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 us to recover Gnedenko and provide a version with effective bounds of Gamkrelidze’s local limit theorem. We further establish by this method a local limit theorem with effective remainder for random walks in random scenery.
"Approximate local limit theorems with effective rate and application to random walks in random scenery." Bernoulli 23 (4B) 3268 - 3310, November 2017. https://doi.org/10.3150/16-BEJ846