This thesis consists of two papers devoted to the asymptotics of random matrix ensembles and measure valued stochastic processes which can be considered as generalizations of the Gaussian unitary ensemble (GUE) of Hermitian matrices H=A+A†, where the entries of A are independent identically distributed (iid) centered complex Gaussian random variables. In the first paper, a system of interacting diffusing particles on the real line is studied; special cases include the eigenvalue dynamics of matrix-valued Ornstein-Uhlenbeck processes (Dyson's Brownian motion). It is known that the empirical measure process converges weakly to a deterministic measure-valued function and that the appropriately rescaled fluctuations around this limit converge weakly to a Gaussian distribution-valued process. For a large class of analytic test functions, explicit formulae are derived for the mean and covariance functionals of this fluctuation process. The second paper concerns a family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of n x n matrices with iid centered complex Gaussian entries. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order n -1/3. / Denna avhandling består av två vetenskapliga artiklar som handlar om gränsvärdessatser för slumpmatriser och måttvärda stokastiska processer. De modeller som studeras kan betraktas som generaliseringar av den gaussiska unitära ensembeln (GUE) av hermiteska n x n-matriser H=A+A†, där A är en matris vars element är oberoende, likafördelade, centrerade, komplexa normalfördelade stokastiska variabler. I artikel I betraktas ett system av växelverkande diffunderande partiklar på reella linjen, vissa specialfall av denna modell kan tolkas som egenvärdesdynamiken för matrisvärda Ornstein-Uhlenbeck-processer (Dysons brownska rörelse). Sedan tidigare är det känt att den empiriska måttprocessen konvergerar svagt mot en deterministisk måttvärd funktion och att fluktuationerna runt denna gräns, i lämplig skalning, konvergerer svagt mot en distributionsvärd gaussisk process. För en stor klass av analytiska testfunktioner härleds explicita formler för medelvärdes- och kovariansfunktionalerna för denna fluktuationsprocess. Artikel II behandlar en familj av slumpmatrisensembler som interpolerar mellan GUE och Ginibre-ensembeln, bestående av matriser A som ovan. För denna modell är egenvärdena komplexa och asymptotiskt likformigt fördelade i en ellips i komplexa planet. Skalningsgränsvärdessatser för egenvärdet med maximal realdel och för egenvärdespunktprocessen kring detta visas för ett allmänt val av interpolationsparametern i modellen. Då förhållandet mellan axlarna i den asymptotiska ellipsen är av storleksordning n-1/3 uppträder en övergångsfas mellan Airypunktprocess- och Poissonprocessbeteendena, typiska för GUE respektive Ginibre-ensembeln. / QC 20100705

This PhD thesis consists of a summary and four papers which deal with stochastic approximation algorithms and first-passage percolation. Paper I deals with the a.s. limiting properties of bounded stochastic approximation algorithms in relation to the equilibrium points of the drift function. Applications are given to some generalized Pólya urn processes. Paper II continues the work of Paper I and investigates under what circumstances one gets asymptotic normality from a properly scaled algorithm. The algorithms are shown to converge in some other circumstances, although the limiting distribution is not identified. Paper III deals with the asymptotic speed of first-passage percolation on a graph called the ladder when the times associated to the edges are independent, exponentially distributed with the same intensity. Paper IV generalizes the work of Paper III in allowing more edges in the graph as well as not having all intensities equal.

Šiame magistriniame darbe yra nagrinėjami nepriklausomi vienodai pasiskirstę atsitiktiniai dydžiai, turintys visus absoliutinius baigtinius momentus. Magistrinio darbo tikslas - atlikti konvergavimo greičio į normalųjį dėsnį įvertinimą. Darbą sudaro aštuoni skyriai. Įvade aprašoma problema ir visi tyrimo parametrai. Antrasis skyrius skirtas teoriniai analizei. Šiame skyriuje pateikiamos svarbiausios teorinės žinios ir metodai, kurie bus taikomi magistrinio darbo uždaviniams bei tikslams įgyvendinti. Trečiame skyriuje nagrinėjami kumuliantai Bernulio schemos atveju, o ketvirtame - analizuojamas Čebyšovo asimptotinis skleidinys ir pasinaudojus matematiniu paketu Maple, grafiniu būdu, tyrinėjamas jo konvergavimas. Aproksimacijos normaliuoju dėsniu tikslumui įvertinti naudojamas charakteristinių funkcijų metodas, todėl penktasis skyrius yra skiriamas suglodinimo nelygybių patikslinimui. Šeštame skyriuje, pasinaudojus turimais rezultatais, realizuojamas magistrinio darbo tikslas, o septintame - patikrinamas absoliutinės paklaidos įvertis Bernulio schemos atveju. Išvados ir rezultatai glaustai išdėstomi aštuntame skyriuje. / This master thesis considers independiant and identically distributed random variables, having absolute finite moments. The main task is to determine error estimate of the normal approximation. The work consists of eight chapters. In the introduction are considered problems and all subjects of research. The second chapter is designed for the theory analysis. Here are placed the main theoretical studies and methods that are used to solve the aims of the master thesis. The third chapter is intended to deal with cumulants in case of the Bernoulli’s distribution, the fourth one - is analyzing the Čebyšova’s asymptotic expansion and it convergence with the help of the mathematical package Maple. The method of characteristic’s functions is used to find the remainder term of the normal approximation, so the fifth chapter is designed to specify smoothing inequalities. Based on these results, the main task of the master thesis was obtained and specified in the sixth chapter. In the seventh one the error estimate in case of Bernoulli’s distribution, was examined with a mathematical package Maple. The short conclusions and results are placed in the eighth chapter.

In the thesis, the joint universality of periodic Hurwitz zeta-functions as well as that jointly with the Riemann zeta-functions of normalized cusp forms is obtained. / Darbe yra įrodomas jungtinis universalumas periodinėms Hurvico dzeta funkcijoms, taip pat bendras universalumas su Rymano dzeta funkcija ir normuotų parabolinių formų dzeta funkcija.

Darbe yra įrodomas jungtinis universalumas periodinėms Hurvico dzeta funkcijoms, taip pat bendras universalumas su Rymano dzeta funkcija ir normuotų parabolinių formų dzeta funkcija. / In the thesis, the joint universality of periodic Hurwitz zeta-functions as well as that jointly with the Riemann zeta-function or zeta functions of normalized cusp forms is obtained.

Magistro darbe įrodyta elipsinių kreivių L-funkcijų jungtinė diskreti universalumo Voronino prasme teorema. / The aim of the master work is to obtain a joint discrete universality theorem in the Voronin sense for L-function of elliptic curves.

Darbe nagrinėjamos Hurvico dzeta funkcijos _dzeta(s; alfa_), s = _alfa +it su algebriniu iracionaliuoju parametru _alfa, 0 < alfa_ ≤ 1 diskretusis reikšmių pasiskirstymas. Įrodyta, jog funkcijai _(s; alfa_) galioja diskrečioji ribinė teorema su svoriu kompleksinėje plokštumoje C. / Master’s work is devoted to the investigation of value distribution of Hurwitz zeta-function _(s; alfa_), s = alfa_ + it with algebraic irrational parameter alfa_, 0 < alfa_ ≤ 1. It is proved that for the function _(s; alfa_) valid discrete limit theorem with weight in the complex plane.

Sakykime, kad F yra normuota parabolinė tikrinė forma pilnosios modulinės grupės atžvilgiu, L(s, F) yra susieta su L funkcijos sąsūka L(s, F, χ) su Dirichlė charakteriu moduliu q, kai q yra pirminis skaičius. Bakalauro darbe įrodyta ribinė teorema L funkcijų sąsūkų argumentui arg L(s, F, χ). / Let F(z) a holomorfic normalized Hecke eigen cups form of weight κ for the full modular group, L(s, F), s = σ + it, be the L-function attached to the form F. Let L(s, F, χ) denote a twist of L(s, F) with a Dirichlet character χ modulo q, by the Dirichlet series and can be analytically continued to an entire function. It has an Euler product over prime numbers. We obtain the weak dinvergence for probability measures μQ(arg L(s, F, χ) ∈ A), A ∈ B(γ), where γ is the unit circle on the complex plane, as Q → ∞. For the proof, the method of characteristic transforms and the limit measures in limit theorems obtained are defined characteristic transforms.

Magistro darbe nagrinėjamos periodinės dzeta funkcijos ir periodinės Hurvico dzeta funkcijos jungtinis reikšmių pasiskirstymas ir jungtinė ribinė teorema tikimybinių matų silpno konvergavimo prasme kompleksinėje plokštumoje. / Master's thesis the periodic zeta functions and zeta functions of periodic Hurwitz joint distribution of the values ​​and the joint limit theorem of probability measures converge weak sense of the complex plane.

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