Antoni uses the following definition of white process of order p:
a process whose all cumulants up to order p are such that $$\text{Cum}\left[X(t),X(t-\tau_1)...,X(t-\tau_{r-1})\right]=C_{rX}\delta(\tau_{1})...\delta(\tau_{r-1})\forall r\le p$$
Where $X(t)$ is a random process of time.
Obviously, the expression collapses to zero for any case other than $\tau_1=..\tau_{r-1}=0$ .
Besides, I can tell very little about this expression and would like anyone to shed some light on its meaning.