Doney, Ronald A.; Jones, Elinor Mair Large deviation results for random walks conditioned to stay positive. (English) Zbl 1252.60041 Electron. Commun. Probab. 17, Paper No. 38, 11 p. (2012). Summary: Let \(X_{1},X_{2},\ldots\) denote independent, identically distributed random variables with common distribution \(F\), and \(S\) the corresponding random walk with \(\rho :=\lim_{n\rightarrow \infty }\text{P}(S_{n}>0)\) and \(\tau :=\inf \{n\geq 1:S_{n}\leq 0\}\). We assume that \(X\) is in the domain of attraction of an \(\alpha \)-stable law, and that \(\text{P}(X\in [ x,x+\Delta))\) is regularly varying at infinity, for fixed \(\Delta >0\). Under these conditions, we find an estimate for \(\text{P}(S_{n}\in [ x,x+\Delta)|\tau >n)\), which holds uniformly as \(x/c_{n}\rightarrow \infty \), for a specified norming sequence \(c_{n}\). This result is of particular interest as it is related to the bivariate ladder height process \(((T_{n},H_{n}),n\geq 0)\), where \(T_{r}\) is the \(r\)th strict increasing ladder time, and \(H_{r}=S_{T_{r}}\) the corresponding ladder height. The bivariate renewal mass function \(g(n,dx)=\sum_{r=0}^{\infty }\text{P}(T_{r}=n,H_{r}\in dx)\) can then be written as \(g(n,dx)=\text{P}(S_{n}\in dx|\tau >n)\text{P}(\tau >n)\), and since the behaviour of \(\text{P}(\tau >n)\) is known for asymptotically stable random walks, our results can be rephrased as large deviation estimates of \(g(n,[x,x+\Delta))\). Cited in 4 Documents MSC: 60G50 Sums of independent random variables; random walks 60F10 Large deviations 60G52 Stable stochastic processes 60E07 Infinitely divisible distributions; stable distributions Keywords:limit theorems; random walks; stable laws × Cite Format Result Cite Review PDF Full Text: DOI