Large deviation results for random walks conditioned to stay positive

Let X1,X2,... denote independent, identically distributed random variables with common distribution F, and S the corresponding random walk with ρ:=lim[subscript: n→∞]P( S[subscript: n] > 0 ) and τ :=inf{n≥1:S[subscript: n]≤0}. We assume that X is in the domain of attraction of an α-stable law, and that P(X∈[x,x+Δ)) is regularly varying at infinity, for fixed Δ>0. Under these conditions, we find an estimate for P(S[subscript: n]∈[x,x+Δ)|τ>n), which holds uniformly as x/cn→∞, for a specified norming sequence c[subscript: n]. This result is of particular interest as it is related to the bivariate ladder height process ((T[subscript: n],H[subscript: n]),n≥0), where T[subscript: r] is the rth strict increasing ladder time, and H[subscript: r]=ST[subscript: r] the corresponding ladder height. The bivariate renewal mass function g(n,dx)=∑∞[subscript: r=0]P(T[subscript: r]=n,H[subscript: r]∈dx) can then be written as g(n,dx)=P(S[subscript: n]∈dx|τ>n)P(τ>n), and since the behaviour of P(τ>n) is known for asymptotically stable random walks, our results can be rephrased as large deviation estimates of g(n,[x,x+Δ)).