I would like to prove, using the definition of autocorrelation, that Gaussian and uniform white noise have zero autocorrelation. I am working on the continuous case (but I think discrete shouldn't be too different).
Using the definition of the autocorrelation of this link shown on equation 9 (https://courses.edx.org/asset-v1:KyotoUx+009x+2T2017+type@asset+block@009x_31.pdf) we have that:
\begin{equation} \phi_Y(t) = \lim_{T\to \infty} \frac{1}{T} \int_{- \infty}^\infty d\tau Y_T(\tau)Y_T(\tau + t)) \end{equation}
We want to prove that $\phi_Y(t) = A^2\delta(t)$.
Let $Y_T(t) = A \xi(t)$ be white noise. If it is uniform white noise I think $\xi(t) = \frac{1}{T}$ for $t \in (-T/2, T/2)$ and zero otherwise. If we substitute this in the definition we would have: $$\phi_Y(t) = A^2 \lim_{T\to \infty} \frac{1}{T} \int_{- \infty}^\infty d\tau \frac{1}{T^2} $$ but this is not the $\delta(t)$ function. Where am I wrong? See bottom for a better attempt, using the time representation.
I also tried with Gaussian white noise. Gaussian noise is given by: $Y_T(t) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(t - \mu)^2}{2\sigma^2}}$, if I include this in the definition of autocorrelation we have:
$$\phi_Y(t) = \lim_{T\to \infty} \frac{1}{T \sigma^2 2 \pi} \int_{- \infty}^\infty d\tau e^{-\frac{(\tau - \mu)^2}{2\sigma^2}} e^{-\frac{(\tau + t- \mu)^2}{2\sigma^2}}$$ which can be re-written as $$\phi_Y(t) = \lim_{T\to \infty} \frac{1}{T \sigma^2 2 \pi} \int_{- \infty}^\infty d\tau e^{-\frac{(\tau - \mu)^2}{\sigma^2}} e^{\frac{-t(t + 2\tau - 2\mu)^2}{2\sigma^2}}$$ Under some conditions I would expect the $\delta(t)$ function to come up, but I don't see where.
Edit The frequency representation of uniform white noise is a constant probability $\tilde{\xi(\omega)} = p \in (0, 1)$. However, to obtain the time representation we must take the inverse Fourier transform.
\begin{equation} Y_T(\tau) = \frac{1}{2\pi} \int_{-\infty}^\infty d\omega e^{-i\omega \tau} Ap = Ap \delta(\tau). \end{equation}
Now we substitute in the definition of $\phi_Y(t)$ and we get
\begin{equation} \phi_Y(t) = \lim_{T\to \infty} \frac{1}{T} \int_{- \infty}^\infty d\tau Ap\delta(\tau)Ap\delta(\tau + t)). \end{equation} Now, I know that $\int_{- \infty}^\infty d\tau Ap\delta(\tau)Ap\delta(\tau + t)) = (Ap)^2 \delta(t)$. However, I am still confused by the term $\lim_{T\to\infty}\frac{1}{T}$ and about the $p$ being introduced (although this might just mean amplitude I guess).