Asymptotic spectral distributions of distance $k$-graphs of star product graphs

Let $G$ be a finite connected graph and let $G^{[\star N,k]}$ be the distance $k$-graph of the $N$-fold star power of $G$. For a fixed $k\geq1$, we show that the large $N$ limit of the spectral distribution of $G^{[\star N,k]}$ converges to a centered Bernoulli distribution, $1/2\delta_{-1}+1/2\delta_1$. The proof is based in a fourth moment lemma for convergence to a centered Bernoulli distribution.