I have seen in class that for some reasons I forgot, the Brownian Motion has a Multivariate normal distribution, but I am unable to prove it easily. Could someone tell me why it's true? From what I understand, I have to take a finite linear combination of values of Brownian motion at different times, and check that it's normally distributed. Could someone help me on this? thanks
edit : the definition I start from is the one from wikipedia : http://en.wikipedia.org/wiki/Brownian_motion#Mathematics points 1 to 4 what I'm trying to prove is that Y = a1*B1 + … + ak*Bk is normally distributed, where Bi are values of the Brownian motion at time Ti
You can choose $\lambda_q,\dots,\lambda_k$ so that $$Y = a_1B_1 + \dots + a_kB_k = \lambda_1B_1 + \lambda_2(B_2-B_1) + \dots + \lambda_k(B_k-B_{k-1}).$$ But from the definition of Brownian motion, you know that $B_1, B_2-B_1, \dots, B_k-B_{k-1}$ are normally distributed and independent, so a linear combination of them is again normally distributed.