Research On A Sum Of Probability Inequalities

Inequalities have always played a very important role in the development of every field of mathematics, not excepting probability and statistics. Choosing or originating an appropriate inequality becomes the key to solving various problems. However, as the theories advanced and the appearance of new problems, many classical inqualities showed up their deficiencies. Therefore, lots of researchers kept extending and generalizing the original inqualities and established a series of new probability inqualities, which made a great contribution to the the development of probability and its practical application.When estimating the bound of probabilities, we would commonly utilize moment estimation methods, such as Markov's inequality, Chebyshev's inequality and Chernoff's inequality. While all these classical inequalities just estamite a single probability rather than a sum of probabilities which render those inqualities inaccurate. Then some research was carried out on a sum of probability inqualities, introducing a sum of Markov's inqualities and a sum of Chebyshev's inqualities. In this paper, we first give a review and summary of existing sum of probability inqualities. After that, Chernoff's inequality is extended to sums and the result with certain singed measures is verified.As Chernoff's inequality in the form of infimum can be obtained by imposing an infimum on the general form, we only concertrate on a sum of the general form here. Also, we give two different inqualities for t﹥0 and t﹤0. By defining a quantity involved with the coefficients of the sum and the interval endpoints, an estimator for the bound of the probability sum is given and some improvement on the result is made.In the proofs, we discuss particularly when the equality in the sum of Chernoff's inequality holds and point out the probability distribution for that, making the estimation more accurate. Finally, we extend the sum of to certain signed measures and explain that the general sum of Chernoff's inequality is just a special case when certain discrete measure is selected.