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 Baldi, An introduction through theory and exercises, 2017; Çinlar, Probability and stochastics, 2011; Durrett, Essentials of stochastic processes, 2018; Jacod and Protter, Probability essentials, 2003; Kolmogorov, Foundations of the theory of probability, 1956; Shiryaev, Probability, 1996, and Williams, Probability and martingales, 1991).