@article{discovery10086030, number = {38}, title = {Large deviation results for random walks conditioned to stay positive}, year = {2012}, volume = {17}, publisher = {Institute of Mathematical Statistics}, note = {This version is the version of record. For information on re-use, please refer to the publisher's terms and conditions.}, pages = {1--11}, journal = {Electronic Communications in Probability}, url = {http://dx.doi.org/10.1214/ecp.v17-2282}, author = {Doney, R and Jones, E}, abstract = {Let X1, X2, ... denote independent, identically distributed random variables with common distribution F, and S the corresponding random walk with {\ensuremath{\rho}} := limn{$\rightarrow$}{$\infty$} P(Sn {\ensuremath{>}} 0) and {\ensuremath{\tau}} := inf\{n {$\ge$} 1 : Sn {$\leq$} 0\}. We assume that X is in the domain of attraction of an {\ensuremath{\alpha}}-stable law, and that P(X {$\in$} [x, x + ?)) is regularly varying at infinity, for fixed ? {\ensuremath{>}} 0. Under these conditions, we find an estimate for P(Sn {$\in$} [x, x + ?){\ensuremath{|}}{\ensuremath{\tau}} {\ensuremath{>}} n), which holds uniformly as x/cn {$\rightarrow$} {$\infty$}, for a specified norming sequence cn. This result is of particular interest as it is related to the bivariate ladder height process ((Tn, Hn), n {$\ge$} 0), where Tr is the rth strict increasing ladder time, and Hr = STr the corresponding ladder height. The bivariate renewal mass function g(n, dx) = P{$\infty$} r=0 P(Tr = n, Hr {$\in$} dx) can then be written as g(n, dx) = P(Sn {$\in$} dx{\ensuremath{|}}{\ensuremath{\tau}} {\ensuremath{>}} n)P({\ensuremath{\tau}} {\ensuremath{>}} n), and since the behaviour of P({\ensuremath{\tau}} {\ensuremath{>}} n) is known for asymptotically stable random walks, our results can be rephrased as large deviation estimates of g(n, [x, x + ?)).}, keywords = {Limit theorems; Random walks; Stable laws} }