Graphical pattern of short memory ARMA-type component The book of "Time Series and its Applications" gives the following example,

The book also includes the following statement regarding the observation from this figure

I do not know what exactly does it mean for the short memory ARMA-type component? Especially, what should be the generic graphic pattern for this so-called ARMA-type component
 A: ARMA type of process will shows wave-like ACF and PACF, kind of like decaying cosine function. AR(p) processes show exponential decay in ACF, and hard cut-off after p lags. The MA(q) processes show hard cutoff after q lags in ACF, and exponential decay in PACF. All these have to be significant, i.e. above the dotted lines on your plot.
On your plot all bars are under the dotted line of significance, except a couple of ticks around 40 in ACF and PACF. They're barely above the line though so I'd be tempted to ignore them as benign, maybe random spikes. This means that your residuals do not show any obvious ARMA, AR or MA behavior, which is great, because it means that your ARIMA(1,1,1) model fits the data quire reasonably.
If this was my data, I'd pay closer attention to those two spikes around 40, of course. They may point to something important, e.g. maybe you do something to your data every 40th time of collection. However, most of the time these spikes are circumstantial, e.g. your sample maybe not too large, and you have a couple of outliers 40 observations apart, they'd show up as correlation in lags too, red herring, if you wish.
Here's an example in MATLAB, that shows AR, MA and ARMA
   % AR
spec=arima('AR',[0.8 0.1],'Variance',0.1,'Constant',1)
y=spec.simulate(1000);
subplot(2,3,1)
autocorr(y,30)
title 'AR(2)'
subplot(2,3,4)
parcorr(y,30)

% MA
spec=arima('MA',[0.8 0.1],'Variance',0.1,'Constant',1)
y=spec.simulate(1000);
subplot(2,3,2)
autocorr(y,30)
title 'MA(2)'
subplot(2,3,5)
parcorr(y,30)

% ARMA
spec=arima('AR',[0.8 0.1],'MA',[0.8 0.1],'Variance',0.1,'Constant',1)
y=spec.simulate(1000);
subplot(2,3,3)
autocorr(y,30)
title 'ARMA(2,2)'
subplot(2,3,6)
parcorr(y,30)


A: (Disclaimer: I haven't read the book.)
In long memory models  ACF decays in slow rate (like hyperbolic rate). I haven't seen a definition for short memory models. I guess it is taken as a complement side of long memory models. ARMA(p,q) with p and q are relatively small (for eg: p=q<=2) with respect to the sample size can be considered as short memory model. In ARMA(p,q) case, ACF/PACF usually decrease in geometric rate after certain lags. 
In the figure it clear that ACF is significant around lag 40 and lag 80. This is one indication that ARMA may not be a good fit on this case.
