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I have a stochastic process completely uncorrelated. Why the autocorrelation function has a delta of dirac in the origin? Which is the reason of that?
$R_{XX}({\tau})=A{\delta(\tau)}$
where $R_{XX}$ is the autocorrelation function of $X$ process and $A$ is a constant dependent by process spectrum
It has a delta at the origin because you are multiplying your time series with itself (i.e. there is no time delay). Any non-zero time series correlated with itself will give a positive value.
As you noted, the wide-sense stationary stochastic process $X(t)$ is uncorrelated with itself for all nonzero time lags $\tau$. For $\tau=0$, it's impossible for a process to be uncorrelated with itself (unless $X(t)$ is the trivial process $X(t)=0$). Look at the definition for $R_{XX}(\tau)$:
$$ R_{XX}(\tau) = E(X(t)X^*(t+\tau) $$
$$ R_{XX}(0) = E(|X(t)|^2) $$
By definition, $|X(t)|^2$ must be $\ge 0$, so its expectation must also satisfy that condition.
