  We show that the Bernoulli part extraction method can be used to obtain
approximate forms of the local limit theorem for sums of independent lattice
valued random variables, with effective error term, that is with explicit
parameters and universal constants. We also show that our estimates allow to
recover Gnedenko and Gamkrelidze local limit theorems. We further establish by
this method a local limit theorem with effective remainder for random walks in
random scenery.
