István Fazekas

A General Approach to the Strong Law of Large Numbers

A general method for obtaining the strong law of large numbers for sequences of random variables is considered. Some applications for dependent summands are given.

Asymptotic properties of an estimator in nonlinear functional errors-in-variables models with dependent error terms

Nonlinear functional errors-in-variables models with error terms satisfying mixing conditions are studied. It is pointed out that under certain conditions the least-squares estimator of regression parameters is not consistent. An alternative estimator for regression parameters is proposed. The consistency of the alternative estimator is established.

Limit theorems for the empirical distribution function in the spatial case

Functional central limit theorems are proved for the empirical distribution function of strictly stationary and weakly dependent random fields. The theorems cover the discrete and the continuous parameter fields and the case when the observations become dense in a sequence of increasing domains.

Scale-Free Property for Degrees and Weights in a Preferential Attachment Random Graph Model

A random graph evolution mechanism is defined. The evolution studied is a combination of the preferential attachment model and the interaction of four vertices. The asymptotic behaviour of the graph is described. It is proved that the graph exhibits a power law degree distribution; in other words, it is scale-free. It turns out that any exponent in can be achieved. The proofs are based on martingale methods.

Asymptotic properties in space and time of an estimator in nonlinear functional errors-in-variables models

Nonlinear functional errors-in-variables models with error terms satisfying mixing conditions are studied. Both variables X and y are allowed to be vector valued. An estimator for regression parameters is studied. Consistency and asymptotic normality of the estimator are established both for temporal and for spatial observations in the case of increasing domain. Properties of the estimator are studied under infill asymptotics. Simulation results are also presented.

ALMOST SURE LIMIT THEOREMS FOR THE PEARSON STATISTIC

Almost sure versions of limit theorems by Kruglov for the Pearson

Infill Asymptotics Inside Increasing Domains for the Least Squares Estimator in Linear Models

\chi^2

Convergence of random step lines to Ornstein-Uhlenbeck-type processes

-statistic are obtained.

An analogue of the generalised allocation scheme: limit theorems for the number of cells containing a given number of particles

A linear model observed in a spatial domain is considered. Consistency and asymptotic normality of the least squares estimator is proved when the observations become dense in a sequence of increasing domains and the error terms are weakly dependent. Similar statements are obtained for the linear errors-in-variables model.

On the rosenthal inequality for mixing fields

The paper deals with random step-line processes defined by sums of independent identically distributed random variables multiplied by independent indicators. These processes describe some models in which random variables are replaced with other ones. We prove the convergence in distribution of such processes to the weighted Ornstein-Uhlenbeck process.