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Homework Statement
Is the process [itex]\{X(t)\}_{t\geq 0}[/itex], where [itex]X(t)=\rho B_1(t)+\sqrt{1-\rho^2}B_2(t)[/itex] Standard Brownian Motion?
Where [itex]\rho\in(0,1), \ B_1(t)[/itex] and [itex]B_2(t) [/itex] are independent standard brownian motions
Homework Equations
The Attempt at a Solution
Obviously [itex]X(0)=0[/itex]. Now let [itex]0\leq t_1<t_2<t_3[/itex]. Then, I can show that [itex]X(t_2)-X(t_1)\sim\mathcal{N}(0, t_2-t_1)[/itex]. My problem is showing that [itex]X(t_3)-X(t_2)[/itex] and [itex]X(t_2)-X(t_1)[/itex] are independent. I can show their covariance is 0, but that only implies independence if the process is Gaussian, which I have not shown. Any help would be much appreciated!