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Tamás Szabados

Publications and source records attributed to Tamás Szabados.

10 recordsLinked to original sources
arXiv2023

Factorization of a spectral density with smooth eigenvalues of a multidimensional stationary time series

The aim of this paper to give a multidimensional version of the classical one-dimensional case of smooth spectral density. A smooth spectral density gives an explicit method to factorize the spectral density and compute the constituents of the Wold representation of a regular weakly stationary time series. These constituents are important to give the best linear predictions of the time series.

math.ST↗
arXiv2022

Regular multidimensional stationary time series

The aim of this paper is to give a simpler, more usable sufficient condition to the regularity of generic weakly stationary time series. Also, this condition is used to show how regular processes satisfying these sufficient conditions can be approximated by a lower rank \emph{regular} process. The relevance of these issues is shown by the ever increasing presence of high-dimensional data in many fields lately, and because of this, low rank processes and low rank approximations are becoming more important. Moreover, regular processes are the ones which are completely influenced by random innovations, so they are primary targets both in the theory and applications.

math.ST↗
arXiv2018

MiStImm: a simulation tool to compare classical nonsef-centered immune models with a novel self-centered model

Our main purpose is to compare classical nonself-centered, two-signal theoretical models of the adaptive immune system with a novel, self-centered, one-signal model developed by our research group. Our model hypothesizes that the immune system of a fetus is capable learning the limited set of self antigens but unable to prepare itself for the unlimited variety of nonself antigens. We have built a computational model that simulates the development of the adaptive immune system. For simplicity, we concentrated on humoral immunity and its major components: T cells, B cells, antibodies, interleukins, non-immune self cells, and foreign antigens. Our model is a microscopic one, similar to the interacting particle models of statistical physics and agent-based models in immunology. Furthermore, our model is stochastic: events are considered random and modeled by a continuous time, finite state Markov process, that is, they are controlled by finitely many independent exponential clocks. We investigate under what conditions can an immune memory be created that results in a more effective immune response to a repeated infection. The simulations show that our self-centered model is realistic. Moreover, in case of a primary adaptive immune reaction, it can destroy infections more efficiently than a classical nonself-centered model. Predictions of our theoretical model were clinically supported by autoimmune-related adverse events in high-dose immune checkpoint inhibitor immunotherapy trials and also by safe and successful low-dose immune checkpoint inhibitor combination treatment of heavily pretreated stage IV cancer patients who had exhausted all conventional treatments. The MiStImm simulation tool and source codes are available at the address https://github.com/kerepesi/MiStImm.

q-bio.MN↗
arXiv2016

A simple wide range approximation of symmetric binomial distributions

The paper gives a wide range, uniform, local approximation of symmetric binomial distribution. The result clearly shows how one has to modify the the classical de Moivre--Laplace normal approximation in order to give an estimate at the tail as well to minimize the relative error.

math.PR↗
arXiv2015

A stochastic model of B cell affinity maturation and a network model of immune memory

Many events in the vertebrate immune system are influenced by some element of chance. The objective of the present work is to describe affinity maturation of B lymphocytes (in which random events are perhaps the most characteristic), and to study a possible network model of immune memory. In our model stochastic processes govern all events. A major novelty of this approach is that it permits studying random variations in the immune process. Four basic components are simulated in the model: non-immune self cells, nonself cells (pathogens), B lymphocytes, and bone marrow cells that produce naive B lymphocytes. A point in a generalized shape space plus the size of the corresponding population represents nonself and non-immune self cells. On the other hand, each individual B cell is represented by a disc that models its recognition region in the shape space. Infection is simulated by an "injection" of nonself cells into the system. Division of pathogens may instigate an attack of naive B cells, which in turn may induce clonal proliferation and hypermutation in the attacking B cells, and which eventually may slow down and stop the exponential growth of pathogens. Affinity maturation of newly produced B cells becomes expressed as a result of selection when the number of pathogens decreases. Under favorable conditions, the expanded primary B cell clones may stimulate the expansion of secondary B cell clones carrying complementary receptors to the stimulating B cells. Like in a hall of mirrors, the image of pathogens in the primary B cell clones then will be reflected in secondary B cell clones. This "ping-pong" game may survive for a long time even in the absence of the pathogen, creating a local network memory. This memory ensures that repeated infection by the same pathogen will be eliminated more efficiently.

q-bio.MN↗
arXiv2014

Strong approximation of Black--Scholes theory based on simple random walks

A basic model in financial mathematics was introduced by Black, Scholes and Merton in 1973 (BSM model). A classical discrete approximation in distribution is the binomial model given by Cox, Ross and Rubinstein in 1979 (CRR model). The BSM and the CRR models have been used for example to price European call and put options. Our aim in this work is to give a strong (almost sure, pathwise) discrete approximation of the BSM model using a suitable nested sequence of simple, symmetric random walks. The approximation extends to the stock price process, the value process, the replicating portfolio, and the greeks. An important tool in the approximation is a discrete version of the Feynman--Kac formula as well. It is hoped that such a discrete pathwise approximation can be useful for example when teaching students whose mathematical background is limited, e.g. does not contain measure theory or stochastic analysis.

math.PR↗
arXiv2014

Kolmogorov complexity and strong approximation of Brownian motion

Brownian motion and scaled and interpolated simple random walk can be jointly embedded in a probability space in such a way that almost surely the $n$-step walk is within a uniform distance $O(n^{-1/2}\log n)$ of the Brownian path for all but finitely many positive integers $n$. Almost surely this $n$-step walk will be incompressible in the sense of Kolmogorov complexity, and all {Martin-Löf random} paths of Brownian motion have such an incompressible close approximant. This strengthens a result of Asarin, who obtained the bound $O(n^{-1/6} \log n)$. The result cannot be improved to $o(n^{-1/2}{\sqrt{\log n}})$.

math.LO↗
arXiv2011

Self-intersection local time of planar Brownian motion based on a strong approximation by random walks

The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka--Rosen--Yor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-intersection local time is more transparent and elementary than other existing ones.

math.PR↗
arXiv2009

Stochastic integration based on simple, symmetric random walks

A new approach to stochastic integration is described, which is based on an a.s. pathwise approximation of the integrator by simple, symmetric random walks. Hopefully, this method is didactically more advantageous, more transparent, and technically less demanding than other existing ones. In a large part of the theory one has a.s. uniform convergence on compacts. In particular, it gives a.s. convergence for the stochastic integral of a finite variation function of the integrator, which is not càdlàg in general.

math.PR↗