What does "AR(p) filtered series" mean? I guess this means that omitting some variables in a certain interval, say, $(x_1, x_2, x_3, x_4, x_5) \to (x_1, x_5)$  in AR(4) model. 
Is it right? Or does this means eliminating autocorrelations by adjusting autocorrealated variables by a certain method?
Thanks in advance!
 A: Without context it's difficult to be completely certain of the intent; I can think of several possible meanings. I will talk about the one I guess to be the most likely.
'Filter' is almost certainly being used in the signal processing sense, about which more shortly.
An AR process is of the following form:
$$Y_t = c + \sum_{i=1}^p \varphi_i Y_{t-i}+ \varepsilon_t \,$$.
In the time-domain analysis of time series data, a statistician might use a model of that form to describe a data-generating process - one that feeds back on itself over time and also has independent noise added at each time period.
The signal-processing-and-filtering people tend to frame things as processes they apply to inputs (for example, to make it smoother by removing high frequency components) and they call that kind of processing of data 'filtering'. The same terminology is sometimes used to refer to such a process more generally (i.e. even when not deliberately applied as a way of removing components from the data process), so the same structure and name can also used by the signal processing fraternity to refer to an AR process that occurs naturally (since it can still be viewed as a filter operating on some input).
See here.
As such, "an AR(p) filtered series" might mean an AR(p) process that has simply been observed, or it might more often mean a series of data that has had a filter of the form in the link applied to it, one which would produce an AR(p) process from white noise.
If you want to investigate signal processing filters, you might start here.
As mentioned at the start there are other possible meanings. Depending on the phrasing of the remainder of the sentence, it's even possible the intent was to refer to a filter that was applied to data from an AR(p) process (since one may well want to filter an AR(p) process). That is, a filter which is applied to an AR(p) input to produce some particular kind of output, or it might be intended to mean a filter which if applied to a given AR(p) should produce white noise.
More context should allow us to eliminate some of these possibilities.
