Given an uncorrelated, Guassian-distributed random number $\xi$ is unitary variance and zero mean, i.e. $\langle \xi\rangle = 0$ and $\langle \xi(t)\xi(0)\rangle = \delta(t)$.
My question is:
If $\xi$ is unitary variance, does this mean $\langle \xi(0)\xi(0)\rangle = \delta(0)=1$?
If so, then could we still have $\int_{-\infty}^{+\infty}\langle \xi(t)\xi(0)\rangle dt =\int_{-\infty}^{+\infty}\delta(t)dt =1$?
Thank you in advance.