ABSTRACT
Inequalities in probability theory are useful tools for estimating probabilities and moments of random variables if their exact calculation is only possible with extremely high effort or is even impossible in view of incomplete information on the underlying probability distribution. There are three large classes of limit theorems in probability theory: The laws of the large numbers, the central limit theorem and its numerous modifications, and the local limit theorems. The central limit theorem justifies the application of the normal distribution as distribution of random variables, which are known to arise by the additive superposition of numerous random influences. Limit theorems in probability theory are subject to certain convergence criteria for sequences of random variables, which next have to be introduced. The central limit theorem provides theoretical reasons for the significant role of the normal distribution in probability theory and its applications.
