Grouping Independence and the Fresh-Start Sigma-Algebra of an Independent-Increment Process

Let $(\Omega,\mathcal{F},P)$ be a \reftext{def:probability-space-random-variable-2026a}{probability space}.

\textbf{(a) (Grouping.)} Let $q\ge1$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $\xi_1,\dots,\xi_q$ be \reftext{def:independence-events-rvs-2026a}{independent} random variables on $(\Omega,\mathcal{F},P)$, and let $I$ and $J$ be disjoint subsets of $\{1,\dots,q\}$. Then the $\sigma$-algebras $\sigma(\xi_h:h\in I)$ and $\sigma(\xi_h:h\in J)$, each \reftext{def:independence-sigma-algebras-2026a}{generated} by the indicated random variables (with the convention that the $\sigma$-algebra generated by the empty collection is $\{\varnothing,\Omega\}$), are independent in the sense of \reftext{def:independence-sigma-algebras-2026a}{independence of $\sigma$-algebras}.

\textbf{(b) (Fresh start.)} Let $X=(X_t)_{t\ge0}$ be a \reftext{def:inhomogeneous-poisson-process-2026b}{stochastic process} on $(\Omega,\mathcal{F},P)$ with \reftext{def:inhomogeneous-poisson-process-2026b}{independent increments}, suppose there is a real number $c$ with $X_0=c$ \reftext{def:almost-surely-2026a}{almost surely}, let $(\mathcal{F}^X_t)_{t\ge0}$ be the \reftext{def:filtration-adapted-process-2026a}{natural filtration} of $X$, and fix a real number $r\ge0$. Then the $\sigma$-algebra

generated by all post-$r$ increments, is independent of $\mathcal{F}^X_r$ in the sense of \reftext{def:independence-sigma-algebras-2026a}{independence of $\sigma$-algebras}.