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

Let $(\Omega,\mathcal{F},P)$ be a probability space.

(a) (Grouping.) Let $q\ge1$ be a natural number, let $\xi_1,\dots,\xi_q$ be 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 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 independence of $\sigma$-algebras.

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

$$\sigma\big(X_{r+u}-X_r:\ u\ge0\big),$$

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