# Autocovariance of White Noise

I'm studying time series, I don't understand the description below.

The white noise series $$w_t$$ has $$E(w_t) = 0$$ and

$$\gamma_w(s, t) = \operatorname{cov}(w_s, w_t)= \begin{cases} \sigma_w^2 & s = t \\ 0 & s\ne t\end{cases}$$

But I don't understand how to get the result.

If $$s = t,$$ then $$\gamma_w(s, t) = \gamma_w(t, t) = E[(w_t- \mu_t)^2] = \operatorname{var}(w_t)$$

How does $$\operatorname{var}(w_t)$$ become $$\sigma_w^2$$? And why if s and t are not equal, the covariance of two sequence will become 0?

$$s$$ and $$t$$ are not two sequences, but two timesteps within one sequence.
If $$s\ne t$$ you are comparing two different elements of the sequence. If $$s = t$$ you are comparing one timestep with itself and in such a case autocovariance reduces to variance. In White Noice different elements of the sequence are independent of each other so their covariance by definition is 0.