I am out of my element with a topic I am working on. I think I have dwindled down the part I am stuck on to the following.
If $\phi_n(t)$ is a sequence of orthonormal (eigenfunctions) functions and $B(t)$ is a standard Brownian Motion (Weiner Process), then
$$\psi_n =\int^c_0 \phi_n(t) dB(t)$$
where
$$\psi_n \overset{iid}{\sim} N(0, 1).$$
Now, I believe I have seen this result before where $c=1$. I am hoping that what I have above is true for an arbitrary positive $c$ that can be greater than one. If so, where/how can I find information that this is true? Much appreciated!