Expectations and convergence of sequences of random variables

In the third chapter asymptotic properties of sequences of random variables are studied. Lemma of Fatou and the Lebesgue dominated convergence theorem are presented as permanent technical tools of stochastic analysis. It is also emphasized the role of a uniform integrability condition of families of random variables. Classical probabilistic inequalities of Chebyshev, Jensen and Cauchy-Schwartz are proved. It is shown how these inequalities work to investigate interconnections between different types of convergence of sequences of random variables. In particular, the large numbers law (LNL) is derived for the case of independent identically distributed random variables (see [1], [7], [10], [15], [19], [40], and [45]).