  We consider a real random walk S_n = X_1 + ... + X_n attracted (without
centering) to the normal law: this means that for a suitable norming sequence
a_n we have the weak convergence S_n / a_n --> f(x) dx, where f(x) is the
standard normal density (this happens in particular by the CLT when X_1 has
zero mean and finite variance \sigma^2, with a_n = \sigma \sqrt{n}). A local
refinement of this convergence is provided by Gnedenko's and Stone's Local
Limit 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 the
random 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 this
paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this
weak convergence. We also consider the particular case when X_1 has an
absolutely continuous law: in this case the uniform convergence of the density
of S_n^+ / a_n towards f^+(x) holds under a standard additional hypothesis, in
analogy to the classical case. We finally discuss an application of our main
results to the asymptotic behavior of the joint renewal measure of the ladder
variables process. Unlike the classical proofs of the LLT, we make no use of
characteristic functions: our techniques are rather taken from the so--called
Fluctuation Theory for random walks.
