Convergence Of Empirical Process Research Articles

Special empirical processes of independence indexed by classes of measurable functions

When analyzing statistical data in biomedical research, insurance, demography, as well as in other areas of practical research, random variables of interest take on certain values depending on the occurrence of certain events, e.g., when testing physical systems (individuals), the values of their operating time depend on the failures of subsystems; in the insurance business, the payments of insurance companies to their customers depends on insured events. In such experimental situations, the problems of studying the dependence of random variables on the corresponding events become rather important thus entailing the necessity to study the limiting properties of empirical processes indexed by classes of functions. The modern theory of empirical processes generalizes the classical results of the laws of large numbers, central and other limit theorems uniformly over the entire class of indexing under the imposition of entropy conditions. These theorems are generalized analogs of the classical theorems of Glivenko – Cantelli and Donsker. Special empirical processes are proposed in the study to check the independence of a random variable and an event. The properties of convergence of empirical processes to the corresponding Gaussian processes are analyzed. The results obtained are used to test the validity of the random right censoring model.

Stable Limit Theorems for Empirical Processes under Conditional Neighborhood Dependence

This paper introduces a new concept of stochastic dependence among many random variables which we call conditional neighborhood dependence (CND). Suppose that there are a set of random variables and a set of sigma algebras where both sets are indexed by the same set endowed with a neighborhood system. When the set of random variables satisfies CND, any two non-adjacent sets of random variables are conditionally independent given sigma algebras having indices in one of the two sets’ neighborhood. Random variables with CND include those with conditional dependency graphs and a class of Markov random fields with a global Markov property. The CND property is useful for modeling cross-sectional dependence governed by a complex, large network. This paper provides two main results. The first result is a stable central limit theorem for a sum of random variables with CND. The second result is a Donsker-type result of stable convergence of empirical processes indexed by a class of functions satisfying a certain bracketing entropy condition when the random variables satisfy CND. When there are high-degree vertices, they potentially hamper normal approximation by causing widespread dependence among the random variables. Thus we generalize the results by approximating the sum of random variables by the sum conditioned on high-degree vertices so that stable limit results continue to hold even when the maximum degree of the neighborhood system diverges to infinity as the size of the system grows.

Entropy conditions for L r -convergence of empirical processes

The law of large numbers (LLN) over classes of functions is a classical topic of empirical processes theory. The properties characterizing classes of functions on which the LLN holds uniformly (i.e. Glivenko-Cantelli classes) have been widely studied in the literature. An elegant sufficient condition for such a property is finiteness of the Koltchinskii-Pollard entropy integral, and other conditions have been formulated in terms of suitable combinatorial complexities (e.g. the Vapnik- Chervonenkis dimension). In this paper, we endow the class of functions F with a probability measure and consider the LLN relative to the associated Lr metric. This framework extends the case of uniform convergence over F , which is recovered when r goes to infinity. The main result is a Lr-LLN in terms of a suitable uniform entropy integral which generalizes the Koltchinskii-Pollard entropy integral.

Convergence of Empirical Processes for Interacting Particle Systems with Applications to Nonlinear Filtering

In this paper, we investigate the convergence of empirical processes for a class of interacting particle numerical schemes arising in biology, genetic algorithms and advanced signal processing. The Glivenko–Cantelli and Donsker theorems presented in this work extend the corresponding statements in the classical theory and apply to a class of genetic type particle numerical schemes of the nonlinear filtering equation.

Weak Convergence of Empirical Processes of Dependent Random Vectors Under Logarithmic Mixing Rates

It is proved that empirical processes of ø-mixing or Ψ-mlxing random vectors conv11rge to appropriate Gaussian processes under logarithmic mixing rates. A Kiefer-type bound on the tail-probabilities of empirical processes is also obtained under the same mixing rates.

Glivenko-Cantelli type theorems: An application of the convergence theory of stochastic suprema

The uniform convergence of empirical processes on certain classes of sets follows from the convergence theory for random lower semicontinuous functions studies in the context of stochastic optimization. In the process, a richer class of sets for which one can prove this type of result is exhibited.

A Note on Weak Convergence of Empirical Processes Based on Mixing Random Variables Subject to Perturbation and Scale Factors

In this note, we prove the weak convergence (WC) of empirical processes to a Gaussian process for a sequence of nonstationary strong mixing random variables subject to perturbations and scale factors under some regularity conditions on the mixing coefficient and the sequence of marginal distribution functions

Extensions of Billingsley's theorems on weak convergence of empirical processes

In [4], Sen proved Theorems 22.1 and 22.2 of Billingsley [1] on the weak convergence of empirical distribution functions for sequences of Φ-mixing random variables to appropriate Gaussian random functions under less stringent regularity conditions. In this note, we shall prove the same theorems under weaker conditions than Sen's ones.

On weak convergence of empirical processes for random number of independent stochastic vectors

AbstractBy the use of a semi-martingale property of the Kolmogorov supremum, the results of Pyke (6) on the weak convergence of the empirical process with random sample size are simplified and extended to the case of p(≥1)-dimensional stochastic vectors.