Study On Distribution Of Random Sequence Sums With Power Law And Its Applications

All kinds of power-law distribution are common phenomena existing in nature and social economy, thus study on them has broad and far-reaching significance. In this paper, we present a new method to approximate the distribution of random sequence sums with power law, which is called a double approximation method, and give its proof and application.The double approximation method is primarily based on the independence of conditional order statistics and Extreme Value Theory. Specifically, the random sequence sums with power law is first rewritten as the sums of the order statistics, and then separate it into three terms. The first one meet the Central Limit Theorem which can be approximated as a normal distribution. And the third one is the sums of limited upper order statistics, including the information about the heavy tail, which we can get its approximating distribution via the Extreme Value Theory. The second one only has one random sequence and play a separate role to make sure another two terms conditional independent. In this thought, we detailed derivate the approximate distribution function of the random sequence sums with power law and apply the method to financial risk measures. By comparing it with traditional methods, we find the error is less than 0.77% and most less than 0.3% when evaluating the Va R. Besides, the results only weekly dependent upon the sample size. Moreover, the estimation of the CVa R with the method is also accurate. Thus, it is convenient, practical and accurate to compute the distribution function of random sequence sums by using this method.