Probability Theory pp 129-151 | Cite as

# On Convergence of Random Variables and Distributions

## Abstract

In this chapter, several different types of convergence used in Probability Theory are defined and relationships between them are elucidated. Section 6.1 deals with convergence in probability and convergence with probability one (the almost sure convergence), presenting some criteria for them and, in particular, discussing the concept of Cauchy sequences (in probability and almost surely). Then the continuity theorem is established (convergence of functions of random variables) and the concept of uniform integrability is introduced and discussed, together with its consequences (in particular, for convergence in mean of suitable orders). Section 6.2 contains an extensive discussion of weak convergence of distributions. The chapter ends with Sect. 6.3 presenting criteria for weak convergence of distributions, including the concept of distribution determining classes of functions and that of tightness.

## References

- 1.Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968) Google Scholar
- 14.Gikhman, I.I., Skorokhod, A.V.: Introduction to the Theory of Random Processes. Saunders, Philadelphia (1969) Google Scholar
- 25.Lamperti, J.: Probability: A Survey of the Mathematical Theory. Wiley, New York (1996) Google Scholar
- 31.Shiryaev, A.N.: Probability. Springer, New York (1984) Google Scholar