The Hajek-Renyi Inequality For Ρ-mixing Sequences And Conditional VaR Estimation Based On Threshold Quantile Regression Model

Probability limit theory is one of the main branches of probability theory, the strong law of large numbers and convergence of the sequence of random variables are its center research topics. In order to get a better nature, the usual method is that applying probability inequalities to prove. For example, many scholars study the Hajek-Renyi type inequality and its application, martingale inequality and its applications, Doob inequality and so on. Inequality plays a very important role in the studying of probability limit theory.With the continuous development of the financial markets, investors are faced with more and more complex risks, how to measure of the risk accuratly placed in front of the investors, The value at risk is referred to VaR, the traditional VaR theory generally assumed that the characteristics of statistical distribution of certain rate of return were stable basically within a certain period, Then studying the VaR. However, market conditions are changing at moments, then the traditional VaR theory will be limited, and in the management of financial risks, we should consider the relationship between economic variables and the interaction between the various risk factors. Risk of conditions under the given conditions can solve the above problems to some extent.In the first part of this article, we first introduces Hajek-Renyi type inequality and its promotion and application, then we lead to the the conditional VaR and threshold quantile regression model form traditional risk theory.In the second part, we promote the Hajek-Renyi type inequality for independent sequence proved by Hajek and Renyi in1955to the p mixing sequence, then study the strong law of large numbers by applying Hajek-Renyi type inequality for p mixing sequences.In the third part, at first, we introduces the conditional VaR; then describes quantile regression method, because of the shortcoming of linear quantile regression model, we put forward the threshold quantile regression model, and get conditional VaR estimates by applying the model. By the analysis, the result obtained based on that model can describe the practical market better, and forecast the market risk better.