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

Let $\mathbb{N}$ be the set of natural numbers, $\mathbb{N}_0=\mathbb{N}\cup\{0\}$, $\mathbb{R}$ the real numbers with Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ and Lebesgue measure $\lambda$. For a Borel set $A$ with $0<\lambda(A)<\infty$, the uniform law on $A$ is the probability measure $\nu_A(B)=\lambda(A\cap B)/\lambda(A)$ on $\mathcal{B}(\mathbb{R})$, the measure with density $\mathbf{1}_A/\lambda(A)$ with respect to $\lambda$, where $\mathbf{1}_A$ is the indicator of $A$. For $\mu\ge0$ write $\mathrm{poi}_\mu(k)=\exp(-\mu)\mu^{k}/k!$ ($k\in\mathbb{N}_0$) for the mass function of the Poisson distribution with parameter $\mu$, with the exponential function $\exp$, the factorial ($0!=1$) and $\mu^{0}=1$.

Let $R>0$ be real, let $J\ge1$ be a natural number, and let $0=b_0<b_1<\dots<b_J=R$ be real numbers; the cells are the intervals $I_j=(b_{j-1},b_j]$, $1\le j\le J$, with lengths $|I_j|=b_j-b_{j-1}$, so that $(0,R]$ is their disjoint union. Let $(\Omega,\mathcal{F},P)$ be a probability space carrying an independent family of random variables

$$K,\qquad (V_i)_{i\in\mathbb{N}},\qquad (U^{j}_{i})_{1\le j\le J,\ i\in\mathbb{N}},$$

where $K$ has the Poisson distribution with parameter $R$, each $V_i$ has distribution $\nu_{(0,R]}$, and each $U^{j}_{i}$ has distribution $\nu_{I_j}$ (such a family exists on some probability space by Existence of Independent Sequences with Prescribed Distributions, after enumerating the countable index set by a bijection with $\mathbb{N}$, independence of a family being unaffected by reindexing). Put $\widetilde K=K$ on $\{K\in\mathbb{N}_0\}$ and $\widetilde K=0$ elsewhere, as in Thinning: Cell Counts of a Poisson Number of Independent Points, so that $\widetilde K=K$ with probability $1$. Write $\mathbb{E}$ for the expectation; sums over an empty index range are $0$ and finite products over an empty index range are $1$. Let $\Omega_0$ be the complement of the union of the countably many events $\{V_i=V_{i'}\}$ ($i\neq i'$), $\{U^{j}_{i}=U^{j'}_{i'}\}$ ($(j,i)\neq(j',i')$), $\{V_i\notin(0,R]\}$ and $\{U^{j}_{i}\notin I_j\}$; on $\Omega_0$ all $V_i$ are pairwise distinct and lie in $(0,R]$, and all $U^{j}_{i}$ are pairwise distinct with $U^{j}_{i}\in I_j$.

Define the uniform Poisson path $p=(p_u)_{u\in[0,R]}$ and, for every $y=(y_1,\dots,y_J)\in\mathbb{N}_0^{J}$, the deterministic-count path $p^{(y)}=(p^{(y)}_u)_{u\in[0,R]}$ by

$$p_u=\sum_{i=1}^{\widetilde K}\mathbf{1}\{V_i\le u\},\qquad p^{(y)}_u=\sum_{j=1}^{J}\sum_{i=1}^{y_j}\mathbf{1}\{U^{j}_{i}\le u\},$$

and the cell counts $C_j=\sum_{i=1}^{\widetilde K}\mathbf{1}\{V_i\in I_j\}$, $C=(C_1,\dots,C_J)$. Let $\mathbb{R}^{[0,R]}$ be the set of all maps $[0,R]\to\mathbb{R}$ with the cylinder $\sigma$-algebra $\mathcal{C}$, the $\sigma$-algebra generated by the coordinate maps $\mathrm{ev}_u:x\mapsto x(u)$, $u\in[0,R]$, that is, by the sets $\mathrm{ev}_u^{-1}(B)$ with $u\in[0,R]$ and $B\in\mathcal{B}(\mathbb{R})$. For $j_0\in\{1,\dots,J\}$ let $e_{j_0}\in\mathbb{N}_0^{J}$ be the vector with $j_0$th entry $1$ and all other entries $0$.

1. (Distinct points) $P(V_i=V_{i'})=0$ for $i\neq i'$ and $P(U^{j}_{i}=U^{j'}_{i'})=0$ for $(j,i)\neq(j',i')$. Consequently $\Omega_0$ is an event with $P(\Omega_0)=1$.

2. (Poisson increments) Every $p_u$, $p^{(y)}_u$ and $C_j$ is a random variable with values in $\mathbb{N}_0$, and $p_0=p^{(y)}_0=0$ on $\Omega_0$. For $0\le s<u\le R$, $p_u-p_s=\sum_{i=1}^{\widetilde K}\mathbf{1}\{V_i\in(s,u]\}$, and for all real $0\le u_0<u_1<\dots<u_r\le R$ ($r\ge1$) the increments $p_{u_1}-p_{u_0},\dots,p_{u_r}-p_{u_{r-1}}$ are independent, $p_{u_q}-p_{u_{q-1}}$ having the Poisson distribution with parameter $u_q-u_{q-1}$. In particular $C_j=p_{b_j}-p_{b_{j-1}}$, the cell counts $C_1,\dots,C_J$ are independent with $C_j$ Poisson with parameter $|I_j|$, and $P(C=y)=\prod_{j=1}^{J}\mathrm{poi}_{|I_j|}(y_j)>0$ for every $y\in\mathbb{N}_0^{J}$.

3. (Conditional law given the cell counts) The maps $p:\Omega\to\mathbb{R}^{[0,R]}$ and $p^{(y)}:\Omega\to\mathbb{R}^{[0,R]}$ are measurable with respect to $\mathcal{F}$ and $\mathcal{C}$, and for every $y\in\mathbb{N}_0^{J}$ and every $F:\mathbb{R}^{[0,R]}\to[0,\infty]$ measurable with respect to $\mathcal{C}$ (in the sense of Lebesgue Integral of a Nonnegative Measurable Function),

$$\mathbb{E}\bigl[F(p)\,\mathbf{1}\{C=y\}\bigr]=P(C=y)\,\mathbb{E}\bigl[F(p^{(y)})\bigr]\qquad\text{in }[0,\infty].$$

4. (Insertion of $\mathsf{m}$ points into a cell) Let $y\in\mathbb{N}_0^{J}$, $j_0\in\{1,\dots,J\}$ and $\mathsf{m}\in\mathbb{N}$. Then for every $\omega\in\Omega$ and $u\in[0,R]$,

$$p^{(y+\mathsf{m}e_{j_0})}_u=p^{(y)}_u+\sum_{i=y_{j_0}+1}^{y_{j_0}+\mathsf{m}}\mathbf{1}\{U^{j_0}_{i}\le u\},$$

and on $\Omega_0$ the $\mathsf{m}$ inserted points $U^{j_0}_{i}$ ($y_{j_0}<i\le y_{j_0}+\mathsf{m}$) lie in $I_{j_0}$, are pairwise distinct, and differ from every point $U^{j}_{i'}$ with $1\le j\le J$, $1\le i'\le y_j$.