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We consider a real random walk S_n = X_1 + ... + X_n attracted (withoutcentering) to the normal law: this means that for a suitable norming sequencea_n we have the weak convergence S_n / a_n --> f(x) dx, where f(x) is thestandard normal density (this happens in particular by the CLT when X_1 haszero mean and finite variance \sigma^2, with a_n = \sigma \sqrt{n}). A localrefinement of this convergence is provided by Gnedenko's and Stone's LocalLimit Theorems, in the lattice and nonlattice case respectively.  Now let C_n denote the event (S_1 > 0, ..., S_n > 0) and let S_n^+ denote therandom variable S_n conditioned on C_n: it is known that S_n^+ / a_n --> f^+(x)\dd x, where f^+(x) := x \exp(-x^2/2) \ind_{x > 0}. What we establish in thispaper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for thisweak convergence. We also consider the particular case when X_1 has anabsolutely continuous law: in this case the uniform convergence of the densityof S_n^+ / a_n towards f^+(x) holds under a standard additional hypothesis, inanalogy to the classical case. We finally discuss an application of our mainresults to the asymptotic behavior of the joint renewal measure of the laddervariables process. Unlike the classical proofs of the LLT, we make no use ofcharacteristic functions: our techniques are rather taken from the so--calledFluctuation Theory for random walks.Comment: 20 pages; new version of the paper in a more general settin

A local limit theorem for random walks conditioned to stay positive