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In this paper we study correlation measures introduced in\cite{emme_asymptotic_2017}. Denote by $\mu_a(d)$ the asymptotic density of theset $\mathcal{E}_{a,d}=\{n \in \mathbb{N}, \ s_2(n+a)-s_2(n)=d\}$ (where $s_2$is the sum-of-digits function in base 2). Then, for any point $X$ in$\{0,1\}^\mathbb{N}$, define the integer sequence $\left(a_X (n)\right)_{n\in\mathbb{N}}$ such that the binary decomposition of $a_X (n) $ is the prefix oflength $n$ of $X$. We prove that for \textit{any} shift-invariant ergodicprobability measure $\nu$ on $\{0,1\}^\mathbb{N}$, the sequence$\left(\mu_{a_X(n)}\right)_{n \in \mathbb{N}}$ satisfies a central limittheorem. This result was proven in the case where $\nu$ is the symmetricBernoulli measure in \cite{emme_central_2018}.

Normal distribution of correlation measures of binary sum-of-digits functions