Fresh-Start Property for Independent Poisson Clocks Read at Levels Satisfying a Clock-Reading Bound

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathsf{A}$ be a nonempty finite set (the clock labels), let $\mathcal{I}\subseteq\mathcal{F}$ be a $\sigma$-algebra (the initial data), let $R$ be either a real number with $R>0$ or $+\infty$, and for each $a\in\mathsf{A}$ let $Y^a=(Y^a_u)_{u\ge0}$ be a Poisson clock with horizon $R$ on $(\Omega,\mathcal{F},P)$ (for $R=+\infty$, a homogeneous Poisson process with rate $1$ every path of which is a counting path), such that the family of $\sigma$-algebras consisting of $\mathcal{I}$ together with the $\sigma$-algebras generated $\sigma(Y^a_u:u\ge0)$, one for each $a\in\mathsf{A}$, is independent. Let $\mathfrak{F}\subseteq\mathcal{F}$ be a $\sigma$-algebra (the past), and for each $a\in\mathsf{A}$ let $\mathcal{T}^a:\Omega\to[0,\infty)$ be an $\mathfrak{F}$-measurable random variable (the consumed level of clock $a$), and assume there is a real number $\bar{c}\ge0$ with $\bar{c}<R$ and $\mathcal{T}^a\le\bar{c}$ everywhere for every $a$. Write $R^-=R-\bar{c}$, which is $+\infty$ when $R=+\infty$.

For a family $\mathbf{c}=(c_a)_{a\in\mathsf{A}}$ of nonnegative real numbers (caps) let $C_{\mathbf{c}}$ be the event that $\mathcal{T}^a\le c_a$ for every $a\in\mathsf{A}$, and let $\mathcal{H}_{\mathbf{c}}$ be the $\sigma$-algebra generated by $\mathcal{I}$ together with the variables $Y^a_u$ for $0\le u\le c_a$ and $a\in\mathsf{A}$. Assume the clock-reading bound: for every family of caps $\mathbf{c}$, the event $C_{\mathbf{c}}$ agrees up to an event of probability zero with an event of $\mathcal{H}_{\mathbf{c}}$, and for every $F\in\mathfrak{F}$ there is an $H\in\mathcal{H}_{\mathbf{c}}$ such that the symmetric difference of $F\cap C_{\mathbf{c}}$ and $H\cap C_{\mathbf{c}}$ is an event of probability zero.

Define the residual clocks

$$\hat{Y}^a_u=Y^a_{\mathcal{T}^a+u}-Y^a_{\mathcal{T}^a}\qquad(u\ge0,\ a\in\mathsf{A}).$$

Then:

(a) Every path of every residual clock is a counting path, each $\hat{Y}^a_u$ is a random variable on $(\Omega,\mathcal{F},P)$, and for every $a\in\mathsf{A}$ and all real $0=u_0<u_1<\dots<u_p<R^-$ the increments $\hat{Y}^a_{u_1}-\hat{Y}^a_{u_0},\dots,\hat{Y}^a_{u_p}-\hat{Y}^a_{u_{p-1}}$ are independent, the $q$-th having the Poisson distribution with parameter $u_q-u_{q-1}$. In particular, when $R=+\infty$, every residual clock is a homogeneous Poisson process with rate $1$ on $(\Omega,\mathcal{F},P)$.

(b) The family of $\sigma$-algebras consisting of $\mathfrak{F}$ together with the $\sigma$-algebras $\sigma(\hat{Y}^a_u:0\le u<R^-)$, one for each $a\in\mathsf{A}$, is independent.