@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}
}