Lehrstuhl IV Stochastik und Analysis
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Item Freezing limits for general random matrix ensembles and applications to classical β-ensembles(2024) Hermann, Kilian; Voit, Michael; Woerner, JeannetteThis thesis studies the asymptotic behavior of eigenvalue distributions in random matrix theory, more specifically that of beta ensembles as the parameter beta tends to infinity. This regime is known as the “freezing limit”. Beta ensembles are often used to describe systems of interacting particles that are influenced by repulsive forces and external potentials. They appear in various physical and mathematical contexts. In physics, beta ensembles play a key role in Calogero-Moser-Sutherland models. Here, beta represents the inverse temperature, and the freezing limit corresponds to rigid particle configurations. In mathematics, beta ensembles describe eigenvalue densities in classical random matrix models. Examples of such models that are studied in this thesis are the Gaussian, Wishart, and MANOVA ensemble. In the classical sense, the value of beta is fixed and depends on the underlying number field. For arbitrary positive beta, a tridiagonal matrix model can be defined such that its eigenvalue distribution is given by the beta ensemble for this beta. The thesis focuses first on analyzing the behavior of beta ensembles for general convex potential functions in the freezing limit. These include the classical ensembles, the beta-Hermite, the beta-Laguerre the and beta-Jacobi ensemble as well as certain edge cases. Afterwards, the interplay of the freezing regime and pushing the number of particles to infinity is investigated in each of the three classical cases. These investigations provide new insights into the structural connection between beta ensembles and dual orthogonal polynomial systems.Item Freezing limits for Calogero–Moser–Sutherland particle models(2023-08-04) Voit, MichaelOne-dimensional interacting particle models of Calogero–Moser–Sutherland type with N particles can be regarded as diffusion processes on suitable subsets of ℝ^N like Weyl chambers and alcoves with second-order differential operators as generators of the transition semigroups, where these operators are singular on the boundaries of the state spaces. The most relevant examples are multivariate Bessel processes and Heckman–Opdam processes in a compact and noncompact setting where in all cases, these processes are related to special functions associated with root systems. More precisely, the transition probabilities can be described with the aid of multivariate Bessel functions, Jack and Heckman–Opdam Jacobi polynomials, and Heckman–Opdam hypergeometric functions, respectively. These models, in particular, form dynamic eigenvalue evolutions of the classical random matrix models like β-Hermite, β- Laguerre, and β-Jacobi, that is, MANOVA, ensembles. In particular, Dyson's Brownian motions and multivariate Jacobi processes are included. In all cases, the processes depend on so-called coupling parameters. We review several freezing limit theorems for these diffusions where, for fixed N, one or several of the coupling parameters tend to ∞. In many cases, the limits will be N-dimensional normal distributions and, in the process case, Gauss processes. However, in some cases, normal distributions on half spaces and distributions related to some other ensembles appear as limits. In all cases, the limits are connected with the zeros of the classical one-dimensional orthogonal polynomials of order N.Item Limit theorems and statistical inference for Bessel and Dunkl processes(2022) Hufnagel, Nicole; Woerner, Jeannette; Voit, MichaelThe starting point of this thesis is the estimation of the multiplicities of a Dunkl process. We note that a Dunkl process can be transformed into a Bessel process or Cox-Ingersoll-Ross process, respectively. We study estimators for these two processes, which we then apply to the Dunkl process. We present well-established methods for inference in cases where the likelihood function is unknown or too complicated. First, we concentrate on martingale estimators at low frequency data for the index parameter of a Bessel process and, as an extension, polynomial processes. Since these processes are non-ergodic and most results for inference are developed for stationary and ergodic diffusions, we transform them into processes with such properties by adding a mean reverting term. Taking a closer look at these estimators, we recognize that they depend only on a particular transformation of the data which is a realization of a Cox-Ingersoll-Ross process. Therefore, in the remainder of the thesis we focus on estimators for the Cox-Ingersoll-Ross process. For this, we apply the Gaussian quasi-likelihood method. In doing so, the density is approximated by the Gaussian density. Even if the Cox-Ingersoll-Ross process has a non-central chi-squared density which is far from being Gaussian, this local approximation works well. We introduce a known preliminary estimator, then prove asymptotic normality for one-step improvements towards the Gaussian quasi-maximum likelihood estimator. We show that all these estimators for the Cox-Ingersoll-Ross process are asymptotically equivalent to the Gaussian quasi-maximum likelihood estimator and compare them in a simulation study. A key difference of the martingale estimators and the estimators for the Cox-Ingersoll-Ross process lies in their asymptotic behaviour. The Gaussian quasi-maximum likelihood estimators converge only when the underlying Bessel process never hits the origin whereas the martingale estimators converge even in these cases. Of particular interest are multivariate generalizations of these critical cases, where the boundary of the Weyl chamber is then considered instead of the origin. For our final result, we present the Hausdorff dimension of the times when a multivariate Bessel process hits the Weyl chamber's boundary.Item Martingale estimation functions for Bessel processes(2021-08-04) Woerner, Jeannette; Hufnagel, NicoleIn this paper we derive martingale estimating functions for the dimensionality parameter of a Bessel process based on the eigenfunctions of the diffusion operator. Since a Bessel process is non-ergodic and the theory of martingale estimating functions is developed for ergodic diffusions, we use the space-time transformation of the Bessel process and formulate our results for a modified Bessel process. We deduce consistency, asymptotic normality and discuss optimality. It turns out that the martingale estimating function based of the first eigenfunction of the modified Bessel process coincides with the linear martingale estimating function for the Cox Ingersoll Ross process. Furthermore, our results may also be applied to estimating the multiplicity parameter of a one-dimensional Dunkl process and some related polynomial processes.Item Estimation of stopping times for stopped self-similar random processes(2021-03-01) Schulmann, ViktorLet X=(Xt)t≥0 be a known process and T an unknown random time independent of X. Our goal is to derive the distribution of T based on an iid sample of XT. Belomestny and Schoenmakers (Stoch Process Appl 126(7):2092–2122, 2015) propose a solution based the Mellin transform in case where X is a Brownian motion. Applying their technique we construct a non-parametric estimator for the density of T for a self-similar one-dimensional process X. We calculate the minimax convergence rate of our estimator in some examples with a particular focus on Bessel processes where we also show asymptotic normality.Item Limit theorems and statistical inference for solutions of some stochastic (partial) differential equations(2019) Shevchenko, Radomyra; Woerner, Jeannette; Reiß, MarkusThe starting point for the thesis is an Ornstein-Uhlenbeck type stochastic differential equation dXt=(L(t)-cXt)dt+dBt with a real number value at time zero. The driving process B is a fractional Brownian motion with Hurst parameter H between 0.5 and 1, L is a bounded periodic deterministic function and c is a real valued parameter. In the thesis the task of estimation of L and c is solved when the solution process X is observed continuously. In the first part L is considered to be a linear combination of known functions with the same period, and the vector of the coefficients together with the parameter c is estimated using the least squares method for c<0, i.e. in the non-ergodic setting. We show strong consistency of the estimator as well as its asymptotic normality in the first p components. For the last component a noncentral limit theorem is proved. In the second part we consider the ergodic setting, i.e. c>0, and construct a nonparametric estimator for the function L. The idea for construction is to decompose L with respect to a known orthonormal basis and to estimate coefficients of a finite number of summands, letting this number tend to infinity afterwards. For this estimator L2 consistency is shown and its rate of convergence is calculated. The next part is dealing with a similar equation, this time driven by a Rosenblatt process, a non-Gaussian process with the covariance structure of a fractional Brownian motion with Hurst parameter H between 0.5 and 1. We consider again the setting in which L is a linear combination of known functions and c is positive, i.e. the ergodic case. We construct three estimators for the vector of parameters and study their first and second order asymptotics. The last part of the thesis deals with a different problem. There the stochastic wave equation is considered driven by a random field that is white in space and fractional in time with Hurst parameter H between 0.5 and 1. The objects we investigate are so-called empirical power variations of the solution, i.e. averages over the weighted differences of the solution process raised to some integer powers, where the weights fulfil certain conditions. We show central and noncentral limit theorems for these averages, determine in some cases the third order asymptotics in terms of the Wasserstein distance and use these results to define strongly consistent and asymptotically normal estimators for H.Item Limit theorems for random walks on non-compact Grassmann manifolds with growing dimensions(2019) Artykov, Merdan; Voit, Michael; Veselić, Ivan