The Conditions Of Asymptotic Equivalence For The Solutions Of Defective Renewal Equations

It is well known that in various of applications, renewal equations can be established, then by solving the equations we can resolve our concern problems, such as risk theory, queueing theory and branching theory, and so on. Although the solution of renewal function exists and be unique, but it is not easy to calculate, so attention is paid to asymptotics and local asymptotics of the solution. Previous results, such as Asmussen et al.(2003), Cui et al.(2009), mostly investigate asymptotics for the solution of one renewal equation, Psarrakos(2009) investigate the asymptotic behavior of two special function that satisfy defective renewal equations, and apply the results to the risk theory, but, the results of Psarrakos(2009) do not hold, therefore, the conditions and conclusions need to be discussed again. This paper will modify, improve and generalize the results in Psarrakos(2009), for this reason, we first briefly introduce the renewal equation and recall the existing results in the first chapter; in the second chapter, we give some equivalent conditions and sufficient conditions for solution asymptotic equivalent to each other among some defective renewal functions. In these results, the related functions are able to be light-tailed, also can be heavy-tailed, and are not be required to be sub-exponential function; in the third chapter, we give some applications in risk theory, such as establishing an asymptotic equivalent relation between the ruin probability with some Gerber's functions.