THE RELATIONS BETWEEN MODES OF CONVERGENCE FOR SEQUENCES OF RANDOM VARIABLES.

In this paper, we are going to analyze the relations between different types of convergence of a random sequence, such as almost sure convergence, convergence in mean square, convergence in distribution and convergence in probability. The convergence in distributions says nothing about the relationship between the random variables X_n and X and X, while for convergence in probability, the joint distribution of X_n and X is relevant. In the main part of the paper, we are going to prove the theorem which argues that the convergence in probability implies convergence in distribution, and the opposite is not true. But if ..., where c is a constant, then ... which mean that convergence in probability to a constant is equivalent to convergence in distributions. Also, we give some interesting examples. [ABSTRACT FROM AUTHOR]