# Autocorrelation of an AR(1) process

I am learning about this AR process. According to the book I'm reading, the autocorrelatio function of a stationary process: $$y_t = c + \phi y_{t-1} + \varepsilon_t, \quad \quad |\phi|< 1$$

is as follows: $$\rho(h)= \frac{\gamma(h)}{\gamma(0)} = \phi^h$$

Which means a slow exponential decay for successive lags, hence revealing that the series does behaves as an AR(1) process. So, I can not understand why in this case the autocorrelation function drops but then grows again.

 set.seed(123456) y <- arima.sim(n = 100, list(ar = 0.9), innov=rnorm(100))  op <- par(no.readonly=TRUE) layout(matrix(c(1, 1, 2, 3), 2, 2, byrow=TRUE)) plot.ts(y, ylab='') acf(y, main='Autocorrelations', ylab='', ylim=c(-1, 1), ci.col = "black") pacf(y, main='Partial Autocorrelations', ylab='', ylim=c(-1, 1), ci.col = "black") par(op) 

Any explanation?

It's perfectly normal when you have only 100 samples. Take it n=1000` and you'll see an exponentially dropping autocorrelation curve. Small samples have less tendency to obey theoretical results.