Is a Wiener process a Gaussian process?

Question

My background in measure theory and stochastic processes is poor. In the literature I have found that a Wiener process can be defined as the integral of a white noise Gaussian process. But does this make a Wiener process a Gaussian process? Could you provide a very simple explanation for the answer?

Answer 1

There are a bunch of different definitions of the Wiener process; the one you gave is actually quite mathematically inconvenient (basically since you have to get everything started in the Fourier domain).

A more convenient one explicitly states that the Wiener process is Gaussian (and in particular specifies its mean function as $m(t)=0$ and its covariance function as $R(s,t)=\min \{ s,t \}$). Another starts from an assumption of independent increments, from which the Gaussian property follows by the central limit theorem.