The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence

In the analysis of statistical data in biomedical treatments, engineering, insurance, demography, and also in other areas of practical researches, the random variables of interest take their possible values depending on the implementation of certain events. So in tests of physical systems (or individuals) on duration of uptime values of operating systems depend on subsystems failures, in insurance business insurance company payments to its customers depend on insurance claims. In such experimental situations, naturally become problems of studying the dependence of random variables on the corresponding events. The main task of statistics of such incomplete observations is estimating the distribution function or what is the same, the survival function of the tested objects. To date, there are numerous estimates of these characteristics or their functionals in various models of incomplete observations. In this paper investigated the asymptotic properties of sequential processes of independence of the integral structure and uniform versions of the strong law of large numbers and the central limit theorem for integral processes of independence by indexed classes are established. The obtained results can be used to construct statistics of criteria for testing a hypothesis of independence of random variables on the corresponding events.


Introduction
The modern asymptotic theory of empirical processes indexed by a class of measurable functions is actively developing and the current results are detailed in monographs [5,6,8,9,[12][13][14], also in articles [4,10,15]. The main results of this theory allow us to establish uniform versions of the laws of large numbers and central limit theorems for empirical measures under the imposition of entropy conditions for a class of measurable functions. These results are essentially generalized analogues of the classical theorems of Glivenko-Cantelli and Donsker. It should be noted the article [15], in which these results are established for a generalized class of random discrete measures under appropriate conditions for uniform entropy numbers. At the same time, such results can be used in applied problems. For example, to generalize Glivenko-Cantelli theorem for a certain class of sets Vapnik and Chervonenkis in 70-s years of the last century made a significant contribution to the development of statistical (machine) learning theory (theory-Vapnik Chervonenkis), which justifies the principle of minimizing empirical risk (for details, see the monograph [14]). In a recently published monography [9] Mason used the main results of the modern theory of empirical processes to study nonparametric the kernel type statistical estimates. In this paper also established properties of empirical processes, which appear in the problems of statistical data analysis.
In papers of the authors [1][2][3]7] the limiting properties of generalized empirical processes of independence of random variables and events indexed by a class of measurable functions were investigated. In this paper, we study the asymptotic properties (uniform strong laws of large numbers and central limit theorems) of sequential processes of independence of the integral structure. Consider Indicators of events are denoted by It is observed the repeated sample of size n : for each Borel set B represented through subdistributions: Our interest is focused on hypothesis H of independence of k X and k A in each experiment. It's easy to see that under validity of H : B , which equal to zero under hypothesis H . Using this measure, we construct an empirical process for testing a hypothesis H . In this regard, we introduce empirical analogues of the above measures by sample ( ) . Moreover, according to the strong law of large numbers, for each B ∈ B and n → ∞ : , with probability 1. From the theory of empirical processes it is known (see, for example, [5, 6, 8-10, 12, 13]), that such results do not occur uniformly by all elements of σ − algebra B and can be performed for a special class J of sets of B . Consequently, the investigation of the limiting (at n → ∞ ) properties of − J indexed processes of form with a possibly random sequence of non-negative normalizing numbers { , 1} n a n ≥ . It is interesting from the point of view of constructing of statistics for criteria to testing of the hypothesis H . In this paper investigation of papers [1][2][3], will be advanced and the following wider classes of sequences of empirical processes indexed by the set F of Borel functions : f → ℝ X will be considered: , .
Note that, the process (1) is a special case of (2), when F is a class of indicators following empirical processes of independence indexed by the class F were investigated: (see. [1,2]), and also generalized sequential Obviously, that for all f ∈ F . Note that, processes (3) and (4) are variants of (2) with a suitable choice of normalizing sequences n a . In papers [1][2][3]7] it was established the uniform (by corresponding indexing classes F and D ) variants of strong laws of large numbers (Glivenko-Cantelli type) and central limit theorems (Donsker type), respectively, for processes (3) and (4). In this paper, we will study another variant of the processes of the form (2) and for it uniform variants of the above limit theorems will be proved.

Information from the Theory of Metric Entropy
To prove the uniform variants of Glivenko-Cantelli and Donsker type theorems it necessary entropy properties of the class of measurable functions F . In this regard, we define the space . ε − brackets (or ε − balls) are denoted as [ , ] ϕ ψ . We say, that the function f ∈ F covered by bracket [ , ] ϕ ψ , if . Moreover, these Donsker Type Theorems for a Sequential Integral Process of Independence functions ϕ and ψ may not belong to the class F , but they must have finite norms. The smallest number is an important characteristic for determining complexity of the class F . Its logarithm called the metric entropy with bracketing of a class F and it allows to control the number of sets needed to cover F . Note that this number at 0 ε → tends to +∞ . Growth of metric entropy to +∞ controlled by its integral The convergence of this integral depends on the number ε − brackets (5). Since the integral r ≥ , therefore, metric entropy should not grow faster than 2 ε − . For example, for Donsker type theorems it necessary that the number (5) tends to +∞ not very fast (see details, for example, [5,12,13]). According to the theorem 2.7.5 in [12], for class F -of monotone functions Then F is a consistent strong Glivenko-Cantelli class, i.e. at n → ∞ almost surely The following theorem from [3] is a uniform variant of the central limit theorem for process (4).
The result of theorem 2 under hypothesis H is a generalized uniform analogue of the Donsker theorem, because from (8)

Sequential Integral Empirical Processes of Independence
Now introduce the following sequential .
Define the processes in a Banach space and it's easy to calculate For processes (13) taking into account the (14) and (15) Under validity of the hypothesis H from (17)   To the sequence (20), the uniform central limit theorem is holds.
Theorem 3. Let the conditions (6) are holds. Then at n → ∞ ( ) ( ) ( ) In general, the proof of theorem 4 repeats the proofs of theorems 2 in [2] and 3.1 in [3]. In this case, the vector field (23) is represented by the following sequence of sums of normalized independent and identically distributed random fields with a covariance structure coinciding with (24): According to the condition (6), [ ] ; } W t f t f ∈ N being a linear combination of two Gaussian fields with zero means and covariance structure, defined by formulas (24) is also a Gaussian field with zero mean. A direct calculation of the covariance of the limiting process in (27) shows, that it exactly coincides with the covariance (17) of the process (13). Consequently, (21) holds. In particular, under validity of the hypothesis H , the limiting process in (27)