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?


1 Answer 1


$s$ 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.

  • $\begingroup$ Thank you! I suddenly understood! $\endgroup$
    – shihs
    Commented Sep 12, 2019 at 21:00

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