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I want to show that if $B_t$ is a Brownian motion then $t B_{1/t}$ is a Gaussian process, i.e. that it has increments which have the normal distribution.

It seems like a trivial fact, since the distribution of $B_{1/t}$ will just be a downscaled version of the distribution of $B_t$ while $t B_{1/t}$ is scaled back again. However, I don't have any ideas about how to prove this in a more rigorous way.

I am somewhat of a newbie when it comes to Brownian motion so I would appreciate seeing a proof of this property.

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Let $X_t=t\,B_{1/t}$. Every linear combination of the random variables $X_t$ is a linear combination of the random variables $B_t$ hence it is normal. Thus, the process $(X_t)$ is gaussian, in particular every increment $X_t-X_s$ is gaussian.

More generally, let $Y_t=a(t)B_{c(t)}+e(t)$. For every functions $(a,c,e)$, the process $(Y_t)$ is gaussian.

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