Mathematics & Physics 2015, 8(2), 125–133 УДК 519.24 A Class of Special Empirical Processes of Independence Abdurahim A.Abdushukurov∗ Leyla R.Kakadjanova† Dpt. <...> Probability Theory and Mathematical Statistics National University of Uzbekistan VUZ Gorodok, Tashkent, 100174 Uzbekistan Received 12.12.2014, received in revised form 13.02.2015, accepted 12.03.2015 In this paper we investigate the asymptotic properties of one class of empirical processes for certain classes of integrable functions. <...> Introduction In this paper we investigate the limit properties of a class of empirical processes of independence indexed on a set of measurable functions. <...> The necessity of considering such processes stems from practical situations where we are interested in joint properties of pairs consisting of random variables (r.v.-s) and events. <...> Let us consider the following sequence of experiments in which observed pairs are consisted of {(Xk,Ak) , k 1}, where Xk are random elements defined on a probability space (Ω,A,P) with values in a measurable space (X,B). <...> Each pair in the sample S(n) induces a statistical model with the sample space X⊗{0, 1}, sigma-algebra of sets of the form BЧD and induces distribution Q∗ (B ЧD) = P(Xk ∈ B, δk ∈ D), where B ∈ B, D ⊂ {0, 1}. <...> All rights reserved c – 125 – n k=1 I (Xk ∈ B) = Q0n (B)+Q1n (B) . n k=1 (1−δk)I (Xk ∈ B) , n k=1 δkI (Xk ∈ B) , Abdurahim A.Abdushukurov, Leyla R.Kakadjanova A Class of Special Empirical of Independence estimate for p is pn = Q1n (X) = 1 n processes of independence {Λn (B)−Λ(B)} for a certain class G sets of B. In this paper we consider general classes of specially normalized empirical processes of independence indexed by under validity of H, Λn (B) a.s. a class of measurable functions. 1. <...> Empirical processes of independence Suppose that F be a set of measurable functions f : X →R. For the signed measure G and function f ∈ F we define the integral Gf = f dG. <...> According to the SLLN and the central limit <...>