k>0$, $x>0$), as used in Uniform Window Discrepancy Bound for a Counting Process with Poisson Increments on the Integer Grid (the letter $\varpi$ denotes here this exponent only, never the profile of the copy setting). Write $\exp$ for the exponential function, $x\cdot y$ for the dot product, $t^{1/2}=\sqrt{t}$ for the nonnegative square root of a real $t\ge0$, $t^{1/4}=(t^{1/2})^{1/2}$, $\mathbf{1}_A$ for the indicator of an event $A$ (claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions), $\lceil w\rceil$ for the least natural number that is $\ge w$ for a real $w\ge0$ (it exists by The Archimedean Property of the Real Numbers and The Natural Numbers Are Well Ordered, and satisfies $\lceil w\rceil\ge1$ since $1$ is the least natural number), and $\mathcal{F}$-measurable for a real-valued map on $\Omega$ that is measurable with respect to $\mathcal{F}$ and the Borel $\sigma$-algebra of the real line. Integrable, square-integrable and the mean-square norm $\lVert X\rVert_2=(\mathbb{E}[X^{2}])^{1/2}$ on $(\Omega,\mathcal{F},P)$ are as in those definitions. The function $w\mapsto\lceil w\rceil$ agrees with the usual ceiling except at $w=0$, where it equals $1$. Put

$$\mathsf{M}_4(R)=\bigl(8\,(4096+17R^{4})\bigr)^{1/4}.$$

1. (Square-integrable majorant of one window discrepancy.) Assume that $R$ is a natural number. Let $c$ be a transition label, let $w\ge0$ and $x>0$ be real numbers, and put $\mathsf{n}=\lceil w\rceil$. Then there is an event $G^{c}_{w,x}\in\mathcal{F}$ with

$$P\bigl(\Omega\setminus G^{c}_{w,x}\bigr)\le2\,(R+1)(\mathsf{n}+3)\exp\bigl(-\varpi_{\mathsf{n}+2}(x)\bigr)$$

such that the map

$$\Xi^{c}_{w,x}=(x+2)\,\mathbf{1}_{G^{c}_{w,x}}+\bigl(\mathsf{P}^{\sharp,c}_R+R\bigr)\mathbf{1}_{\Omega\setminus G^{c}_{w,x}}:\Omega\to[0,\infty)$$

is $\mathcal{F}$-measurable and square-integrable, satisfies $\mathrm{Disc}_{w}\bigl(\mathsf{P}^{\sharp,c}(\omega)\bigr)\le\Xi^{c}_{w,x}(\omega)$ for every $\omega\in\Omega$ (where $\mathsf{P}^{\sharp,c}(\omega)$ is the counting path $u\mapsto\mathsf{P}^{\sharp,c}_u(\omega)$), and

$$\bigl\lVert\Xi^{c}_{w,x}\bigr\rVert_2\le x+2+\mathsf{M}_4(R)\,\Bigl(2\,(R+1)(\mathsf{n}+3)\exp\bigl(-\varpi_{\mathsf{n}+2}(x)\bigr)\Bigr)^{1/4}.$$

Moreover $\mathsf{P}^{\sharp,c}_R$ has the Poisson distribution with parameter $R$, and $\mathbb{E}\bigl[(\mathsf{P}^{\sharp,c}_R+R)^{4}\bigr]\le\mathsf{M}_4(R)^{4}$.

2. (Majorant of a sum of two window discrepancies.) Assume that $R$ is a natural number, let $w_1\ge0$, $w_2\ge0$, $x_1>0$ and $x_2>0$ be real numbers, and for every transition label $c$ and $i\in\{1,2\}$ let $G^{c}_{w_i,x_i}$ be an event as furnished by claim 1 (for the data $c$, $w_i$, $x_i$) and $\Xi^{c}_{w_i,x_i}$ the corresponding map; put $\Xi^{c}=\Xi^{c}_{w_1,x_1}+\Xi^{c}_{w_2,x_2}$. Then, for every such choice, each $\Xi^{c}:\Omega\to[0,\infty)$ is $\mathcal{F}$-measurable and square-integrable, $\mathrm{Disc}_{w_1}(\mathsf{P}^{\sharp,c}(\omega))+\mathrm{Disc}_{w_2}(\mathsf{P}^{\sharp,c}(\omega))\le\Xi^{c}(\omega)$ for every $\omega\in\Omega$, and

$$\lVert\Xi^{c}\rVert_2\le x_1+x_2+4+\mathsf{M}_4(R)\sum_{i=1}^{2}\Bigl(2\,(R+1)(\lceil w_i\rceil+3)\exp\bigl(-\varpi_{\lceil w_i\rceil+2}(x_i)\bigr)\Bigr)^{1/4}.$$

Consequently, in any instance of the setting and of the hypotheses other than (DM) of Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter whose probability space, copy clocks, clock horizon $R$ (a natural number), cells and bijection of $\mathsf{L}$ with $\{1,\dots,d\}$ are the present ones, hypothesis (DM) of that lemma holds for the family $(\Xi^{c})_{c}$ formed with $w_1=\mathsf{w}^{\mathrm{clk}}$ (the clock tolerance of that lemma, a real number $\ge0$) and $w_2=\mu_{\max}$, whatever the event $G$ of that instance.

3. (Second and fourth moments of a weighted centred cell-count sum.) The cell counts $\mathsf{K}_q$ ($q\in\mathsf{L}$) are independent, and $\mathsf{K}_q$ has the Poisson distribution with parameter $\mu_q$. For every $\alpha=(\alpha_q)_{q\in\mathsf{L}}\in\mathbb{R}^d$ the random variable $\alpha\cdot(\mathsf{K}-\mu)=\sum_{q\in\mathsf{L}}\alpha_q(\mathsf{K}_q-\mu_q)$ has an integrable fourth power, and

$$\mathbb{E}\bigl[\alpha\cdot(\mathsf{K}-\mu)\bigr]=0,\qquad \mathbb{E}\bigl[(\alpha\cdot(\mathsf{K}-\mu))^{2}\bigr]=\sum_{q\in\mathsf{L}}\alpha_q^{2}\mu_q,\qquad \mathbb{E}\bigl[(\alpha\cdot(\mathsf{K}-\mu))^{4}\bigr]\le3\Bigl(\sum_{q\in\mathsf{L}}\alpha_q^{2}\mu_q\Bigr)^{2}+\sum_{q\in\mathsf{L}}\alpha_q^{4}\mu_q .$$

Consequently, in any instance of the setting and of the hypotheses other than (FM) and (DM) of Mean-Square Assembly of the Estimand Linearisation on the Synthetic Copy: Approximation of the Recentred Copy Endpoint by an Affine Function of the Parameter whose probability space, cell-count vector, cells and bijection of $\mathsf{L}$ with $\{1,\dots,d\}$ are the present ones, the quantity $\mathsf{k}_4$ of that lemma satisfies $\mathsf{k}_4^{4}\le3(\sum_q\alpha_q^{2}\mu_q)^{2}+\sum_q\alpha_q^{4}\mu_q$ for its vector $\alpha\in\mathbb{R}^d$ of cell coefficients.