Asymptotic Theories Of Random Sums And Their Applications

This paper mainly focuses on the asymptotic theories of random sums and theirapplications in random walks and finance and insurance. Firstly, assuming that thesupport of the distribution of the summands is supported on the real line, and divid-ing the random sums into heavy-tailed case and light-tailed case, we investigate theasymptotic behaviour of the random sums, and obtain the lower bound and upperbound for the limits of the ratio of the tail distribution of the random sums and that ofthe summands, moreover, we illustrate that these results are optimal in certain sense.Similarly, we investigate the asymptotic behaviour of the local distribution and thedensity of the random sums. We obtain upper bound for the lower limits and lowerbound for the upper limits of the ratio of the local distribution of the random sums andthat of the summands, as well as upper bounds for the lower lmits and lower boundfor the upper limits of the ratio of the density of the random sums and that of thesummands. These results enrich the existing asymptotic theories of random sums, andclarify some related problems in distribution theories, and to a great extent solve aproblem which was put forward by Watanabe (2008)[78].Secondly, in classical results, the conclusions that the tail distribution of the radomsums has asymptotic expressions, the distribution of the summands is convolutionequivalent, as well as the the distribution of the random sums is convolution equivalentare equivalent under certain conditions. However, in theories and applications, thedistribution of the summands of the random sums does not necessarily belong to theconvolution equivalence distribution classes. So, to study the asympotic theory ofrandom sums whose summands have a distribution belonging to a larger class is ofinterest. The third chapter of this paper systematically studies the properties of theexponential classes and the generalized subexponential classes. In the bisis of theseproperties, assuming that the distribution of the summands belongs to the intersectionof the exponential classes and the generalized subexponential classes, we obtain lowerlimits and upper bound for the upper limits of the ratio of the tail distribution ofthe random sums and that of the summands. Similarly, we study the corresponding properties of local distributions and densities, and give the lower limits and upperbound for the upper limits of the ratio of the local distribution of the random sumsand that of the summands, as well as the lower limits and upper bound for the upperlimits of the ratio of the density of the random sums and that of the summands. Theseresults generalize the classic results.In the basis of the investigation to the asymptotic theories of the random sums, westudy the asymptotic theories for the supremum of the random walk, and get the lowerlimits and and upper bound for the upper limits of the ratio of the tail distributionof the supremum and the integrated tail distribution of the increments. These resultsenrich the classic theories of random walks. In the end, we apply these results into risktheory, and obtain the lower limits and upper bound for the upper limits of the ratioof the ruin probability and the integrated tail distribution of the claim sizes.