A local limit theorem for random walks conditioned to stay positive
Consider a random walk \(S_{n}=X_{1}+\cdots +X_{n},\;n\geq 1,\) where \( X_{i},\;i\geq 1,\) are i.i.d. real-valued random variables. Assume that for suitable norming constants \(a_{n}\) the ratio \(S_{n}/a_{n}\;\)converges weakly to the standard normal distribution. Let \(\mathcal{C}_{n}\) denote the random event \((S_{1}>0,\dots ,S_{n}>0)\). The author's main result is that if \(X_{1}\) is nonlattice, then \[ a_{n}P( S_{n}\in [ x,\;x+h)\mid \mathcal{C}_{n}) =h\varphi ^{+}( x/a_{n}) +o(1) \] as \(n\rightarrow \infty \), where \(\varphi ^{+}\left( x\right) =xe^{-x^{2}/2}1_{\left( x\geq 0\right) }\,.\) The cases of a lattice or an absolutely continuous \(X_{1}\) are also considered, and an application to the asymptotic behaviour of the joint renewal measure of the ladder variables process is discussed in some detail. The author's approach is of a combinatorial nature and makes use of the random walk fluctuation theory.
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