On the increments of Brownian motion Let $B_t$ be a Brownian motion on some probability space.
It is known that the increments of the Brownian motion are independent, meaning that $\sigma(B_{t_1}-B_{s_1})$ is independent from $\sigma(B_{t_2}-B_{s_2})$ for any $0 \leq s_{1}<t_{1} \leq s_{2}<t_{2}$.
Can we use that fact to show that $\sigma(\frac{B_{t_1}}{t_1}-\frac{B_{s_1}}{s_1})$ is independent from $\sigma(\frac{B_{t_2}}{t_2}-\frac{B_{s_2}}{s_2})$ for any $0 \leq s_{1}<t_{1} \leq s_{2}<t_{2}$ ? Of course those are independent (we can show that they have covariance $0$), but how to show it using the fact that the increments of Brownian motion are independent (i.e using a $\sigma$-algebra argument, not computing covariances)?
Can this be extended for arbitrary non-zero coefficients, i.e for $a,b,c,d \in \mathbb R_0^+$, $\sigma(aB_{t_1}-bB_{s_1})$ is independent from $\sigma(cB_{t_2}-dB_{s_2})$ for any $0 \leq s_{1}<t_{1} \leq s_{2}<t_{2}$ ?
Can this be extend to arbitrary number of times, for example using 3 times: for $a_1,a_2,a_3,a_4,a_5,a_6 \in \mathbb R_0^+$, $\sigma(a_1B_{t_1}-a_2B_{s_1} -a_3B_{q_1})$ is independent from $\sigma(a_3B_{t_2}-a_4B_{s_2} -a_6B_{q_2})$ or any $0 \leq s_{1}<t_{1}<q_{1} \leq s_{2}<t_{2} <q_2$ ?
 A: It turns out that the independence of $\frac{B_{t_1}}{t_1} - \frac{B_{s_1}}{s_1}$ and $\frac{B_{t_2}}{t_2} - \frac{B_{s_2}}{s_2}$ relies crucially on the fact that these two random variables are jointly Gaussian; independence of the increments alone does not imply that these random variables are independent.
For example, consider a rate-1 Poisson process $(N_t)_{t \geq 0}$. This process has independent increments, and for $s < t$, $N_t - N_s$ is distributed as a Poisson random variable with parameter $t-s$. It also has the same covariance function as Brownian motion, namely $\mathrm{Cov}(B_s, B_t) = \min\{s,t\}$, so that
$$ \mathrm{Cov} \left( \frac{N_{t_1}}{t_1} - \frac{N_{s_1}}{s_1}, \frac{N_{t_2}}{t_2} - \frac{N_{s_2}}{s_2} \right) = 0 $$
just as above. However, I claim that $\frac{N_{t_1}}{t_1} - \frac{N_{s_1}}{s_1}$ and $\frac{N_{t_2}}{t_2} - \frac{N_{s_2}}{s_2}$ are not in general independent. Consider $N_1 - 2N_{1/2} = U$ and $\frac{1}{2} N_2 - N_1 = V$. Denote the increments $X = N_{1/2} - N_{0}$, $Y = N_{1} - N_{1/2}$, and $Z = N_2 - N_1$. All three are independent Poisson, the first two have parameter $1/2$, and the last has parameter $1$. Write
$$ U = X-Y, \qquad V = \frac 1 2 (-X-Y+Z) ,$$
and use the characteristic function for Poisson random variables to find a value $(s,t)$ for which
$$ E(e^{isU + itV}) \neq E(e^{isU}) \, E(e^{isV}) . $$
If my calculation is correct, for $(s,t) = (\pi, 2 \pi)$, the above reads $e^{-3} \neq e^{-6}$. I would appreciate if someone could double-check this!
