Proof of 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

Throughout, sums, scalar multiples, products, absolute values and maxima of $\mathcal{F}$-measurable real-valued maps are $\mathcal{F}$-measurable by claims 2, 3 and 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and constants and indicators of events by claim 1 there; expectations of nonnegative random variables are monotone and additive by Linearity and Monotonicity of the Lebesgue Integral, and $\mathbb{E}[\mathbf{1}_A]=P(A)$ by The Integral of an Indicator Function is the Measure of the Set. We use the elementary inequalities $(a+b)^{2}\le2a^{2}+2b^{2}$ (recorded in Square-Integrable Random Variables and the Mean-Square Inner Product), hence $(a+b)^{4}\le8(a^{4}+b^{4})$ (apply it twice), and $\sqrt{s+t}\le\sqrt{s}+\sqrt{t}$ for real $s,t\ge0$ (square both sides), together with the monotonicity of $t\mapsto t^{1/2}$ on $[0,\infty)$ (if $0\le s\le t$ and $\sqrt{s}>\sqrt{t}$ then $s>t$), hence also of $t\mapsto t^{1/4}=(t^{1/2})^{1/2}$, and the identity $\sqrt{st}=\sqrt{s}\sqrt{t}$ for $s,t\ge0$ (both sides are nonnegative with equal squares). For $\omega\in\Omega$ and a transition label $c$ write $p_\omega=\mathsf{P}^{\sharp,c}(\omega)$ for the counting path $u\mapsto\mathsf{P}^{\sharp,c}_u(\omega)$; each $\mathsf{P}^{\sharp,c}_u$ is $\mathcal{F}$-measurable by claim 2 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record.

A crude bound on every window discrepancy. Let $p$ be a counting path and $w\ge0$. For $0\le u\le u'\le R$, conditions 1 and 2 of Counting Path and Its Jump Times give $0\le p(u)\le p(u')\le p(R)$, hence $0\le p(u')-p(u)\le p(R)$, and $0\le u'-u\le R$; for nonnegative reals $a,b$ one has $|a-b|\le\max(a,b)\le a+b$, so $|p(u')-p(u)-(u'-u)|\le p(R)+R$. Thus $p(R)+R$ is an upper bound of the set whose least upper bound is $\mathrm{Disc}_w(p)$ (as also recorded in Cell-Count Form of a Compensated Counting-Path Functional Along a Time Change: Window-Discrepancy Bounds for the Time-Change and Partial-Cell Errors), and $\mathrm{Disc}_w(p)\le p(R)+R$.

Claim 1. Assume $R\in\mathbb{N}$, fix $c$, $w\ge0$, $x>0$ and $\mathsf{n}=\lceil w\rceil\in\mathbb{N}$, so that $w\le\mathsf{n}$. By claim 3 of 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, applied with $R$ in the role of the natural number called $n$ there (admissible since $R\le R$), with $\mathsf{n}$ in the role of the window length called $m$ there, and with the real number $x$, the process $\mathsf{P}^{\sharp,c}$ satisfies the hypotheses of Uniform Window Discrepancy Bound for a Counting Process with Poisson Increments on the Integer Grid. That lemma provides an event $G=G^{c}_{w,x}$ belonging to the $\sigma$-algebra generated by $\mathsf{P}^{\sharp,c}_0,\dots,\mathsf{P}^{\sharp,c}_R$ (a sub-$\sigma$-algebra of $\mathcal{F}$ by that definition, these random variables being $\mathcal{F}$-measurable; so $G\in\mathcal{F}$), with

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

such that for every $\omega\in G$ and all real $0\le u\le u'\le R$ with $u'-u\le\mathsf{n}$, $|\mathsf{P}^{\sharp,c}_{u'}(\omega)-\mathsf{P}^{\sharp,c}_u(\omega)-(u'-u)|\le x+2$.

Majorisation. Let $\omega\in G$. Every pair $(u,u')$ in the index set of $\mathrm{Disc}_w(p_\omega)$ satisfies $0\le u\le u'\le R$ and $u'-u\le w\le\mathsf{n}$, so every element of that set is at most $x+2$, and $\mathrm{Disc}_w(p_\omega)\le x+2=\Xi^{c}_{w,x}(\omega)$. Let $\omega\notin G$. By the crude bound, $\mathrm{Disc}_w(p_\omega)\le p_\omega(R)+R=\mathsf{P}^{\sharp,c}_R(\omega)+R=\Xi^{c}_{w,x}(\omega)$. Hence $\mathrm{Disc}_w(p_\omega)\le\Xi^{c}_{w,x}(\omega)$ for every $\omega\in\Omega$. The map $\Xi^{c}_{w,x}$ is $\mathcal{F}$-measurable (indicators of $G$ and $\Omega\setminus G$, the measurable $\mathsf{P}^{\sharp,c}_R$, constants, products and sums) with values in $[0,\infty)$, since $x+2>0$ and $\mathsf{P}^{\sharp,c}_R\ge0$.

The Poisson law and the fourth moment. By claim 2 of 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, $\mathsf{P}^{\sharp,c}_0(\omega)=0$ for every $\omega$ and the increment $\mathsf{P}^{\sharp,c}_R-\mathsf{P}^{\sharp,c}_0=\mathsf{P}^{\sharp,c}_R$ has the Poisson distribution with parameter $R-0=R$. By claim (b) of Factorial Moments and Moments of Every Order of the Poisson Distribution in order $4$, $(\mathsf{P}^{\sharp,c}_R)^{4}$ is integrable with $\mathbb{E}[(\mathsf{P}^{\sharp,c}_R)^{4}]\le8^{4}+2^{4}R^{4}=4096+16R^{4}$. Hence, by $(a+b)^{4}\le8(a^{4}+b^{4})$ and monotonicity and linearity of the expectation,

$$\mathbb{E}\bigl[(\mathsf{P}^{\sharp,c}_R+R)^{4}\bigr]\le8\bigl(4096+16R^{4}+R^{4}\bigr)=8\,(4096+17R^{4})=\mathsf{M}_4(R)^{4}.$$

In particular $(\mathsf{P}^{\sharp,c}_R+R)^{2}$ is square-integrable.

The mean-square norm. Since $\mathbf{1}_G\mathbf{1}_{\Omega\setminus G}=0$ and $\mathbf{1}_G^{2}=\mathbf{1}_G$, $\mathbf{1}_{\Omega\setminus G}^{2}=\mathbf{1}_{\Omega\setminus G}$, one has pointwise

$$(\Xi^{c}_{w,x})^{2}=(x+2)^{2}\,\mathbf{1}_G+(\mathsf{P}^{\sharp,c}_R+R)^{2}\,\mathbf{1}_{\Omega\setminus G}.$$

Taking expectations, using $\mathbb{E}[\mathbf{1}_G]=P(G)\le1$ and the Cauchy--Schwarz inequality (claim 1 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm) for the square-integrable random variables $(\mathsf{P}^{\sharp,c}_R+R)^{2}$ and $\mathbf{1}_{\Omega\setminus G}$,

$$\mathbb{E}\bigl[(\Xi^{c}_{w,x})^{2}\bigr]\le(x+2)^{2}+\Bigl(\mathbb{E}\bigl[(\mathsf{P}^{\sharp,c}_R+R)^{4}\bigr]\Bigr)^{1/2}\bigl(P(\Omega\setminus G)\bigr)^{1/2}<\infty ,$$

so $\Xi^{c}_{w,x}$ is square-integrable, and by $\sqrt{s+t}\le\sqrt{s}+\sqrt{t}$, the monotonicity of the square root and the bounds just proved,

$$\lVert\Xi^{c}_{w,x}\rVert_2\le x+2+\Bigl(\mathbb{E}\bigl[(\mathsf{P}^{\sharp,c}_R+R)^{4}\bigr]\Bigr)^{1/4}\bigl(P(\Omega\setminus G)\bigr)^{1/4}\le x+2+\mathsf{M}_4(R)\Bigl(2\,(R+1)(\mathsf{n}+3)\exp\bigl(-\varpi_{\mathsf{n}+2}(x)\bigr)\Bigr)^{1/4}.$$

This proves claim 1.

Claim 2. Fix $c$ and events $G^{c}_{w_i,x_i}$ as furnished by claim 1 with the corresponding maps $\Xi^{c}_{w_i,x_i}$ ($i=1,2$). By claim 1 applied to each, $\Xi^{c}=\Xi^{c}_{w_1,x_1}+\Xi^{c}_{w_2,x_2}$ is a sum of two $\mathcal{F}$-measurable, $[0,\infty)$-valued, square-integrable maps, hence is $\mathcal{F}$-measurable, $[0,\infty)$-valued and square-integrable (closure of square-integrability under sums, Square-Integrable Random Variables and the Mean-Square Inner Product); for every $\omega$, $\mathrm{Disc}_{w_1}(p_\omega)+\mathrm{Disc}_{w_2}(p_\omega)\le\Xi^{c}_{w_1,x_1}(\omega)+\Xi^{c}_{w_2,x_2}(\omega)=\Xi^{c}(\omega)$; and by the triangle inequality (claim 2 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm) and the bound of claim 1 for each summand,

$$\lVert\Xi^{c}\rVert_2\le\lVert\Xi^{c}_{w_1,x_1}\rVert_2+\lVert\Xi^{c}_{w_2,x_2}\rVert_2\le x_1+x_2+4+\mathsf{M}_4(R)\sum_{i=1}^{2}\Bigl(2\,(R+1)(\lceil w_i\rceil+3)\exp\bigl(-\varpi_{\lceil w_i\rceil+2}(x_i)\bigr)\Bigr)^{1/4}.$$

In an instance 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 as described in the statement, hypothesis (DM) there asks, for every label $c$, for an $\mathcal{F}$-measurable map $\Omega\to[0,\infty)$ with finite second moment that majorises $\mathrm{Disc}_{\mathsf{w}^{\mathrm{clk}}}(p_\omega)+\mathrm{Disc}_{\mu_{\max}}(p_\omega)$ for every $\omega$ in its event $G$, where $\mathsf{w}^{\mathrm{clk}}=N(\Lambda_1T\varepsilon_S+\varepsilon_{\mathrm{ctl}})\ge0$ is a real number, $\mu_{\max}$ is the maximal cell length of the same cells, and the window discrepancies there are those of Cell-Count Form of a Compensated Counting-Path Functional Along a Time Change: Window-Discrepancy Bounds for the Time-Change and Partial-Cell Errors formed with the same clock horizon $R$ and the same copy clocks; with $w_1=\mathsf{w}^{\mathrm{clk}}$ and $w_2=\mu_{\max}$ (both $\ge0$, so admissible in claim 1), the family $(\Xi^{c})_c$ has these properties for every $\omega\in\Omega$, hence for every $\omega\in G$ whatever the event $G$. This proves claim 2.

Claim 3. Law and independence of the cell counts. By claim 1 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record, each $\mathsf{K}_q$ is a random variable with values in $\mathbb{N}_0$, and $P(\mathsf{K}=y)=\prod_{q\in\mathsf{L}}\mathrm{poi}_{\mu_q}(y_q)$ for every $y\in\mathbb{N}_0^{\mathsf{L}}$, where, for a real $s\ge0$ and $k\in\mathbb{N}_0$, $\mathrm{poi}_{s}(k)=\exp(-s)s^{k}/k!$ is the mass function of the Poisson distribution with parameter $s$ (the notation of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), with values in $[0,1]$ since $\mathrm{poi}_{s}(k)=P_{s}(\{k\})$ for the Poisson probability measure $P_{s}$ of Poisson Distribution. Moreover, for $q=(c,j)$ the cell count $\mathsf{K}_{c,j}$ is the cell count $C_j$ of Uniform Representation of the Rate-One Poisson Counting Path on a Bounded Interval: Distinct Points, Poisson Increments, Conditional Law Given the Cell Counts, and Point Insertion for the data of the label $c$ (so identified in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), which by claim 2 of that lemma has the Poisson distribution with parameter $|I_{c,j}|=\mu_q$. By Poisson Distribution, the masses $\mathrm{poi}_{\mu_q}(k)$, $k\in\mathbb{N}_0$, sum to $1$ (the total mass of the Poisson distribution); this can also be read off claim 1 of Series Formula, Exponential Moments, and Chernoff Tail Bounds for the Poisson Distribution applied to $\mathsf{K}_q$ with the constant map $g\equiv1$, which gives $\sum_{k\in\mathbb{N}_0}\mathrm{poi}_{\mu_q}(k)=\mathbb{E}[1]=1$. Hence, indexing $\mathsf{L}$ by $\{1,\dots,d\}$ through the fixed bijection, the random variables $\mathsf{K}_1,\dots,\mathsf{K}_d$ satisfy the hypotheses of Factorized Joint Probability Mass Function Implies Independence with $g_q=\mathrm{poi}_{\mu_q}$ (the event $\{\mathsf{K}=y\}$ being $\bigcap_q\{\mathsf{K}_q=y_q\}$), and its claim 2 shows that $\mathsf{K}_1,\dots,\mathsf{K}_d$ are independent.

Moments of a centred Poisson variable. Let $K$ be Poisson with parameter $\lambda\ge0$ (a scalar; the letter $\mu$ is reserved for the vector of cell lengths, and the bare letter $\lambda$ here is unrelated to the Lebesgue measure $\lambda_d$ and to the intensities $\lambda^{\sharp,\omega}$ of the adopted setting) and put $Y=K-\lambda$. By claim (b) of Factorial Moments and Moments of Every Order of the Poisson Distribution, $K^{i}$ is integrable for $i\in\{1,2,3,4\}$, hence so is every polynomial in $K$ of degree at most $4$, in particular $Y^{i}$ for $i\le4$; and by claim (a) there, $\mathbb{E}[K]=\lambda$, $\mathbb{E}[K(K-1)]=\lambda^{2}$, $\mathbb{E}[K(K-1)(K-2)]=\lambda^{3}$ and $\mathbb{E}[K(K-1)(K-2)(K-3)]=\lambda^{4}$. Since $K(K-1)=K^{2}-K$, $K(K-1)(K-2)=K^{3}-3K^{2}+2K$ and $K(K-1)(K-2)(K-3)=K^{4}-6K^{3}+11K^{2}-6K$, linearity of the expectation gives successively

$$\mathbb{E}[K^{2}]=\lambda^{2}+\lambda,\qquad\mathbb{E}[K^{3}]=\lambda^{3}+3\lambda^{2}+\lambda,\qquad\mathbb{E}[K^{4}]=\lambda^{4}+6\lambda^{3}+7\lambda^{2}+\lambda .$$

Expanding $Y^{2}=K^{2}-2\lambda K+\lambda^{2}$ and $Y^{4}=K^{4}-4\lambda K^{3}+6\lambda^{2}K^{2}-4\lambda^{3}K+\lambda^{4}$ and inserting these values,

$$\mathbb{E}[Y]=0,\qquad\mathbb{E}[Y^{2}]=\lambda,\qquad\mathbb{E}[Y^{4}]=(\lambda^{4}+6\lambda^{3}+7\lambda^{2}+\lambda)-4\lambda(\lambda^{3}+3\lambda^{2}+\lambda)+6\lambda^{2}(\lambda^{2}+\lambda)-4\lambda^{4}+\lambda^{4}=3\lambda^{2}+\lambda .$$

These apply to $K=\mathsf{K}_q$ with $\lambda=\mu_q$ for every $q\in\mathsf{L}$.

Induction over the summands. Put $Y_q=\mathsf{K}_q-\mu_q$ and, for $k\in\{0,1,\dots,d\}$, $S_k=\sum_{q=1}^{k}\alpha_qY_q$ (so $S_0=0$) and $v_k=\sum_{q=1}^{k}\alpha_q^{2}\mu_q$, $\mathsf{F}_k=\sum_{q=1}^{k}\alpha_q^{4}\mu_q$. We show by induction on $k$ that $S_k^{4}$ is integrable and

$$\mathbb{E}[S_k]=0,\qquad\mathbb{E}[S_k^{2}]=v_k,\qquad\mathbb{E}[S_k^{4}]\le3v_k^{2}+\mathsf{F}_k .\qquad(\mathrm{H}_k)$$

For $k=0$ all quantities vanish. Let $0\le k<d$, assume $(\mathrm{H}_k)$, and write $S=S_k$, $a=\alpha_{k+1}$, $Y=Y_{k+1}$, $\lambda'=\mu_{k+1}$, so that $S_{k+1}=S+aY$. Factorisation of the mixed moments. We claim that for $i,j\in\{1,2,3\}$ the product $S^{i}Y^{j}$ has finite expectation equal to $\mathbb{E}[S^{i}]\,\mathbb{E}[Y^{j}]$. If $k=0$ then $S=0$, so $S^{i}Y^{j}=0$ identically and both sides vanish. Let $k\ge1$, and let $\mathcal{G}_1=\sigma(\mathsf{K}_1,\dots,\mathsf{K}_k)$ and $\mathcal{G}_2=\sigma(\mathsf{K}_{k+1})$ be the generated $\sigma$-algebras. Each $\mathsf{K}_q$ with $q\le k$ is $\mathcal{G}_1$-measurable, since by that definition every $\mathsf{K}_q^{-1}(B)$ with $B$ Borel and $q\le k$ belongs to $\mathcal{G}_1$; so $S$, $S^{2}$ and $S^{3}$ (linear combinations, with the constants $\mu_q$, and powers) are $\mathcal{G}_1$-measurable by claims 1, 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions applied on the measurable space $(\Omega,\mathcal{G}_1)$; likewise $Y$, $Y^{2}$ and $Y^{3}$ are $\mathcal{G}_2$-measurable. Grouping Lemma for Independent Random Variables, applied to the independent family $(\mathsf{K}_q)_{1\le q\le d}$ and the disjoint nonempty index sets $I_1=\{1,\dots,k\}$, $I_2=\{k+1\}$, shows that $\mathcal{G}_1$ and $\mathcal{G}_2$ are independent and hence that any $\mathcal{G}_1$-measurable random variable is independent of any $\mathcal{G}_2$-measurable one; in particular $S^{i}$ and $Y^{j}$ are independent, and, both having finite expectation (see below), Expectation of a Product of Independent Random Variables gives the claim. Expansion. Pointwise,

$$(S+aY)^{4}=S^{4}+4a\,S^{3}Y+6a^{2}\,S^{2}Y^{2}+4a^{3}\,SY^{3}+a^{4}\,Y^{4},\qquad(S+aY)^{2}=S^{2}+2a\,SY+a^{2}Y^{2}.$$

By $(\mathrm{H}_k)$ and the integrability of $Y^{4}$, every power $S^{i}$ and $Y^{j}$ with $i,j\le4$ has finite expectation (as $|t|^{i}\le1+t^{4}$ for $i\le4$), so the factorisation of the mixed moments applies. Using $\mathbb{E}[S]=0$ and $\mathbb{E}[Y]=0$, $\mathbb{E}[S^{2}]=v_k$, $\mathbb{E}[Y^{2}]=\lambda'$ and $\mathbb{E}[Y^{4}]=3\lambda'^{2}+\lambda'$, linearity of the expectation gives that $S_{k+1}^{4}$ is integrable, $\mathbb{E}[S_{k+1}]=\mathbb{E}[S]+a\mathbb{E}[Y]=0$, $\mathbb{E}[S_{k+1}^{2}]=v_k+0+a^{2}\lambda'=v_{k+1}$, and

$$\mathbb{E}[S_{k+1}^{4}]=\mathbb{E}[S^{4}]+0+6a^{2}v_k\lambda'+0+a^{4}(3\lambda'^{2}+\lambda')\le3v_k^{2}+\mathsf{F}_k+6a^{2}\lambda'v_k+3a^{4}\lambda'^{2}+a^{4}\lambda'=3\,(v_k+a^{2}\lambda')^{2}+\mathsf{F}_k+a^{4}\lambda'=3v_{k+1}^{2}+\mathsf{F}_{k+1},$$

which is $(\mathrm{H}_{k+1})$. Taking $k=d$ gives the three assertions of claim 3 for $\alpha\cdot(\mathsf{K}-\mu)=S_d$. The final consequence for $\mathsf{k}_4$ 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 follows because, in an instance as described in the statement, $\mathsf{k}_4^{4}=\mathbb{E}[(\alpha\cdot(\mathsf{K}-\mu))^{4}]$ there with the same cell-count vector and cell lengths, and its vector $\alpha$ of cell coefficients is an element of $\mathbb{R}^d$, to which the bound just proved applies. $\blacksquare$