Local Precise Large Deviations And Local Asymptotic Behavior Of Random Sums

In this paper, we obtain some results on local precise large deviations of partial sums.We note that many results have been obtained based on some cases that the types of randomvariables are restricted. The applications of the existing results are greatly limited in thesecases. We do not distinguish the types of random variables in this paper, and our resultscontain the results of Doney(1989), Baltruˉnas(1996), Lin(2008), Yang et al.(2010) etc; for theabsolutely continuous random variables or integer-valued random variables, our conditionsare slightly weaker than that in the above results. Moreover, we also discuss the situationthat random variables have an infinite mean.On the other hand, by using the results of local precise large deviations of partial sums,we establish some local versions of inequalities, and get the local asymptotic tail behavior ofrandom sums with heavy-tailed random numbers. The results we get are similar to Theorem1of Denisov et al.(2010)(global version). In particular, if the random variables have an infinitemean, our results are also new in the global version.Furthermore, assuming a distribution F has a density function f, we discuss the rela-tionships between f∈OR and F (x+)∈OR, between f and F (x+) upper and lowerMatuszewska’s indices and so on, and give some properties.