Local Asymptotics For Random Walks

It is well-known that the local asymptotics for random walks are extensively used in queues, risk theory and Bellman-Harris branch processes, so cause our interest. As-mussen, Foss and Korshunov (2003)~[1] put forward the concept of the class (?)_△ and S_△ and formed an integrated theoretical system. Naturally, we can put forward the concept of the class (?)_△(γ) and S_△(γ) γ≥ 0 and discuss their properties. In Chap 2 of this paper we discuss the local asymptotics for random walks, mainly the equivalent conditions of (?) _△(γ) and S_△(γ) and the local asymptotics and closure of convolutions of (?)_△(γ) and S_△(γ)- In Chap 3 we extend the properties about the subexponential density in [1] to a wider range, mainly discuss the closure and asymptotics of density convolutions.