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I am analyzing time series concerning daily water temperatures in rivers and I stubmled upon a series with autocorrelation like this:

Autocorrelation function with maximum lag = 730

Moreover the series is stationary (I checked it with the Augmented Dickey-Fuller test). Partial autocorrelation plots looks like this (and I guess it indicates that it is AR(2) process):

enter image description here

So my problem is: what does this weird autocorrelation plot indicates? It seems to be telling me something, but I can not understand what it is exactly, so any help will be much appreciated.

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What the autocorrelation plot indicates is that a good model for the stochastic process in question is $$X(t) = \cos(2\pi t + \Theta), -\infty < t < \infty$$ where $\Theta$ is a random variable that is uniformly distributed on $[0,2\pi)$. For a proof that the autocorrelation function of this process is indeed $\cos(2\pi \tau)$ where $\tau$ is the lag, see this answer on dsp.SE.

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