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István Fazekas

Publications and source records attributed to István Fazekas.

11 recordsLinked to original sources
arXiv2026

Strong law of large numbers for $φ$-sub-Gaussian random variables under sub-linear expectation spaces

We introduce the notions of sub Gaussian random variables in sub-linear expectation spaces. To avoid the problem caused by the existence of two different expectations, i.e., the upper expectation and the lower expectation, we divide the definition of the sub-Gaussian property into an upper part and a lower part. It turns out that this approach fits well to the sub-linear setting; it provides a proper framework for extending Zajkowski's general result to sublinear expectation spaces. Within our framework, we establish a strong law of large numbers for sub-Gaussian sequences. We present an example showing the usefulness of our results.

math.PR↗
arXiv2023

Limit theorems for runs containing two types of contaminations. Paper with detailed proofs

In this paper, sequences of trials having three outcomes are studied. The outcomes are labelled as success, failure of type I and failure of type II. A run is called at most 1+1 contaminated, if it contains at most 1 failure of type I and at most 1 failure of type II. The limiting distribution of the first hitting time and the accompanying distribution for the length of the longest at most 1+1 contaminated run are obtained. This paper contains the detailed mathematical proofs. Simulation results supporting the theorems are also presented.

math.PR↗
arXiv2023

Convergence rate for the longest T-contaminated runs of heads. Paper with detailed proofs

We study the length of $T$-contaminated runs of heads in the well-known coin tossing experiment. A $T$-contaminated run of heads is a sequence of consecutive heads interrupted by $T$ tails. For $T=1$ and $T=2$ we find the asymptotic distribution for the first hitting time of the $T$ contaminated run of heads having length $m$; furthermore, we obtain a limit theorem for the length of the longest $T$-contaminated head run. We prove that the rate of the approximation of our accompanying distribution for the length of the longest $T$-contaminated head run is considerably better than the previous ones. For the proof we use a powerful lemma by Csáki, Földes and Komlós.

math.PR↗
arXiv2023

Diffusion-based Data Augmentation for Skin Disease Classification: Impact Across Original Medical Datasets to Fully Synthetic Images

Despite continued advancement in recent years, deep neural networks still rely on large amounts of training data to avoid overfitting. However, labeled training data for real-world applications such as healthcare is limited and difficult to access given longstanding privacy, and strict data sharing policies. By manipulating image datasets in the pixel or feature space, existing data augmentation techniques represent one of the effective ways to improve the quantity and diversity of training data. Here, we look to advance augmentation techniques by building upon the emerging success of text-to-image diffusion probabilistic models in augmenting the training samples of our macroscopic skin disease dataset. We do so by enabling fine-grained control of the image generation process via input text prompts. We demonstrate that this generative data augmentation approach successfully maintains a similar classification accuracy of the visual classifier even when trained on a fully synthetic skin disease dataset. Similar to recent applications of generative models, our study suggests that diffusion models are indeed effective in generating high-quality skin images that do not sacrifice the classifier performance, and can improve the augmentation of training datasets after curation.

cs.LG↗
arXiv2020

Taylor's power law for the $N$-stars network evolution model

Taylor's power law states that the variance function decays as a power law. It is observed for population densities of species in ecology. For random networks another power law, that is, the power law degree distribution is widely studied. In this paper the original Taylor's power law is considered for random networks. A precise mathematical proof is presented that Taylor's power law is asymptotically true for the $N$-stars network evolution model.

math.PR↗
arXiv2019

The $N$-stars network evolution model

A new network evolution model is introduced in this paper. The model is based on co-operations of $N$ units. The units are the nodes of the network and the co-operations are indicated by directed links. At each evolution step $N$ units co-operate which formally means that they form a directed $N$-star subgraph. At each step either a new unit joins to the network and it co-operates with $N-1$ old units or $N$ old units co-operate. During the evolution both preferential attachment and uniform choice are applied. Asymptotic power law distributions are obtained both for the in-degrees and the out-degrees.

math.PR↗
arXiv2014

Weights of Cliques in a Random Graph Model Based on Three-Interactions

A random graph evolution rule is considered. The graph evolution is based on interactions of three vertices. The weight of a clique is the number of its interactions. The asymptotic behaviour of the weights is described. It is known that the weight distribution of the vertices is asymptotically a power law. Here it is proved that the weight distributions both of the edges and the triangles are also asymptotically power laws. The proofs are based on discrete time martingale methods. Some numerical results are also presented.

math.PR↗
arXiv2014

On a General Approach to the Strong Laws of Large Numbers

A general method to obtain strong laws of large numbers is studied. The method is based on abstract Hájek-Rényi type maximal inequalities. The rate of convergence in the law of large numbers is also considered. Some applications for weakly dependent sequences are given.

math.PR↗
arXiv2014

The asymptotic behaviour of the weights and the degrees in an N-interactions random graph model

A random graph evolution based on the interactions of N vertices is studied. During the evolution both the preferential attachment method and the uniform choice of vertices are allowed. The weight of a vertex means the number of its interactions. The asymptotic behaviour of the weight and the degree of a fixed vertex, moreover the limit of the maximal weight and the maximal degree are described. The proofs are based on martingale methods.

math.PR↗
arXiv2013

Scale-free property for degrees and weights in an N-interactions random graph model

A general random graph evolution mechanism is defined. The evolution is a combination of the preferential attachment model and the interaction of N vertices (N>=3). A vertex in the graph is characterized by its degree and its weight. The weight of a given vertex is the number of the interactions of the vertex. The asymptotic behaviour of the graph is studied. Scale-free properties both for the degrees and the weights are proved. It turns out that any exponent in (2,\infty) can be achieved. The proofs are based on discrete time martingale theory.

math.PR↗