Some Inequalities In Probability Theory

Inequalities are always the active field in the 20th century, since the importance in sovling many problems. Please refer to Hardy, Littlewood and Polya's classic book "In-equalities". Say nothing of the Probability and Mathematical Statistics, some inequalities are frequently used, such as Chebyshev inequality, Kolmogorov inequality, Talagrand in-equality, Azuma martingale inequality and logarithmic Sobolev inequality and so on Under this framework, we study some probability inequalities in this thesis.In chapter 2, applying the technique of large deviations, we give a simple proof of the classic Stirling formula. As the extension from independence to dependence, we first consider the p-uniform ergodic Markov chains in chapter 3, and obtain a moment inequality for the tail probability of the additive functional of this Markov chains by the technique of Poisson equation and the martingale method. In the last chapter, we consider a self-similar process:mixed fractional Brownian motion (no increment independence and no Markov property), and prove a maximum inequality that is similar to the B-D-G inequality of Brownian motion.