# Showing that if $B_t$ is a Brownian motion then $t B_{1/t}$ is Gaussian

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.

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.