Proof 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

Preliminaries. Uniform laws. For a Borel set $A$ with $0<\lambda(A)<\infty$, $\nu_A$ is a measure by claim 3 of Image Measures, Measures with Densities, and Change of Variables with $\nu_A(\mathbb{R})=1$, so it is a probability measure; $\nu_A(A)=1$; and $\nu_A(\{x\})=\lambda(\{x\})/\lambda(A)=0$ for every real $x$, a one-point set being an interval of length $0$ (claim 4 of Existence of Lebesgue Measure on the Real Line). For a Borel set $B$, $\nu_{(0,R]}(I_j\cap B)=\lambda(I_j\cap B)/R=(|I_j|/R)\,\nu_{I_j}(B)$, as $\lambda(I_j)=|I_j|$ and $\lambda((0,R])=R$ (claim 4 of Existence of Lebesgue Measure on the Real Line). Independence. Every subfamily of an independent family is independent (Independence of Events and of Random Variables), and for finitely many independent random variables the joint distribution is the product of the distributions and expectations of nonnegative or bounded jointly Borel functions are integrals against it (claims 1 and 2 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables); there $\mathcal{B}_r$ is the $r$-fold product Borel $\sigma$-algebra, which coincides with that of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$ (same recursion) and is generated by the Borel rectangles (claim 1 there). Integrals of nonnegative functions are handled with claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and Tonelli's theorem (Tonelli and Fubini Theorems) applies to products of probability measures. Countable subadditivity of $P$ is claim 4 of Basic Properties of a Measure. Indicators of Borel sets composed with random variables are random variables (preimages), and finite sums and products of random variables are random variables (Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); $\widetilde K$ is a random variable with values in $\mathbb{N}_0$ as in Thinning: Cell Counts of a Poisson Number of Independent Points, and $\sum_{i=1}^{\widetilde K}Z_i=\sum_{n\in\mathbb{N}_0}\mathbf{1}\{\widetilde K=n\}\sum_{i=1}^{n}Z_i$ is a random variable whenever the $Z_i$ are (a pointwise limit of finite sums, claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions).

Step 1: claim 1. Let $X,Y$ be independent random variables with distributions $\nu,\nu'$ each vanishing on one-point sets (this covers $(V_i,V_{i'})$ and $(U^{j}_{i},U^{j'}_{i'})$ for distinct index pairs, which are independent as subfamilies). The diagonal $\Delta=\{(x,x'):x=x'\}$ is a closed subset of $\mathbb{R}^{2}$, hence belongs to $\mathcal{B}_2$ (claim 4 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets), so by claims 1 and 2 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables and Tonelli,

$$P(X=Y)=\int_{\mathbb{R}^{2}}\mathbf{1}_\Delta\,d(\nu\otimes\nu')=\int_{\mathbb{R}}\nu'(\{x\})\,\nu(dx)=0 .$$

Also $P(V_i\notin(0,R])=1-\nu_{(0,R]}((0,R])=0$ and $P(U^{j}_{i}\notin I_j)=0$. The complement of $\Omega_0$ is, by definition, the union of countably many events each of probability $0$, so it is an event of probability $0$ by countable subadditivity, and $P(\Omega_0)=1$.

Step 2: claim 2. By the preliminaries, each $p_u$, $p^{(y)}_u$ and $C_j$ is a random variable with values in $\mathbb{N}_0$ (a sum of indicators); and $p_0=\sum_{i\le\widetilde K}\mathbf{1}\{V_i\le0\}$ and $p^{(y)}_0=\sum_{j}\sum_{i\le y_j}\mathbf{1}\{U^{j}_{i}\le0\}$ vanish on $\Omega_0$, where all points are positive. For $0\le s<u\le R$, $\mathbf{1}\{V_i\le u\}-\mathbf{1}\{V_i\le s\}=\mathbf{1}\{V_i\in(s,u]\}$, giving the increment formula. Now let $0\le u_0<\dots<u_r\le R$ and apply Thinning: Cell Counts of a Poisson Number of Independent Points to $K$ (Poisson with parameter $\mu=R$), the probability measure $\nu=\nu_{(0,R]}$, the variables $V_i$ and the pairwise disjoint Borel sets $A_q=(u_{q-1},u_q]$, $1\le q\le r$: the cell counts of that lemma are exactly $p_{u_q}-p_{u_{q-1}}$, and by its claim 3 they are independent with Poisson distributions with parameters $R\,\nu_{(0,R]}(A_q)=R\,\lambda(A_q)/R=u_q-u_{q-1}$. Taking $u_q=b_q$ ($0\le q\le J$) gives $C_j=p_{b_j}-p_{b_{j-1}}$ (the same thinning count with $A_j=I_j$), independence of $C_1,\dots,C_J$, and $C_j$ Poisson with parameter $|I_j|$; by claim 2 of Thinning: Cell Counts of a Poisson Number of Independent Points, $P(C=y)=\prod_j\mathrm{poi}_{|I_j|}(y_j)$, which is positive since $\exp>0$ and $|I_j|>0$.

Step 3: claim 3. Measurability. By the generator criterion of Measurable Function and Real-Valued Measurable Function, a map $Z:\Omega\to\mathbb{R}^{[0,R]}$ is measurable with respect to $\mathcal{C}$ if and only if $\mathrm{ev}_u\circ Z$ is a random variable for every $u$; this holds for $p$ and $p^{(y)}$ by claim 2. For $n\in\mathbb{N}$ define $\Pi_n:\mathbb{R}^{n}\to\mathbb{R}^{[0,R]}$ by $\Pi_n(v)(u)=\sum_{i=1}^{n}\mathbf{1}\{v_i\le u\}$; each $\mathrm{ev}_u\circ\Pi_n$ is jointly Borel (a sum of indicators of the sets $\{v:v_i\le u\}$, preimages of Borel sets under the coordinate projections, claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets), so $\Pi_n$ is measurable from $\mathcal{B}_n$ to $\mathcal{C}$ by the same criterion, and $F\circ\Pi_n$ is $\mathcal{B}_n$-measurable for every $\mathcal{C}$-measurable $F$. On the event $\{\widetilde K=n\}$ one has $p=\Pi_n(V_1,\dots,V_n)$ pointwise ($n\ge1$).

The case $y=0$. On $\Omega_0$, $\sum_jC_j=\widetilde K$ (every $V_i$ with $i\le\widetilde K$ lies in exactly one cell), so $\Omega_0\cap\{C=0\}=\Omega_0\cap\{\widetilde K=0\}$, on which $p$ is the zero path; and $p^{(0)}$ is the zero path everywhere (empty sums). Hence $\mathbb{E}[F(p)\mathbf{1}\{C=0\}]=F(0)P(K=0)=F(0)\exp(-R)$, while $P(C=0)\mathbb{E}[F(p^{(0)})]=\prod_j\exp(-|I_j|)\,F(0)=\exp(-R)F(0)$ by claim 2 and the functional equation of $\exp$ (Basic Properties of the Exponential Function), where $0$ denotes the zero path. So the claim holds for $y=0$; from now on let $n=\sum_jy_j\ge1$.

Patterns. Fix $y\in\mathbb{N}_0^{J}$ with $n=\sum_jy_j\ge1$. Let $\Sigma_y$ be the (finite, nonempty) set of maps $\sigma:\{1,\dots,n\}\to\{1,\dots,J\}$ with exactly $y_j$ indices mapped to $j$ for every $j$. On $\Omega_0\cap\{\widetilde K=n\}$, every $V_i$ with $i\le n$ lies in exactly one cell, say $I_{\sigma(i)}$, and then $C_j=\#\{i\le n:\sigma(i)=j\}$; hence $\Omega_0\cap\{C=y\}=\Omega_0\cap\{\widetilde K=n\}\cap\bigcup_{\sigma\in\Sigma_y}E_\sigma$, where $E_\sigma=\bigcap_{i=1}^{n}\{V_i\in I_{\sigma(i)}\}$, the union being disjoint since the cells are disjoint. As $P(\Omega_0)=1$ and $\widetilde K=K$ with probability $1$,

$$\mathbb{E}\bigl[F(p)\mathbf{1}\{C=y\}\bigr]=\sum_{\sigma\in\Sigma_y}\mathbb{E}\Bigl[F\bigl(\Pi_n(V_1,\dots,V_n)\bigr)\,\mathbf{1}\{K=n\}\prod_{i=1}^{n}\mathbf{1}\{V_i\in I_{\sigma(i)}\}\Bigr],\tag{3.1}$$

nonnegative integrands agreeing off a null event $N$ having equal integrals (split each by $\mathbf{1}_N$ and $\mathbf{1}_{N^c}$: every simple function below $h\mathbf{1}_N$ is bounded by a multiple of $\mathbf{1}_N$, so $\int h\mathbf{1}_N\,dP=0$ for every nonnegative measurable $h$ by Lebesgue Integral of a Nonnegative Measurable Function and Simple Function and Its Integral).

Integrating out $K$ and the cell indicators. Fix $\sigma\in\Sigma_y$ and first let $F$ be bounded. The finite family $(V_1,\dots,V_n,K)$ is independent with distributions $\nu_{(0,R]}$ ($n$ times) and the Poisson law $\mathrm{Poi}_R$, and $\varphi(v,k)=F(\Pi_n(v))\mathbf{1}\{k=n\}\prod_i\mathbf{1}_{I_{\sigma(i)}}(v_i)$ is jointly Borel (the projection $(v,k)\mapsto v$ is measurable from $\mathcal{B}_{n+1}=\mathcal{B}_n\otimes\mathcal{B}(\mathbb{R})$ to $\mathcal{B}_n$, preimages being the rectangles $A\times\mathbb{R}$, and products of measurable functions are measurable) and bounded, so by claim 2 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables and Tonelli (integrating the last coordinate $k$ first),

$$\mathbb{E}[\varphi(V_1,\dots,V_n,K)]=P(K=n)\int_{\mathbb{R}^{n}}F(\Pi_n(v))\prod_{i=1}^{n}\mathbf{1}_{I_{\sigma(i)}}(v_i)\,d\nu_{(0,R]}^{\otimes n}(v).$$

The measure with density $\prod_i\mathbf{1}_{I_{\sigma(i)}}(v_i)$ with respect to $\nu_{(0,R]}^{\otimes n}$ (claim 3 of Image Measures, Measures with Densities, and Change of Variables) and the measure $c_\sigma\,\nu_{I_{\sigma(1)}}\otimes\dots\otimes\nu_{I_{\sigma(n)}}$ with $c_\sigma=\prod_{i=1}^{n}\bigl(|I_{\sigma(i)}|/R\bigr)$ are finite measures on $\mathcal{B}_n$ that agree on every Borel rectangle $B_1\times\dots\times B_n$ (both give $\prod_i\nu_{(0,R]}(I_{\sigma(i)}\cap B_i)=\prod_i(|I_{\sigma(i)}|/R)\nu_{I_{\sigma(i)}}(B_i)$ by the preliminaries and the rectangle values of product measures, Existence and Uniqueness of the Product Measure), in particular on $\mathbb{R}^{n}$; Borel rectangles form a $\pi$-system generating $\mathcal{B}_n$, so the two measures coincide by claim 1 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law, and integrals of nonnegative measurable functions against them agree. Hence the last integral equals $c_\sigma\int F(\Pi_n(v))\,d(\nu_{I_{\sigma(1)}}\otimes\dots\otimes\nu_{I_{\sigma(n)}})(v)$. For $i\le n$ let $r_\sigma(i)$ be the number of indices $i'\le i$ with $\sigma(i')=\sigma(i)$, and put $W_i=U^{\sigma(i)}_{r_\sigma(i)}$; the map $i\mapsto(\sigma(i),r_\sigma(i))$ is a bijection from $\{1,\dots,n\}$ onto $\{(j,i'):1\le j\le J,\ 1\le i'\le y_j\}$, so $W_1,\dots,W_n$ are distinct members of the independent family, hence independent, with distributions $\nu_{I_{\sigma(i)}}$, and by claim 2 of Joint Distribution, Expectations, and Block Independence for Independent Random Variables again the last integral equals $\mathbb{E}[F(\Pi_n(W_1,\dots,W_n))]$. Finally, reindexing the finite sum along that bijection (claim 2 of Properties of a Sum over a Finite Index Set) gives $\Pi_n(W_1,\dots,W_n)(u)=\sum_{i\le n}\mathbf{1}\{U^{\sigma(i)}_{r_\sigma(i)}\le u\}=\sum_j\sum_{i'\le y_j}\mathbf{1}\{U^{j}_{i'}\le u\}=p^{(y)}_u$ pointwise on $\Omega$. Altogether, for bounded $F$,

$$\mathbb{E}\bigl[F(p)\mathbf{1}\{C=y\}\bigr]=P(K=n)\Bigl(\sum_{\sigma\in\Sigma_y}c_\sigma\Bigr)\,\mathbb{E}\bigl[F(p^{(y)})\bigr].\tag{3.2}$$

Taking $F\equiv1$ in (3.2) gives $P(C=y)=P(K=n)\sum_\sigma c_\sigma$, and substituting this back yields the claim for bounded $F$. For general $F:\mathbb{R}^{[0,R]}\to[0,\infty]$ apply the bounded case to $\min(F,M)$, $M\in\mathbb{N}$, and let $M\to\infty$ on both sides by the monotone convergence theorem.

Step 4: claim 4. Splitting the inner sum over $i\le y_{j_0}+\mathsf{m}$ at $i=y_{j_0}$ (Splitting a Finite Sum at an Index when $y_{j_0}\ge1$; trivially when $y_{j_0}=0$, the first block being empty) gives the displayed identity for every $\omega$ and $u$. On $\Omega_0$ the points $U^{j_0}_{i}$ with $y_{j_0}<i\le y_{j_0}+\mathsf{m}$ lie in $I_{j_0}$ and all the $U^{j}_{i'}$ are pairwise distinct, by claim 1, which gives the remaining assertions. $\blacksquare$