Why do ACF and PACF plots tend closer to each other with the increase of observations? I have an AR(2) process: $Y_t=0.6Y_{t-1}+0.1Y_{t-2}+\varepsilon_t$.  
When I plot its autocorrelation function & partial autocorrelation function on the same plane, I notice that their separate lines tend to become closer to each other until they merge together completely and become one line. How can I explain it?
Here are the pictures:





 A: This should not be the case since the partial auto correlation(PACF) after lag 2 onwards should be 0, and will not be the same as the ACF, which should decay to 0 exponentially. If you are seeing this, then you are not generating data that follows an AR(2) process. You'll get better help if you post a reproducible example.
A: Do you mean this?

plot(0:20,ARMAacf(c(.6, .1), lag.max = 20),type="l",lwd=3,col="lightblue",ylim=c(0,1))
lines(1:20,ARMAacf(c(.6, .1), lag.max = 20, pacf = TRUE),col="purple",lwd=3)

This follows from standard properties of stationary AR processes:


*

*The ACF (blue) of a stationary process decays to zero.

*The PACF (purple) of an $AR(p)$ process is zero for any $q>p$

A: It appears to me that there is something amiss in your Data Generating Function (DGF). Your ACF suggests that the variance of the errors might be too large and is thus hiding/masking the (nearly) expected structure of .6 , .36  , .216 , ... OR your generating equation is not what you think it is. There is another possibility , you might have inadvertently introduced an anomaly (read:pulse outlier) which can distort both the covariance and the variance of a series this leading to a diminshed ACF ( note the ACF = covariance/variance ) which explains why outliers(pulses) /level shifts/seasonal pulses/time trends) play hovoc (distort ) with simple ARIMA model identification ala auto.arima.
