We show that the Bernoulli part extraction method can be used to obtainapproximate forms of the local limit theorem for sums of independent latticevalued random variables, with effective error term, that is with explicitparameters and universal constants. We also show that our estimates allow torecover Gnedenko and Gamkrelidze local limit theorems. We further establish bythis method a local limit theorem with effective remainder for random walks inrandom scenery.
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