Transition phenomena for ladder epochs of random walks with small negative drift

2009/05/08 by Wachtel, Vitali
#60G50 #60G52 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0905.1186

Abstract

For a family of random walks \S^(a)\ satisfying ES₁^(a)=-a<0 we consider ladder epochs τ^(a)=min\k≥1: S_k^(a)<0\. We study the asymptotic, as a→0, behaviour of P(τ^(a)>n) in the case when n=n(a)→∞. As a consequence we obtain also the growth rates of the moments of τ^(a).

Related

• Local limit theorems for ladder epochs
• Ladder epochs and ladder chain of a Markov random walk with discrete driving chain
• Local probabilities for random walks conditioned to stay positive
• Small drift limit theorems for random walks
• Exact asymptotics for the instant of crossing a curve boundary by an asymptotically stable random walk