It has been suggested that Radius of convergence pdf convergence in probability be merged into this article. In probability theory, there exist several different notions of convergence of random variables. Stochastic convergence” formalizes the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle into a pattern. That the variance of the random variable describing the next event grows smaller and smaller.
These other types of patterns that may arise are reflected in the different types of stochastic convergence that have been studied. While the above discussion has related to the convergence of a single series to a limiting value, the notion of the convergence of two series towards each other is also important, but this is easily handled by studying the sequence defined as either the difference or the ratio of the two series. This result is known as the weak law of large numbers. Other forms of convergence are important in other useful theorems, including the central limit theorem. The first few dice come out quite biased, due to imperfections in the production process. As the factory is improved, the dice become less and less loaded, and the outcomes from tossing a newly produced die will follow the uniform distribution more and more closely. With this mode of convergence, we increasingly expect to see the next outcome in a sequence of random experiments becoming better and better modeled by a given probability distribution.
Convergence in distribution is the weakest form of convergence, since it is implied by all other types of convergence mentioned in this article. In general, convergence in distribution does not imply that the sequence of corresponding probability density functions will also converge. However, according to Scheffé’s theorem, convergence of the probability density functions implies convergence in distribution. The portmanteau lemma provides several equivalent definitions of convergence in distribution. Although these definitions are less intuitive, they are used to prove a number of statistical theorems. A natural link to convergence in distribution is the Skorokhod’s representation theorem.
First, pick a random person in the street. Xn will converge in probability to the random variable X. Let Xn be his score in n-th shot. No matter how professional the archer becomes, there will always be a small probability of making an error. The concept of convergence in probability is used very often in statistics.