The Second-order Reglar Variation Of Randomly Weighted Sums And Order Statistics

Second-order regular variation （2RV） is a refinement of the concept of regular variation （RV） which appears in a natural way in applied probability, statistics, risk management, telecommunication networks, and other fields.2RV provides a nice theoretical platform for studying the rates of convergence to extreme value and stable distributions, characterizing asymptotic normality of Hill’s estimator, and establishing the second-order asymptotics of risk measures and their risk con-centrations based on quantiles of extreme risks.Randomly weighted sum Σ_i=1¹ï¼ˆ?）iXi appears naturally in insurance, finance and risk management. A well-known example in ruin theory interprets the weights as discount factors and{Xi,1≤i≤n} as the net losses of an insurance company. In term of a portfolio consisting of n obligors that are subject to possible default over a period, Xi can be interpreted as the loss given default of obligor i and （?）i as the default indicator for obligor i which is a Bernoulli random variable.Due to the important role of the heavy-tailed distributions, several studies were conducted to obtain the first-order asymptotics of the tail probability of Σ_i=1ⁿ（?）iXi under different assumptions on{X1,..., Xn} and{（?）1,...,（?）n}-The first part of this thesis is to investigate the closure property of2RV under randomly weighted sums. We first prove that the sum Σ_i=1ⁿ（?） Xi preserves the2RV property, where X1,..., Xn are independent random variables with2RV survival functions, which may have different second order parameters. Then, the closure property of2RV under randomly weighted sums Σ_i=1ⁿ（?）iXi is considered, where X1,..., Xn are assumed to be iid with2RV survival function, and （?）1,...,（?）n are independent and nonnegative random variables, independent of X1,..., Xn and satisfying cer-tain moment condition. Finally, we generalize the above for n=2to dependence case. It is shown that randomly weighted sum （?）1X1+(?02X2preserves the2RV, where X1and X2are independent, and （（?）1,（?）2） is independent of（X1,X2）, but no assumption is made on the dependence structure of （（?）1,（?）2）.In the second part of the thesis, we consider the closure property of2RV for order statistics and the spacings of two order statistics. If the samples are nonnegative iid2RV random variables, it is shown that both the order statistics and the difference of two order statistics preserve the2RV property.The main results in this thesis strengthen and complement some results in Barbe and McCormick （2005）, Mao and Hu （2012）, Geluk et al.（1997）, and Lv et al.（2012）.