  We prove the convergence of the Bergman kernels and the $L^2$-Hodge numbers
on a tower of Galois coverings $\{X_j\}$ of a compact K\"ahler manifold $X$
converging to an infinite Galois (not necessarily universal) covering
$\widetilde{X}$. We also show that, as an application, sections of canonical
line bundle $K_{X_j}$ for sufficiently large $j$ give rise to an immersion into
some projective space, if so do sections of $K_{\widetilde{X}}$.
