One dimensional Ising model MCMC simulation strong fluctuations in autocorrelation functions

I’m performing some MCMC simulation of the 1D Ising model, but when it comes to computing the integrated autoroccelation time I observe strong fluctuations in the autocorrelations functions for long times . The following plots represent the correlations functions of magnetisation computed from an already thermalized configuration. The system is a linear chain with PBC and $N=1000$ spins, and $beta=0.1$.

As you can see near $t~N$ strong fluctuaions arises in autocorrelation functions, thus giving a wrong estimate of the integrated correlation time
$$ tau_{rm int} = 1+2sum_{t=1}^{N-1}left(1-frac{t}{N}right)frac{C(t)}{C(0)}.$$

What is happening? Is there some nice reference for thermalization of 1D Ising model and the decorrelation of configurations as a function of time?