Solved – What method to test if the mean is constant

You may want to look at a nonparametric approach to this problem, since you cannot rely on normality. Break your sample into two groups, $x_1, dots, x_{400}$ and $x_{401}, dots, x_{800}$ and compare them in pairs ($x_1, x_{401}$), $dots$, ($x_{400}, x_{800}$).

You now have the framework of a two-sample test of equal means/medians. Under the null hypothesis of a constant mean, we should expect the mean/median of these two groups to be the same.

Since we expect the medians to be the same, some NP tests will assert that in the pair $(x_i, x_{i+400})$, the chance that $x_i$ is bigger is a random variable, Bernoulli distributed with probability $frac{1}{2}$. Two hypothesis tests proceed from this. Tests to look into would be: Wilcoxon Signed Rank Test and the Signed Test.