Limit Theorems In Random Structures

This dissertation consists of three parts.We establish some limit laws for the insurance, reinsurance,and random graphs models,respectively.In insurance and finance,the product Z=XY of two random variables X and Y is one of basic elements in stochastic modelling.There were many works about the tail behavior of the product under the assumption that the random variables are independent.However,this assumption is far too unrealistic for most applied problems.Therefore,it is more interesting and practical to study the case that X and Y are dependent.We assume that X and Y follow a generalized FGM distribution.We are interested in the question how to capture the impact of the dependence of X and Y in this model on the tail behavior of their product Z.We shall derive an explicit asymptotic formula for the tail probability of Z.In comparison to the asymptotic formula for the independent case,ours contains an extra factor representing the impact of the dependence of X and Y.We shall also investigate the asymptotic behavior of the tail probabilities of L_ι(t) and E_ι(t),which are reinsurance amounts in large claims reinsurance models LCR and ECOMOR, respectively.We will establish precise asymptotic estimates for the tail probabilities of L_ι(t) and E_ι(t),withιand t fixed.Our results show that,when F,the the distribution of claim size,is an exponential distribution,these tail probabilities are both asymptotic to a multiple of the tail of a gamma distribution with suitable parameters,while when F has a convolution-equivalent tail,they are both asymptotic to a multiple of the tail of F.The prefactors involved are completely explicit and transparent.At last,we will obtain some limit theorems in random graphs,for example,the maximal gap in division of one-sided interval trees,the degree sequences of Buckley-Osthus scale-free random graphs,and so on.