Proving a process is Brownian Motion

[IniquiTrance]

Homework Statement

Is the process $\{X(t)\}_{t\geq 0}$, where $X(t)=\rho B_1(t)+\sqrt{1-\rho^2}B_2(t)$ Standard Brownian Motion?

Where $\rho\in(0,1), \ B_1(t)$ and $B_2(t)$ are independent standard brownian motions

Homework Equations

The Attempt at a Solution

Obviously $X(0)=0$. Now let $0\leq t_1<t_2<t_3$. Then, I can show that $X(t_2)-X(t_1)\sim\mathcal{N}(0, t_2-t_1)$. My problem is showing that $X(t_3)-X(t_2)$ and $X(t_2)-X(t_1)$ 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!

 

[Ray Vickson]

Homework Statement

Is the process $\{X(t)\}_{t\geq 0}$, where $X(t)=\rho B_1(t)+\sqrt{1-\rho^2}B_2(t)$ Standard Brownian Motion?

Where $\rho\in(0,1), \ B_1(t)$ and $B_2(t)$ are independent standard brownian motions

Homework Equations

The Attempt at a Solution

Obviously $X(0)=0$. Now let $0\leq t_1<t_2<t_3$. Then, I can show that $X(t_2)-X(t_1)\sim\mathcal{N}(0, t_2-t_1)$. My problem is showing that $X(t_3)-X(t_2)$ and $X(t_2)-X(t_1)$ 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!

For i = 1,2 and j = 1,2, are the $B_i(t_3)-B_i(t_2)$ independent of the $B_j(t_2)-B_j(t_1)$?

 

[IniquiTrance]

For i = 1,2 and j = 1,2, are the $B_i(t_3)-B_i(t_2)$ independent of the $B_j(t_2)-B_j(t_1)$?

Hi Ray, yes they are.

 

[IniquiTrance]

Not sure how to proceed.

 

[IniquiTrance]

Are you implying that if $A\perp B\perp C \perp D$, then $A+B \perp C+D$, where $\perp$ means independent?

 

[Ray Vickson]

Are you implying that if $A\perp B\perp C \perp D$, then $A+B \perp C+D$, where $\perp$ means independent?

More than that: if $A\perp B\perp C \perp D$, then $f(A,B) \perp g(C,D)$ for any (measurable) functions f(.,.) and g(.,.).

 

[IniquiTrance]

More than that: if $A\perp B\perp C \perp D$, then $f(A,B) \perp g(C,D)$ for any (measurable) functions f(.,.) and g(.,.).

Thanks. How can I go about proving that?

 

[Ray Vickson]

Thanks. How can I go about proving that?

It depends on what you know already. Anyway, all you need is the result for A+B and C+D, and you can use any number of standard tools, but first you need to know what those tools are. Do you know about characteristic functions? What can you say about the density (or cdf) of a set of independent random variables? How can you test whether two random variables (A+B and C+D in this case) are independent?

All of these things can be found in books and articles---many freely available on-line---so I will leave it to you to search for the answers if you do not know them already. If you do know them already, just think about how you would use them.