Some Inequalities Of Martingale

Martingale is one of the most important fundamental tools in probability theory and statistics for modeling and studying sequences of random variables. Taking advantage of martingale’s properties we can solve many problems. So it is widely used in mathematica and attracted many scholars attention.In this thesis, we mainly study some inequalities of martingale, then two problems are investigated, which are respectively discussed in section 2 and section 3.In Section 1, the research background and problems of exponential inequality and PAC-Bayesian inequalities for martingales in this paper are simply presented, what’s more, we will give some classic results.In Section 2, firstly, we give some important definitions and theorems. Secondly, we not only establish Two-side exponential inequalities for martingales in the spirit of original work of B. Bercu and A. Touati [3], but also obtain an exponential upper bound for the tail probability for self-normalized martingales in the work of De la Pena and Pang[8]. In the end, we consider the application and establish deviation inequality of the least-squares estimate of the unknown paperter in linear regressive model.In Section 3, we mainly inwestigate PAC-Bayesian inequalities and its application. On the one hand, we establish some PAC-Bayesian inequalities for a sequence of ran-dom variables, which include conditionally symmetric random variables and martingales. Owing to PAC-Bayesian analysis can be extended to martingale, a combination of PAC-Bayesian analysis with inequality was applied in our life to slove problems. Therefore, for the particular random variables, making use of the transformation of measure inequality we can derive the better results. On the other hand, we establish the PAC-Bayesian in-equalities for the bounded martingale difference sequencewhich controls the concentration of weighted averages of multiple (possibly uncountably many) simultaneously evolving and interdependent martingales. The results presented here are both tighter and more gen- eral, so it can widely used in mew domains.