Grenzwertsätze

Grenzwertsätze, also known as limit theorems, are fundamental results in probability theory and statistics that describe the behavior of the sum or average of a large number of random variables. These theorems provide a bridge between the properties of individual random variables and the properties of their collective behavior, forming the bedrock of many statistical inferential methods.

The most well-known of these is the Law of Large Numbers. In its weak form, it states that the average of a large number of independent and identically distributed random variables will converge in probability to the expected value of those variables. The strong form of the Law of Large Numbers strengthens this by stating that the average will converge to the expected value almost surely. This means that in the long run, the observed average of an experiment is likely to be close to its theoretical average.

Another crucial set of theorems is the Central Limit Theorem. This theorem states that the distribution of the sum (or average) of a large number of independent and identically distributed random variables, regardless of their original distribution, will approximate a normal (Gaussian) distribution. This approximation becomes more accurate as the number of variables increases. The Central Limit Theorem is particularly powerful because it allows us to make inferences about population parameters using sample averages, even when the underlying population distribution is unknown. These theorems are essential tools for understanding randomness, making predictions, and analyzing data in various scientific and practical applications.