Shumway Stoffer Time_Series_Analysis_and_Its_Applications__With_R_Examples 3rd edition (1)

24 1 Characteristics of Time Series
ACovF
0.00 0.15 0.30
−4 −2 0 2 4
Fig. 1.12. Autocovariance function of a three-point moving average.
Lag
Definition 1.8 The autocovariance function of a stationary time series
will be written as
γ(h) = cov(x t+h , x t ) = E[(x t+h − µ)(x t − µ)]. (1.22)
Definition 1.9 The autocorrelation function (ACF) of a stationary
time series will be written using (1.14) as
ρ(h) =
γ(t + h, t)
√ = γ(h)
γ(t + h, t + h)γ(t, t) γ(0) . (1.23)
The Cauchy–Schwarz inequality shows again that −1 ≤ ρ(h) ≤ 1 for all
h, enabling one to assess the relative importance of a given autocorrelation
value by comparing with the extreme values −1 and 1.
Example 1.19 Stationarity of White Noise
The mean and autocovariance functions of the white noise series discussed
in Examples 1.8 and 1.16 are easily evaluated as µ wt = 0 and
{
σw 2 h = 0,
γ w (h) = cov(w t+h , w t ) =
0 h ≠ 0.
Thus, white noise satisfies the conditions of Definition 1.7 and is weakly
stationary or stationary. If the white noise variates are also normally distributed
or Gaussian, the series is also strictly stationary, as can be seen by
evaluating (1.18) using the fact that the noise would also be iid.
Example 1.20 Stationarity of a Moving Average
The three-point moving average process of Example 1.9 is stationary because,
from Examples 1.13 and 1.17, the mean and autocovariance functions
µ vt = 0, and